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\hypersetup{
  pdftitle={The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8},
  pdfauthor={OpenAI},
  pdfsubject={A two-stage proof using cubic theta reflection and character moments}
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\title{The Quasi-Riemann Hypothesis\\[0.5em]
\large A Zero-Free Half-Plane \texorpdfstring{$\Re s>7/8$}{Re(s)>7/8}}
\author{OpenAI}
\date{30 September 2026}

\begin{document}
\maketitle

\begin{abstract}
We prove that all finite-order Hecke $L$-functions over
$\mathbb Q(\sqrt{-3})$ and all Dirichlet $L$-functions are zero-free in
the half-plane $\Re s>7/8$, with the principal pole at $s=1$ allowed.
In particular, the Riemann zeta function is zero-free in this half-plane,
proving the quasi-Riemann hypothesis.
\end{abstract}

\tableofcontents
\clearpage

\section{Introduction}\label{sec:introduction}

Let $F=\mathbb Q(\sqrt{-3})$, let $\mathcal O$ be its ring of integers,
and write $N\mathfrak a=|\mathcal O/\mathfrak a|$ for the norm of a
nonzero integral ideal. A finite-order Hecke character $\eta$ modulo
a nonzero integral ideal $\mathfrak f$ is a character of the ray class group
modulo $\mathfrak f$, extended by zero to ideals not coprime to $\mathfrak f$.
This agrees with the usual finite-order idelic definition: the complex
place contributes no nontrivial finite-order continuous character
\cite[Chapter VI, Section 1]{MilneCFT}. For $\Re s>1$, put
\[
 L_F(s,\eta)=\sum_{\mathfrak a\ne0}
       \frac{\eta(\mathfrak a)}{(N\mathfrak a)^s}
 =\prod_{\mathfrak p}
       \left(1-\eta(\mathfrak p)(N\mathfrak p)^{-s}\right)^{-1}.
\]
The same notation denotes its meromorphic continuation. For a
Dirichlet character $\chi$ modulo $q$, extended by zero on nonunits, we
likewise write
\[
 L(s,\chi)=\sum_{n\ge1}\frac{\chi(n)}{n^s}
 \qquad(\Re s>1)
\]
and then continue meromorphically; the character modulo $1$ gives the
Riemann zeta function $\zeta(s)$. The problem here is uniform exclusion:
to find a constant $\sigma_0<1$, independent of the character, its
conductor, and the height, such that these functions have no zeros in
$\Re s>\sigma_0$.

For $\zeta$ alone, the existence of a fixed $\sigma_0<1$ for which
$\zeta(s)\ne0$ in $\Re s>\sigma_0$ is called the quasi-Riemann
hypothesis \cite[Section 1]{BCSS}. It asks for one gap valid at every
height, not merely nonvanishing on $\Re s=1$. The corresponding uniform
formulation over all Dirichlet characters appears in
\cite[p.~288, Equation (1.4) and the following paragraph]{FG}.

The connection between these functions and prime distribution has a
long history. Dirichlet's 1837 proof of infinitude of primes in every
reduced arithmetic progression introduced the character $L$-series
that separate residue classes \cite{Dirichlet}. Riemann's 1859 memoir
related the zeros of $\zeta$ to the distribution of primes and formulated
the critical-line conjecture \cite{Riemann}. Hadamard and de la
Vall\'ee Poussin independently proved in 1896 that $\zeta$ has no zero
on $\Re s=1$, obtaining the prime number theorem
\cite{Hadamard,ValleePoussin}. Hecke subsequently developed the
character zeta functions and their analytic theory over number fields
\cite{Hecke}.

Classical zero-free regions for Hecke $L$-functions approach the line
$\Re s=1$ as the conductor or height increases and can allow one
simple real exceptional zero; a precise form for abelian extensions is given
by \cite[Theorem 3.1]{TZ}. Such regions do not give uniform exclusion in
a fixed half-plane. Zero-density estimates address a different question:
they bound the number of zeros to the right of a given vertical line.
For example, Guth and Maynard's large-value estimates improve zero-density
bounds for $\zeta$ \cite[Theorem 1.2]{GM}, but these zero-density bounds
do not exclude every zero.

\begin{theorem}\label{thm:main}
Every finite-order Hecke $L$-function over
$F=\mathbb Q(\sqrt{-3})$ has no zero in $\Re s>7/8$.
The same holds for every Dirichlet $L$-function, including $\zeta(s)$.
A pole at $s=1$ for a principal character is allowed.
\end{theorem}

In particular, Theorem~\ref{thm:main} resolves the quasi-Riemann
hypothesis affirmatively: the supremum of the real parts of the
nontrivial zeros of $\zeta$, namely its zeros in $0<\Re s<1$, is at most
$7/8$. The boundary $\Re s=7/8$ is not included, while any exceptional
real zero in $(7/8,1)$ is excluded. The theorem does not establish the
Riemann hypothesis or its generalized versions, which place nontrivial
zeros on $\Re s=1/2$. The Riemann hypothesis remains open \cite{Clay}.

Kubota's metaplectic theory and Patterson's cubic theta series provide
the automorphic setting \cite{Kubota,Patterson}. We use the unconditional
explicit cusp expansions recorded by Dunn and Radziwi\l\l{}
\cite[Section 5 and Appendix A]{DR}; their GRH-conditional prime asymptotic
is not an input. The corresponding first-moment asymptotic is proved
unconditionally in an independent manuscript
\cite[Theorem~1.1]{OpenAICubicFirstMoment}; that result is also not used
here. Proposition~\ref{prop:completed-reflection} derives the
reflection used here while retaining the characters' zero-on-nonunit
restrictions.

The character large-sieve arguments build on the quadratic Hecke-family
estimate of Goldmakher and Louvel \cite[Definition 1 and Theorem 1.1]{GL},
the higher-order norm recursion of Blomer, Goldmakher, and Louvel
\cite[Theorem 1.3 and Section 3]{BGL}, and Heath-Brown's cubic estimate
\cite[Theorem 2]{HB}. We prove the precise sextic specialization needed
here in Lemma~\ref{lem:sextic-large-sieve}, using paired residue characters
to carry out that recursion in the primary-generator convention. An
Eisenstein-integer form is also recorded by Gao and Zhao
\cite[Lemma 2.9]{GZSextic}. The recursive
moment arguments are related to the framework of Heath-Brown's quadratic
method \cite[Section 2]{HBQuadratic}. We also use the ordinary Hecke
functional equation \cite[Equation (1.1)]{GZ} and prime counting in a
fixed ray class \cite[Theorem 1.1]{TZ}.

The planar additive large sieve used in the first stage is classical;
compare Huxley's multivariable and number-field inequality \cite{Huxley}
and the Poisson proof in \cite[Section 3, Theorem 3]{BaierBansal}.
We include a direct Eisenstein-lattice proof to record the normalization
needed for the reduced-fraction expansion. David, de Faveri, Dunn, and
Stucky combine Patterson's coefficients, the cubic large sieve, and
mollified moments to prove nonvanishing at $s=1/2$ for a positive
proportion of a cubic Hecke family
\cite[Theorem 1.1 and Section 1.2]{DFDS}. The zero detector below uses
the classical truncated-inverse mechanism; compare
\cite[Appendix C]{MP}, and for a higher-order-character
density application \cite[Corollary 1.6 and Section 5]{BGL}.

The proof has two stages, each comparing two representations of a
completed cubic-theta sum but using a different normalized sum in the
continuation argument. Part~\ref{part:eleven-twelfths} proves the corresponding
$11/12$ assertion in Theorem~\ref{thm:eleven-twelfths}, already obtaining a
fixed zero-free half-plane for both families.
Part~\ref{part:seven-eighths} starts from that conclusion and introduces
prime compensation, asymmetric scales, and two additional moment estimates.
The contribution developed here is the construction of these compatible
reflected and Poisson comparisons, including the principal residues, with
positive exponent margins chosen independently of the target character.

\subsection*{Proof overview}

\paragraph{A common continuation principle.}
The analytic argument begins with the family of primitive finite-order
Hecke characters over $F$. Let $\beta_*$ be the supremum of $1/2$ and the
real parts of their zeros in $1/2\le\Re s\le1$; poles are not included.
Imprimitive characters have the same possible zeros in $\Re s>0$, because
the finitely omitted Euler factors are nonzero there. If $\beta_*$ exceeds
a proposed boundary $\sigma_0$, the task is to continue every target
reciprocal across a common positive distance to the left of $\beta_*$.

Section~\ref{sec:analytic-reduction} gives the precise criterion. For each
target $\eta$ and large real scale $Z$, it compares a normalized character
sum with a Mellin integral containing $1/L_F(s,\eta)$, after deletion of
finitely many Euler factors and multiplication by a holomorphic factor
bounded away from zero. A direct bound for the sum and a power-saving bound
for its difference from that integral imply the required continuation.
The two parts use this same principle with different affine powers of $Z$
in the Mellin integral:
\[
 C_{\mathrm I}(s)=s-\frac23,\qquad
 C_{\mathrm{II}}(s)=s-\frac{11}{16}.
\]
Thus the common analytic principle does not identify the two normalized
sums. Only the positive power margins must be independent of $\eta$;
fixed-character constants and lower thresholds may depend on it.

\paragraph{The completed sum.}
Section~\ref{sec:arithmetic} sets up the sextic residue characters over
$\mathcal O$, always retaining their zero values on nonunits. The base
sum in Section~\ref{sec:probe} averages smoothed cubic-theta Fourier
coefficients against these characters and a fixed target $\eta$.
After the fixed rescaling in the theta expansion, the completed indices
are $cn^3$, where $c,n\in\mathcal O$ are congruent to $1$ modulo $3$ and
$c$ is squarefree. The variables $c$ and $n$ may share prime factors.
Here completion means retaining the full cubic factor $n^3$, rather than
restricting the index to its squarefree part.
The target character and a finite ray-class phase act on the whole product
$cn^3$, rather than separately on its two factors.

The resulting completed base sum has two exact representations.
The reflection in Proposition~\ref{prop:completed-reflection}
transforms its theta coefficients while retaining the character zeros in
the resulting formula.
Poisson summation transforms the averaging variable. Its nonzero frequencies
have the form $ua^6$, where $u,a\in\mathcal O$ and $u$ is sixth-power-free:
every prime valuation of $u$ is at most five.
Section~\ref{sec:local-euler}
then expresses the contribution of each such row through a quotient of
Hecke $L$-functions and a controlled Euler product. For $u=1$, this
quotient contains the reciprocal of the target function.

\paragraph{The balanced first stage.}
In Part~\ref{part:eleven-twelfths}, the two averaging scales are both
$Z^{1/2}$. On the reflected side, the base sum separates into a completed
theta row and an additive polynomial. The quadratic large sieve bounds
the mean square of the reflected rows. Expanding the additive polynomial
produces reduced fractions in $\mathbb C/\mathcal O$; their separation and
coefficient mass give the required planar large-sieve bound. The
Cauchy--Schwarz inequality combines the two norms in
Proposition~\ref{prop:balanced-low}.
This direct estimate does not use the later inverse-moment recursion.

On the Poisson side, the intermediate rows are grouped by their norm and by
the location of zeros in bounded-height rectangles for the finite family
of Hecke twists that each row determines. Section~\ref{sec:detector} assigns
zero-free rectangles to these families. When a row has a selected zero
above the detector's fixed floor for the real part, it produces two large
Dirichlet polynomials: one with
ideal M\"obius coefficients, representing a truncated reciprocal, and one
without those coefficients. They have one common row character and one
common twist height. Part~\ref{part:eleven-twelfths} uses only the inverse
polynomial in its row count. After the part of the row with prime valuations
at least two is fixed, the sextic large sieve applies to its squarefree
factor; Proposition~\ref{prop:sextic-row-count} gives the resulting count.
Rows at the floor and the remaining small and large norm ranges are bounded
directly.

The principal row is treated separately. Its residues, together with the
local Euler identity, give a nonzero scalar multiple of the target Mellin
integral. After normalization, the direct estimate and the remaining-row
estimate verify the common continuation criterion with
$C_{\mathrm I}(s)=s-2/3$. This proves
Theorem~\ref{thm:eleven-twelfths}. The entire family is needed even for the
consequence about $\zeta$, because the Poisson rows introduce finite-order
Hecke twists of the target.

\paragraph{The refined second stage.}
Part~\ref{part:seven-eighths} retains the completed support and the shared
arithmetic identities, but it modifies the base sum. Selected prime factors
provide a local compensation that cancels an unwanted Euler contribution,
and the two averaging scales are no longer equal. After the new residue
calculation and scalar normalization, the corresponding Mellin signal uses
$C_{\mathrm{II}}(s)=s-11/16$. The low estimate for the normalized sum again
follows directly from reflected energy and an additive mean-square estimate;
Section~\ref{sec:compensated-low} proves it for the modified sum.

The high estimate uses both polynomials supplied by the zero detector.
For a smooth compactly supported $W$ on $(0,\infty)$, a scale $D>0$,
and a finite-order Hecke character $\psi$, their basic forms are
\[
 D^{-1/2}\sum_{\mathfrak a\ne0}\mu(\mathfrak a)\psi(\mathfrak a)
       W(N\mathfrak a/D),\qquad
 D^{-1/2}\sum_{\mathfrak a\ne0}\psi(\mathfrak a)W(N\mathfrak a/D),
\]
where $\mu$ is the ideal M\"obius function. In the specified intermediate
norm ranges above the detector floor, each retained row has a large inverse polynomial and a large plain
polynomial with one common row character and twist height. Bounds for
their moments limit the number of such rows. Section~\ref{sec:refined-row-counts}
combines the two bounds, using integer powers and selected prime factors
in the larger intermediate norm ranges. Small and large norm ranges,
and the floor class, remain direct estimates.

The two moment bounds require separate inductions.
Section~\ref{sec:inverse} proves the second-moment estimate for an inverse
polynomial multiplied by sums over disjoint prime sets
(Lemma~\ref{lem:marked}). Its recursive step uses two finite Poisson
transformations to shorten the row range, with reflected energy providing
the terminal bound. Section~\ref{sec:plain} proves a mean-square estimate
for products of two plain polynomials, again with permitted prime factors
(Lemma~\ref{lem:plain}). Ordinary Hecke reflection reduces the length
ranges at each stage; the recursive step uses two finite Poisson
transformations.

The plain estimate retains different row families according to whether
prime factors are present. With them, it excludes rows whose primitive
inducing character belongs to the fixed finite group generated by the
target and the auxiliary ray characters; without them, it excludes only
principal inducing characters. Transformed rows in that finite family
are treated separately within the induction. Direct volume bounds handle
uncentered products. For a difference of two products with the same two
profiles and equal products of scales, the common main terms cancel.
This cancellation is used only for that centered expression, not for every
row in the finite inducing-character family.

Section~\ref{sec:conclusion} combines the resulting row counts with the
compensated expansion, its local errors, the contour tails, and the
principal residue. After normalization, their bound verifies the second
comparison in the continuation criterion at $7/8$.

\paragraph{Transfer to Dirichlet $L$-functions.}
The final transfer is the same at both boundaries. Composing a Dirichlet
character $\chi$ with the ideal norm gives a Hecke character over $F$.
Away from finitely many Euler factors, quadratic factorization writes its
$L$-function as the product of the Dirichlet $L$-functions attached to $\chi$
and to $\chi\chi_{-3}$, where $\chi_{-3}$ is the quadratic character of
$F/\mathbb Q$. The omitted factors are nonzero in $\Re s>0$, and the
principal pole is handled separately. Thus each Hecke half-plane transfers
to all Dirichlet characters, completing the two stages.

\subsection*{Least nonresidues and square roots over prime fields}

The fixed Dirichlet zero-free half-plane also makes the following
classical consequences unconditional.

\begin{corollary}\label{cor:nonresidues-square-roots}
There are absolute constants $A,C>0$ such that, for every odd prime $p$,
the least positive quadratic nonresidue $n(p)$ satisfies
\[
 n(p)\le C(\log p)^A.
\]
In particular, $n(p)\ll_\delta p^\delta$ for every $\delta>0$, proving
Vinogradov's least quadratic nonresidue conjecture
\cite[Conjecture 1.1]{TaoNonresidues}.
Given an odd prime $p$ and $a\in\mathbb F_p$ in binary representation,
there is a deterministic algorithm, with running time polynomial in
$\log p$, that returns a square root of $a$ or reports that none exists.
\end{corollary}

\begin{proof}
By Theorem~\ref{thm:main} and the functional equation, the nontrivial
zeros of every primitive Dirichlet $L$-function lie in
$1/8\le\Re s\le7/8$. They therefore lie in the strictly larger strip
$1/16<\Re s<15/16$. This supplies the weak-GRH hypothesis of
Bhargava, Ivanyos, Mittal, and Saxena
\cite[Conjecture 6.3 and Theorem 6.7]{BIMS} with $\epsilon=7/16$.
Their bound for the least quadratic nonresidue gives the asserted
inequality, for example with $A=32$; no optimization is needed here.
The assertion for each $\delta>0$ follows because every fixed power
of $\log p$ is $O_\delta(p^\delta)$.

For the algorithm, first handle $a=0$ and test a nonzero $a$ by Euler's
criterion. If $a$ is a square, scan $2,3,\ldots$ until a quadratic
nonresidue is found, testing each candidate by its Legendre symbol.
The bound just proved makes this a polynomial-time deterministic
search; its stopping rule does not require knowing $C$.
Use the resulting nonresidue in the Tonelli--Shanks algorithm
\cite[Section 2.9, Algorithm 3 and Lemma 2.9.5]{Galbraith}.
All remaining steps are deterministic and polynomial in $\log p$.
Writing $p-1=2^e u$ with $u$ odd only requires removing factors of two,
not factoring $u$.
\end{proof}

\section{From a common signal to a zero-free half-plane}
\label{sec:analytic-reduction}

Both parts of the proof use the same analytic principle. A character sum
is bounded directly and is also compared with a Mellin integral containing
the reciprocal of a target $L$-function. A power saving in both comparisons
then continues that reciprocal across the rightmost possible zeros. We prove
this principle for a variable boundary, so that it can be applied at
$11/12$ and at $7/8$ without repeating the argument.

Throughout the paper, $F=\mathbb Q(\sqrt{-3})$. It suffices to consider
primitive target characters. Indeed, a character induced from a primitive
character $\eta$ has $L$-function differing from $L_F(s,\eta)$ by finitely
many factors $1-\eta(\mathfrak p)N\mathfrak p^{-s}$, all nonzero for
$\Re s>0$. Define
\begin{equation}
 \beta_* = \sup\left(\{1/2\}\cup
 \left\{\Re\rho:\begin{array}{l}
  1/2\le\Re\rho\le1,\quad L_F(\rho,\eta)=0\\
  \text{for some primitive finite-order Hecke character }\eta
 \end{array}\right\}\right).
 \label{old-eq:1.1b}
\end{equation}
Poles are not included. Absolute convergence of the Euler product excludes
zeros in $\Re s>1$, so $1/2\le\beta_*\le1$.

For a finite set $\mathcal S$ of prime ideals, a superscript $\mathcal S$
means that the corresponding Euler factors have been deleted:
\[
 L_F^{\mathcal S}(s,\eta)
 =L_F(s,\eta)\prod_{\mathfrak p\in\mathcal S}
       (1-\eta(\mathfrak p)N\mathfrak p^{-s}).
\]
The local value is zero at a ramified prime. Each displayed factor is nonzero
for $\Re s>0$, so deleting finitely many factors neither creates nor removes
a zero there.

\begin{proposition}[Continuation from a common signal]
\label{lem:continuation-criterion}
Fix $\sigma_0\in(1/2,1)$ and suppose $\Delta_0=\beta_*-\sigma_0>0$.
Let $C(s)=s+c$, where $c\in\mathbb R$ is fixed. Suppose that numbers
$\omega,\sigma$ satisfying
\[
 0<\omega<\Delta_0,\qquad \sigma>0
\]
can be chosen independently of the target character. For every primitive
finite-order Hecke character $\eta$, suppose there exist a finite set
$\mathcal S$, a holomorphic function $H_\eta$ on $\Re s>\sigma_0$, and a
function $J_\eta(Z)$ defined for all sufficiently large real $Z$, such that
\begin{equation}
 \sup_{\Re s>\sigma_0}|H_\eta(s)-1|\le\frac12.
 \label{eq:signal-euler-contract}
\end{equation}
For $Z>0$, define
\begin{equation}
 f_\eta(Z)=\frac1{2\pi i}\int_{\Re s=2}
 Z^{C(s)}e^{(s-5/6)^2}
 \frac{H_\eta(s)}{L_F^{\mathcal S}(s,\eta)}\,ds.
 \label{eq:common-residue}
\end{equation}
Assume that, as $Z\to\infty$,
\begin{align}
 |J_\eta(Z)|&\ll_\eta Z^{C(\sigma_0)+\omega},
       \label{eq:low-probe-contract}\\
 |J_\eta(Z)-f_\eta(Z)|&\ll_\eta Z^{C(\beta_*)-\sigma}.
       \label{eq:high-probe-contract}
\end{align}
The implied constants, lower thresholds, excluded set, and function $H_\eta$
may depend on $\eta$. Then these assumptions contradict $\beta_*>\sigma_0$.
\end{proposition}

\begin{proof}
Put
\[
 \epsilon_*:=\min\{\Delta_0-\omega,\sigma\}>0.
\]
Because $C$ has slope one, the triangle inequality gives
\begin{equation}
 |f_\eta(Z)|\ll_\eta Z^{C(\beta_*)-\epsilon_*}
 \qquad (Z\ge Z_{0,\eta}).
 \label{eq:uniform-saving}
\end{equation}
Moreover $\epsilon_*\le\Delta_0-\omega<\Delta_0$, so
$\beta_*-\epsilon_*>\sigma_0$.

We also need control as $Z\downarrow0$. By
Equation~\eqref{eq:signal-euler-contract}, $H_\eta$ is bounded on
$\Re s>\sigma_0$. The reciprocal Euler product is absolutely and uniformly
bounded on $\Re s\ge2$. On every fixed strip $2\le\Re s\le B$, the Gaussian
in Equation~\eqref{eq:common-residue} is $O_B(e^{-(\Im s)^2})$. Cauchy's
theorem on rectangles therefore moves the contour to any fixed $B>2$, with
horizontal integrals tending to zero. Hence
\[
 |f_\eta(Z)|\ll_{\eta,B}Z^{B+c}\qquad(0<Z\le1).
\]
Since $B$ is arbitrary, the signal has arbitrarily rapid power decay at zero.

This bound and Equation~\eqref{eq:uniform-saving} show that
\[
 F_\eta(s):=\int_0^\infty f_\eta(Z)Z^{-C(s)}\,\frac{dZ}{Z}
\]
converges locally uniformly on $\Re s>\beta_*-\epsilon_*$. On each compact
subset, choose $B$ larger than all occurring real parts for the integral near
zero, and use Equation~\eqref{eq:uniform-saving} near infinity. Thus $F_\eta$
is holomorphic on that half-plane.

We identify this Mellin transform without moving a contour across a zero.
Set
\[
 A_\eta(s)=e^{(s-5/6)^2}
             \frac{H_\eta(s)}{L_F^{\mathcal S}(s,\eta)}
 \qquad(\Re s>1).
\]
The function $A_\eta(2+it)$ is continuous and integrable in $t$. Writing
$Z=e^u$, Equation~\eqref{eq:common-residue} becomes
\[
 e^{-(2+c)u}f_\eta(e^u)
   =\frac1{2\pi}\int_{\mathbb R}e^{itu}A_\eta(2+it)\,dt.
\]
The left side is integrable in $u$ by the two endpoint estimates. Ordinary
Fourier inversion therefore gives $F_\eta(2+it)=A_\eta(2+it)$ for every real
$t$. Both sides are holomorphic on $\Re s>1$, so the identity theorem gives
\[
 F_\eta(s)=e^{(s-5/6)^2}
             \frac{H_\eta(s)}{L_F^{\mathcal S}(s,\eta)}
 \qquad(\Re s>1).
\]

Equation~\eqref{eq:signal-euler-contract} implies $|H_\eta(s)|\ge1/2$ on
$\Re s>\sigma_0$. Consequently
\[
 e^{-(s-5/6)^2}\frac{F_\eta(s)}{H_\eta(s)}
 \qquad(\Re s>\beta_*-\epsilon_*)
\]
is a holomorphic continuation of $1/L_F^{\mathcal S}(s,\eta)$.
The number $\epsilon_*$ was chosen independently of $\eta$. By the definition
of $\beta_*$, some target has a zero $\rho$ with
$\Re\rho>\beta_*-\epsilon_*$. Its reciprocal has a pole at $\rho$, and the
deleted factors are nonzero there. This contradicts the continuation.
\end{proof}

The high estimate in Proposition~\ref{lem:continuation-criterion} is measured
relative to $C(\beta_*)$, not to $C(\sigma_0)$. This distinction matters when
a row contribution reaches the low scale but still has a power saving relative
to the hypothetical rightmost zero. Part~\ref{part:eleven-twelfths} uses
\[
 \sigma_0=\frac{11}{12},\qquad C(s)=s-\frac23,
 \qquad C(\sigma_0)=\frac14,
\]
whereas Part~\ref{part:seven-eighths} uses
\[
 \sigma_0=\frac78,\qquad C(s)=s-\frac{11}{16},
 \qquad C(\sigma_0)=\frac3{16}.
\]
Only the positive power margins must be uniform in the target. Fixed-character
constants, excluded sets, and sufficiently large lower thresholds may depend
on it in both applications.

\clearpage
\part{The quasi-Riemann hypothesis}
\label{part:eleven-twelfths}

\section{A first zero-free half-plane}
\label{sec:first-stage}

We first prove a fixed zero-free half-plane using the basic completed
cubic-theta sum. This isolates the mechanism that excludes zeros before the
additional estimates needed for the sharper boundary are introduced.

\begin{theorem}[The $11/12$ half-plane]
\label{thm:eleven-twelfths}
Every finite-order Hecke $L$-function over $F=\mathbb Q(\sqrt{-3})$ has no
zero in $\Re s>11/12$. The same holds for every Dirichlet $L$-function,
including $\zeta(s)$. A pole at $s=1$ for a principal character is allowed.
\end{theorem}

For the Hecke assertion, suppose for contradiction that
\begin{equation}
 \Delta_1:=\beta_*-\frac{11}{12}>0.
 \label{eq:first-stage-gap}
\end{equation}
The proof will construct one character sum with two exact representations.
Cubic-theta reflection bounds the sum after an elementary additive
large-sieve estimate. Poisson summation expresses the same sum as a principal
Mellin signal and a family of remaining character rows. A zero detector and
the sextic large sieve control those rows. These estimates verify
Proposition~\ref{lem:continuation-criterion} with boundary $11/12$.

The fixed family in Equation~\eqref{old-eq:1.1b} is essential even when the
desired consequence concerns $\zeta$: the Poisson representation introduces
finite-order Hecke twists of the target. Proving the family-wide assertion
also supplies the precise input used at the beginning of
Part~\ref{part:seven-eighths}.

\section{Arithmetic and analytic preliminaries}\label{sec:arithmetic}

The two parts use the same arithmetic conventions and analytic estimates.
This section fixes the residue symbols, retaining their zero values even for
principal powers, and proves the Gauss and reciprocity identities used to
transform character sums.  It then establishes a calculus for smooth norm
profiles and uniform bounds for a Hecke $L$-function on a disk known to be
zero-free.  These are the common preliminaries for the balanced argument.

\subsection{Arithmetic notation and coefficient classes}

Let $F=\mathbb Q(\sqrt{-3})$, let
$\omega=e^{2\pi i/3}$, and put
\[
 \mathcal O=\mathbb Z[\omega],\qquad
 \lambda=\sqrt{-3}=\omega-\omega^2=1+2\omega.
\]
For an element $a$ write $q_a=|a|^2=N_{F/\mathbb Q}(a)$, and for $a\ne0$
write $\alpha(a)=a/|a|$.  The same notation $q_{\mathfrak a}$ denotes the norm of
a nonzero ideal.  The ring $\mathcal O$ is Euclidean for this norm: a point of
$\mathbb C$ is at distance at most $1/\sqrt3<1$ from the triangular lattice
$\mathcal O$, which gives Euclidean division.  In particular every ideal is
principal.  The six units are $\{\pm1,\pm\omega,\pm\omega^2\}$, and their
images are the six units of $\mathcal O/3\mathcal O$.  Consequently every
ideal coprime to $3$ has a unique generator congruent to $1\pmod3$; we call
this generator primary.  Products of primary generators are primary.

In the character-polynomial estimates below, a row is an outer index for a
character polynomial, while a column is an ideal index, or a tuple of ideal indices,
summed inside it.  A label is an auxiliary index distinguishing parts of the
family.  Whether a label is averaged or locally frozen refers to the current
sum; freezing it does not make it part of the fixed arithmetic datum.

For the arithmetic datum in any given invocation, fix from the outset a
finite set $S$ of prime ideals containing the primes above $6$ and the prime
supports of the defining moduli of every finite-order character presentation
in that datum and of the fixed finite ray group used to present them.  For a
character presentation, its defining-modulus support consists exactly of the
primes at which it is extended by zero, including any redundant primes of an
imprimitive presentation.  A fixed character includes its entire zero-extended
presentation and the finite ray group through which it is presented, all
fixed independently of $Z$, the current rows, and the averaged labels.  The
fixed datum may depend on a target fixed beforehand.  In particular every
such character has modulus one at every prime outside $S$.  We also write
$\mathcal S=S$.  We call primes outside $S$ good, and call an ideal good if
all its prime divisors lie outside $S$.  Unless a different support is
specified, ideal sums exclude $S$ and use primary generators.  We write
``sf'' for squarefree and $\mu$ for the ideal M\"obius function.  Element rows,
in contrast, may have arbitrary
prime powers and unit factors.

For $z\in F$ define
\[
 e(z)=\exp\bigl(2\pi i\operatorname{Tr}_{F/\mathbb Q}(z/\lambda)\bigr).
\]
Its extension to $\mathbb C$ is
$e(z)=\exp(4\pi i\operatorname{Im}z/\sqrt3)$.  The measure
$d\mu(z)=(2/\sqrt3)\,dx\,dy$ makes $\mathcal O$ self-dual for the pairing
$(z,y)\mapsto e(zy)$.  Indeed, writing $y=u+v\omega$ with real $u,v$, the
conditions $e(y)=e(\omega y)=1$ say $v,u-v\in\mathbb Z$, hence
$y\in\mathcal O$; and the covolume of $\mathcal O$ for $d\mu$ is one.  Lattice point counting
in a disk, followed by division by the six units, gives the unrestricted
ideal count
\[
 \#\{\mathfrak a:q_{\mathfrak a}\le H\}
 =\frac{\pi H}{3\sqrt3}+O(\sqrt H+1)\qquad(H\ge0).
\]
For example, the error follows by covering the boundary of the disk with
$O(\sqrt H+1)$ fixed fundamental parallelograms.

If $p\notin S$ is prime, its residue field has order $P=q_p\equiv1\pmod6$:
the six roots of unity remain distinct in that field.  Define the sextic
symbol $\chi_p(a)=(a/p)_6$ to be the unique sixth root of unity satisfying
\[
 \chi_p(a)\equiv a^{(P-1)/6}\pmod p\quad(p\nmid a),
 \qquad \chi_p(a)=0\quad(p\mid a).
\]
For a primary $c=\prod p^{v_p(c)}$ outside $S$, define
$\chi_c(a)=\prod_{p\mid c}\chi_p(a)^{v_p(c)}$, and put $\chi_1(a)=1$ for
every $a$.  For every integer $j$, including $j=0$ and negative $j$, the
notation $\chi_c(a)^j$ means its usual power when $(a,c)=1$ and means zero
otherwise.  Thus an exponent divisible by six is the function
$1_{(a,c)=1}$, not the constant function one.  The square $\chi_c^2$ is the
cubic symbol.  For squarefree $c$ set
\[
 \gamma_j(c)=q_c^{-1/2}\sum_{v\bmod c}\chi_c(v)^j e(v/c),
 \qquad \gamma_j(1)=1.
\]

\begin{lemma}[Fixed numerators give ray characters]
\label{lem:fixed-numerator-ray}
Fix $0\ne a\in\mathcal O$.  At a prime ideal $\mathfrak p\nmid6a$, let
$(a/\mathfrak p)_6$ be the unique sixth root of unity congruent to
$a^{(q_{\mathfrak p}-1)/6}\pmod{\mathfrak p}$; at a prime outside $S$ this is
$\chi_p(a)$.  The extension $F(a^{1/6})/F$ is finite abelian and unramified
outside the primes dividing $6a$, and
\[
 \frac{\operatorname{Frob}_{\mathfrak p}(a^{1/6})}{a^{1/6}}
       =(a/\mathfrak p)_6,
\]
where $\operatorname{Frob}_{\mathfrak p}$ is arithmetic Frobenius.  Hence the
multiplicative extension of this symbol to ideals coprime to $6a$ is a
finite-order ray character whose conductor is supported on the primes
dividing $6a$.  No bound on its conductor exponents at those primes is
asserted.  This identification is only on ideals coprime to $6a$; it does
not erase the prescribed zero of $\chi_A(a)$ when a good ideal $A$ meets $a$.

In particular, choose one generator $\pi_{\mathfrak p}$ for each
$\mathfrak p\in S$.  On primary ideals outside $S$, the characters
\[
 A\longmapsto\chi_A\left(u\prod_{\mathfrak p\in S}
                       \pi_{\mathfrak p}^{v_{\mathfrak p}}\right),
 \qquad u\in\mathcal O^\times,\quad v_{\mathfrak p}\ge0,
\]
belong to one finite family of ray characters with a common modulus
supported on $S$; this family is determined by $u$ and the residues
$v_{\mathfrak p}\pmod6$.
\end{lemma}

\begin{proof}
Let $\xi^6=a$.  Since $F$ contains all sixth roots of unity, all roots of
$X^6-a$ lie in $F(\xi)$, and
$\sigma\mapsto\sigma(\xi)/\xi$ embeds its Galois group into $\mu_6$.
The extension is therefore abelian.  A prime outside $6a$ is unramified;
for example this follows from the discriminant of $X^6-a$, which is supported
on $6a$.  At such a prime, arithmetic Frobenius is characterized on the
residue field by $x\mapsto x^{q_{\mathfrak p}}$.  Consequently its quotient
on $\xi$ reduces to $a^{(q_{\mathfrak p}-1)/6}$.  Reduction is injective on
$\mu_6$ outside $6$, proving the displayed identity.  Multiplicativity of
the Artin map gives the assertion for ideals.  Artin reciprocity makes this
map factor through a ray group with modulus supported on the primes dividing
$6a$; these are the power-residue and ray-group assertions in
\cite[Chapter VIII, (5.3) and (5.5)]{MilneCFT}.

For the last statement, an ideal $A$ outside $S$ is coprime to every
$\pi_{\mathfrak p}$ and to every unit.  Its symbol therefore has no zero
there, and the power of each $\chi_A(\pi_{\mathfrak p})\in\mu_6$ depends
only on $v_{\mathfrak p}\pmod6$.  There are at most
$6^{|S|+1}$ possible displayed numerators after this reduction.  Apply the
first assertion to each and take a common multiple of their ray moduli.
Their prime supports are all contained in $S$ because $S$ contains the
primes above $6$.
\end{proof}

The word fixed in this lemma is essential: a numerator containing a moving
good prime does not thereby enter a ray group fixed independently of $Z$.
Its character and its natural zero support remain moving data.

At a base $Z>1$, a plain polynomial of real log-length $n$ is
\begin{equation}
 S_\psi(n;W)=Z^{-n/2}\sum_l\psi(l)W(q_l/Z^n),
 \label{old-eq:1.1}
\end{equation}
and the centrally normalized inverse polynomial of log-length $r$ is
\begin{equation}
 M_\psi(r;W)=Z^{-r/2}\sum_n\mu(n)\psi(n)W(q_n/Z^r).
 \label{old-eq:1.1a}
\end{equation}
Every mask in $\psi$ is retained in both definitions.  At base $U$, when
$\psi(n)=\nu(n)\chi_n(u)$, we also denote the latter polynomial by
$M_u(U^r;W)$, or by $M_u(D;W)$ when $D=U^r$.

An annular test is a smooth function with support in a fixed compact
subinterval of $(0,\infty)$.  Families of annular or coupled profiles are
used only with fixed logarithmic support and uniform bounds for every
logarithmic derivative that is invoked.  The precise separation norm is
given in Lemma~\ref{lem:smooth-calculus}.  In a product of two plain factors,
both factors have the same row character, including the same fixed finite-ray
twist.  Conjugating a whole factor inside its absolute value permits the
opposite orientation; conjugating only part of its coefficients does not.

We use the following uniformity convention.  Log-lengths range over fixed
bounded sets, and every estimate allows any specified positive power loss.
The exponents of the norm scales depend only on those real parameter ranges,
the strict margins, and the specified losses.
Write $\mathcal A$ for this fixed arithmetic datum: the complete zero-extended
presentations of the fixed finite characters, their defining moduli and the
fixed finite ray group used to present them, the excluded set, and any fixed
arithmetic normalization.  It is chosen independently of $Z$, the current
rows, and the averaged labels, though it may depend on a previously fixed
target.  Finite seminorm orders, polynomial height orders,
implied constants, and lower thresholds may depend on $\mathcal A$.  They
are uniform over the declared moving moduli and outer labels in their stated
ranges, even when an outer label is fixed during one row sum.

For a nonzero ideal $\mathfrak a$, let
$d_{\mathcal O}(\mathfrak a)=\#\{\mathfrak d:\mathfrak d\mid\mathfrak a\}$,
where the count includes all integral ideal divisors, including those meeting
$S$, and for $0\ne f\in\mathcal O$ put
$d_{\mathcal O}(f)=d_{\mathcal O}((f))$.
A nonnegative multiplicity $w(f)$ is called divisor-bounded only if
\[
 w(f)\le C\,d_{\mathcal O}(f)^C
\]
for one fixed $C\ge1$, independent of $Z$, the current rows, and the averaged
labels.  The constant $C$ may depend on $\mathcal A$.
This convention does not relax any separate condition that $w$ depend only on
$f$.  For every $\delta>0$ it implies $w(f)\ll_{C,\delta}q_f^\delta$
uniformly.  Indeed, for prime ideals $\mathfrak p$ of sufficiently large norm,
$(e+1)^C\le q_{\mathfrak p}^{\delta e}$ for every integer $e\ge0$, by
$e+1\le2^e$ for $e\ge1$.  For each of the finitely many remaining prime ideals,
$\sup_{e\ge0}(e+1)^Cq_{\mathfrak p}^{-\delta e}<\infty$; multiplying these
bounds over the prime factorization of $f$ proves the assertion.

\subsection{Gauss sums and the finite reciprocity phase}

We first evaluate the prime Gauss sums.  This also fixes the orientation of
the cubic symbol in all subsequent formulas.

\begin{lemma}[Prime Gauss identities]\label{lem:prime-gauss-identities}
Let $p$ be the primary generator of a prime ideal outside $S$, and put
$H=\chi_p$ and $P=q_p$.  Then
\[
 \gamma_2(p)^3=-\alpha(p),\qquad
 \gamma_1(p)\gamma_2(p)
   =\overline{H(4)}\gamma_3(p)\gamma_2(p)^3.
\]
For $j\not\equiv0\pmod6$, one has $|\gamma_j(p)|=1$.
\end{lemma}

\begin{proof}
Let $k=\mathcal O/(p)$ and let $\psi(x)=e(x/p)$ be its nontrivial additive
character.  For a multiplicative character $A$ of $k^\times$, extended by
zero, put
\[
 \tau(A)=\sum_{x\in k}A(x)\psi(x),\qquad
 J(A,B)=\sum_{x\in k}A(x)B(1-x).
\]
If $A$ is nonprincipal, the change of variables $x=ty$ gives
\[
 |\tau(A)|^2
 =\sum_{t\in k^\times}A(t)\sum_{y\in k^\times}\psi((t-1)y)=P.
\]
The inner sum is $P-1$ for $t=1$ and $-1$ otherwise, and
$\sum_{t\in k^\times}A(t)=0$.  Conjugation also gives
$\tau(A)\tau(\overline A)=A(-1)P$.  When $AB$ is nonprincipal, grouping
the product $\tau(A)\tau(B)$ by $x+y$ gives
\[
 \tau(A)\tau(B)=J(A,B)\tau(AB).
\]
The group with $x+y=0$ vanishes because $AB$ is nonprincipal; for a nonzero
sum, division by $x+y$ gives the displayed Jacobi factor.  In particular
$|J(A,B)|=\sqrt P$ if $A,B,AB$ are all nonprincipal.

The number of solutions of $4x(1-x)=y$ is $1+H^3(1-y)$.  Summing $H(y)$
times this identity and using $\sum_yH(y)=0$ gives
$J(H,H^3)=H(4)J(H,H)$.  Write $\tau_j=\tau(H^j)$.
The Gauss--Jacobi identity and $\tau_2\tau_4=P$ now give
\[
 \frac{\tau_1\tau_3}{\tau_4}
   =H(4)\frac{\tau_1^2}{\tau_2},\qquad
 \tau_1\tau_2=\overline{H(4)}\frac{\tau_3\tau_2^3}{P}.
\]
Here $H^2(-1)=1$ because $-1$ is a cube.

It remains to fix the cubic Jacobi sum, including its unit.  Put
$m=(P-1)/3$.  The definition of $H^2$ gives, in $k$,
\[
 J(H^2,H^2)\equiv\sum_{x\in k}x^m(1-x)^m=0\pmod p.
\]
Indeed the polynomial has degree $2m<P-1$, and the sum over $k$ of each
monomial of degree less than $P-1$ is zero in $k$ (the constant case is
$P=0$ in $k$).  The Jacobi sum belongs to $\mathcal O$ and has absolute
value $\sqrt P$.

For $x\ne0,1$, choose $j_x\in\{0,1,2\}$ such that
$H^2(x(1-x))=\omega^{j_x}$.  The products of $x$ and of $1-x$ over these
$x$ are both $-1$, so $\prod_{x\ne0,1}x(1-x)=1$.  Hence
$\sum_{x\ne0,1}j_x\equiv0\pmod3$.  Since
\[
 \omega^j\equiv1+j(\omega-1)\pmod{(\omega-1)^2},
 \qquad ((\omega-1)^2)=(3),
\]
we obtain $J(H^2,H^2)\equiv P-2\equiv-1\pmod3$.  Divisibility by $p$
and equality of norms imply $J(H^2,H^2)=up$ for a unit $u$.  As $p$ is
primary and the six units have distinct residues modulo $3$, the congruence
forces $u=-1$.  Finally,
$\tau_2^3=J(H^2,H^2)\tau_2\tau_4=-pP$.  Dividing the two Gauss identities
by the appropriate powers of $\sqrt P$ proves the lemma.
\end{proof}

The remaining phase is quadratic.  Its evaluation below is valid even for
odd elements that are not squarefree or primary.

\begin{lemma}[Quadratic four-term formula]\label{lem:quadratic-four-term}
For a nonzero odd $c\in\mathcal O$, define
\[
 \Gamma(c)=|c|^{-1}\sum_{x\bmod c}e(x^2/c).
\]
Then
\begin{equation}\label{eq:quadratic-four-term}
 \Gamma(c)=\frac12\sum_{y\bmod2\mathcal O}e(-cy^2/4).
\end{equation}
For $c=a+b\omega$ the right side is
$(1+i^{-b}+i^a+i^{b-a})/2$.  In particular $\Gamma$ depends only on
$c\bmod4\mathcal O$ and $\Gamma(cv^2)=\Gamma(c)$ for every odd $v$.
The units modulo $4$ have square subgroup $\{1,\omega,\omega^2\}$ and
square-class representatives $1,-1,\lambda,-\lambda$.  The function
$\Gamma$ never vanishes on these units, and
$\mathfrak r(a,b)=\Gamma(ab)/(\Gamma(a)\Gamma(b))$ satisfies
\begin{equation}\label{eq:reciprocity-four-class}
 \begin{gathered}
 \begin{array}{c|rrrr}
  c&1&-1&\lambda&-\lambda\\ \hline
  \Gamma(c)&1&1&i&-i
 \end{array}\\
 \mathfrak r((-1)^e\lambda^f,(-1)^g\lambda^h)
       =(-1)^{eh+fg+fh},\qquad e,f,g,h\in\{0,1\}.
 \end{gathered}
\end{equation}
Thus $|\Gamma(c)|=1$ and $\mathfrak r$ is a symmetric
$\{\pm1\}$-valued bicharacter of the square-class group.
\end{lemma}

\begin{proof}
Use the Fourier transform with kernel $e(-zy)$ and measure $d\mu$.
For $\varepsilon>0$ apply Poisson summation on $\mathcal O$ to
$f_\varepsilon(z)=e(z^2/c)e^{-\pi\varepsilon|z|^2}$.  Direct Gaussian
integration gives, with $r_\varepsilon=1+3q_c\varepsilon^2/16$,
\[
 \widehat f_\varepsilon(y)
 =\frac{|c|}{2\sqrt{r_\varepsilon}}
    e^{-\pi\varepsilon q_c|y|^2/(4r_\varepsilon)}
    e\bigl(-cy^2/(4r_\varepsilon)\bigr).
\]
To check the normalization, rotate $z$ by half the argument of $c$.  The
quadratic matrix of the resulting real two-variable Gaussian is
\[
 \begin{pmatrix}
 \varepsilon&-4i/(\sqrt3|c|)\\
 -4i/(\sqrt3|c|)&\varepsilon
 \end{pmatrix},
 \qquad \det=\frac{16r_\varepsilon}{3q_c}.
\]
Its eigenvalues are conjugates with positive real part.  The positive square
root of the determinant, the factor $2/\sqrt3$ in $d\mu$, and completion of
the square give the formula above.

Put $I_\varepsilon=\int_{\mathbb C}e^{-\pi\varepsilon|z|^2}d\mu(z)
=2/(\sqrt3\varepsilon)$.  For a fixed ideal $(b)$, a fixed class $v\bmod b$,
and $a>0$, Gaussian Poisson summation on $b\mathcal O$ gives
\[
 \frac1{I_\varepsilon}\sum_{z\in v+b\mathcal O}
             e^{-\pi a\varepsilon|z|^2}
 \longrightarrow\frac1{a q_b}\qquad(\varepsilon\downarrow0).
\]
The zero dual vector gives the limit and every nonzero dual vector is
exponentially small.  The phase $e(z^2/c)$ is periodic modulo $c$.
Consequently the normalized left side of Poisson's identity tends to
$\Gamma(c)/|c|$.

On the right side one may replace $r_\varepsilon$ by $1$.  For fixed $c$,
the total error from the phase is at most
\[
 C_c\varepsilon^2\sum_{y\in\mathcal O}|y|^2
                         e^{-C_c^{-1}\varepsilon|y|^2}=O_c(1),
\]
because the sum is $O_c(\varepsilon^{-2})$.  The amplitude error is
$O_c(\varepsilon)$, as is the damping error, by the same lattice estimate.
All three are $o(I_\varepsilon)$.  The phase $e(-cy^2/4)$ is periodic modulo
$2\mathcal O$.  In the preceding Gaussian mean take $b=2$ and $a=q_c/4$;
each of its four classes has normalized mean $1/q_c$.  The normalized right
side therefore tends to $(2|c|)^{-1}\sum_{y\bmod2}e(-cy^2/4)$, which proves
Equation~\eqref{eq:quadratic-four-term}.

The representatives $0,1,\omega,\omega^2$ modulo $2$ give the asserted
four-term expression.  Multiplication by an odd $v$ permutes these classes,
so the same formula gives $\Gamma(cv^2)=\Gamma(c)$.  The group of units
modulo $4$ has order $12$.  Squaring a lift modulo $4$ depends only on its
class modulo $2$, and its three possible squares are $1,\omega,\omega^2$.
The four representatives in the statement are distinct modulo this subgroup;
also $\lambda^2=-3\equiv1\pmod4$.  Evaluating the four-term expression on
them gives the table.  Evaluating $\Gamma(ab)/(\Gamma(a)\Gamma(b))$ on the
two generators $-1,\lambda$ gives the displayed exponent, proving the last
claims.
\end{proof}

\begin{lemma}[Sextic reciprocity and the fixed Gauss phase]
\label{lem:fixed-gauss-phase}
For coprime primary $a,b$ outside $S$,
\[
 \chi_b(a)=\mathcal R(a,b)\chi_a(b),\qquad
 \mathcal R(a,b)=\mathfrak r(a,b).
\]
On all primary pairs outside $S$, including noncoprime pairs, define
$\mathcal R$ by the bicharacter $\mathfrak r$ of
Equation~\eqref{eq:reciprocity-four-class}.  Define, on every primary index
outside $S$,
\[
 G(c)=\overline{\chi_c(4)}\Gamma(c).
\]
It has modulus one and factors through a fixed ray group supported at $2,3$.
For all such $v,w$,
\begin{equation}\label{eq:G-quadratic-refinement}
 G(vw)=G(v)G(w)\mathcal R(v,w),\qquad G(1)=1.
\end{equation}
Moreover $G(v^2)=\chi_v(4)$, and at good primes
\begin{equation}\label{eq:fixed-gauss-phases}
 G(p^3)=\gamma_3(p),\qquad
 \mathcal R(p,p)=\gamma_3(p)^2=\chi_p(-1).
\end{equation}
On every primary $n$ outside $S$, one has
$\chi_n(-1)=\mathcal R(n,n)$.  The diagonal $t\mapsto\mathcal R(t,t)$ is a
character of the fixed ray group.  In the square-class notation of
Equation~\eqref{eq:reciprocity-four-class}, it is $\mathfrak r(-1,n)$, where
$-1$ denotes its residue square class modulo $4$, not the ideal ray class of
the unit ideal $(-1)$.
For squarefree $c$ the complete Gauss identities are
\begin{equation}\label{eq:signal}
 \gamma_2(c)^3=\mu(c)\alpha(c),\qquad
 \gamma_1(c)\gamma_2(c)=\mu(c)\alpha(c)G(c),\qquad
 G(c)=\overline{\chi_c(4)}\gamma_3(c),\quad |G(c)|=1.
\end{equation}
All occurrences of $G$ on nonsquarefree indices mean the finite-ray function
just defined, not a nonsquarefree Gauss sum.
\end{lemma}

\begin{proof}
At a good prime, counting square roots in the residue field gives
\[
 \sum_{x\bmod p}e(x^2/p)
 =\sum_{y\bmod p}(1+\chi_p(y)^3)e(y/p)
 =\sum_{y\bmod p}\chi_p(y)^3e(y/p).
\]
The additive sum without the character is zero.  Hence
$\Gamma(p)=\gamma_3(p)$, and replacing $x^2/p$ by $ux^2/p$ for a unit $u$
multiplies the sum by $\chi_p(u)^3$.  For squarefree $c$, representing a
residue as $\sum_{p\mid c}(c/p)x_p$ gives, in both sums,
\[
 \Gamma(c)=\prod_{p\mid c}\chi_p(c/p)^3\Gamma(p),\qquad
 \gamma_3(c)=\prod_{p\mid c}\chi_p(c/p)^3\gamma_3(p).
\]
The cross terms in the square are integral in the additive character.
Thus $\Gamma(c)=\gamma_3(c)$ for squarefree $c$.  In particular, for distinct
good primes $p,q$,
\[
 \frac{\Gamma(pq)}{\Gamma(p)\Gamma(q)}
 =\chi_p(q)^3\chi_q(p)^3.
\]

We use cubic reciprocity in the following precise form: for coprime primary
$a,b$ one has $(a/b)_3=(b/a)_3$; see \cite[Equation (1.4)]{DR}.  Thus the
quotient $\chi_b(a)/\chi_a(b)$ is a sign.  At $p,q$ its cube is
$\chi_q(p)^3\chi_p(q)^3$, so this sign equals $\mathfrak r(p,q)$.
Multiplicativity in both arguments and the bicharacter property extend the
equality to every coprime primary pair.  The definition by $\mathfrak r$ on
noncoprime pairs involves no division of zero symbols.

The element $-2$ is primary and $-1$ is a cube.  Cubic reciprocity therefore
gives
\[
 \chi_c(4)=(2/c)_3=(-2/c)_3=(c/(-2))_3.
\]
This is a character of $c\bmod2$.  Together with the dependence of $\Gamma$
on $c\bmod4$, this proves that $G$ is a function on a fixed ray group (for
example the ray group modulo $12$ suffices here).  Indeed, if two primary
generators represent the same ray class modulo $12$, their quotient differs
from an element congruent to one modulo $12$ by a unit; reduction modulo $3$
forces that unit to be one.  The generators therefore have the same residue
modulo $4$.  Its multiplicative
refinement is the definition of $\mathfrak r$.  Since
$\Gamma(v^2)=1$ and $\chi_v(4)$ has order dividing three,
$G(v^2)=\overline{\chi_v(4)}^{\,2}=\chi_v(4)$.  The table also gives
\[
 \Gamma(p)^2=(-1)^{(q_p-1)/2}=\chi_p(-1).
\]
For the last equality use
$\chi_p(-1)=(-1)^{(q_p-1)/6}$ and the equality of these parities.
Now $\Gamma(p^3)=\Gamma(p)$, and
$\overline{\chi_p(4)}^{\,3}=1$, proving $G(p^3)=\gamma_3(p)$.  Finally
$\mathcal R(p,p)=1/\Gamma(p)^2=\Gamma(p)^2$, as this square is a sign.

For a symmetric sign-valued bicharacter its diagonal is multiplicative:
\[
 \mathcal R(vw,vw)
 =\mathcal R(v,v)\mathcal R(v,w)^2\mathcal R(w,w)
 =\mathcal R(v,v)\mathcal R(w,w).
\]
The prime identity consequently gives $\mathcal R(n,n)=\chi_n(-1)$ for
every primary $n$ outside $S$.  On the square class $(-1)^e\lambda^f$, the
table gives both this diagonal and $\mathfrak r(-1,n)$ the value $(-1)^f$.
This use of $-1$ is in the residue square-class group; the ideal $(-1)$ is
the identity in an ideal ray class group and is not being substituted there.

For coprime squarefree primary $a,b$ outside $S$, the Chinese remainder theorem gives
\[
 \gamma_j(ab)=\chi_a(b)^j\chi_b(a)^j\gamma_j(a)\gamma_j(b).
\]
One obtains this by representing a residue as $bx+ay$ with $x\bmod a$ and
$y\bmod b$.  Cubing the factor for $j=2$ gives one.  The product of the
factors for $j=1$ and $j=2$ is
$(\chi_a(b)\chi_b(a))^3=\mathcal R(a,b)$.  The prime identities from
Lemma~\ref{lem:prime-gauss-identities}, followed by
Equation~\eqref{eq:G-quadratic-refinement}, therefore prove
Equation~\eqref{eq:signal} by induction on the number of prime factors.
\end{proof}

For later element rows, fix the generators $\pi_{\mathfrak p}$ from
Lemma~\ref{lem:fixed-numerator-ray}.  Every $0\ne m\in\mathcal O$ has a
unique expression
\[
 m=u m_S m_{\mathrm{good}},\qquad
 m_S=\prod_{\mathfrak p\in S}\pi_{\mathfrak p}^{v_{\mathfrak p}(m)},
\]
where $u$ is a unit and $m_{\mathrm{good}}$ is the primary generator of the
part of $(m)$ outside $S$.  On every primary $A$ outside $S$, multiplicativity
and sextic reciprocity give the zero-preserving identity
\begin{equation}\label{eq:row-fixed-ray-reduction}
 \chi_A(m)=\chi_A(u m_S)\mathcal R(A,m_{\mathrm{good}})
       \prod_{p\mid m_{\mathrm{good}}}\chi_p(A)^{v_p(m)}.
\end{equation}
For $(A,m_{\mathrm{good}})=1$ this is the reciprocity formula just proved;
if they meet, both sides are zero and $\mathcal R$ is evaluated as its
separately defined bicharacter.  Once the good part's fixed ray class and
the $S$-valuations modulo six are fixed, the first two factors on the right
range over a fixed finite family of characters of $A$.  The product over
good primes retains the moving character factors and all their zeros.

Every function on a fixed finite abelian ray group has a finite Fourier
expansion in its characters.  For example, if $\phi:T\to\mathbb C$, then
\[
 \phi(v)=\sum_{\theta\in\widehat T}a_\theta\theta(v),\qquad
 a_\theta=\frac1{|T|}\sum_{u\in T}\phi(u)\overline{\theta(u)}.
\]
Parseval and Cauchy--Schwarz bound $\sum_\theta|a_\theta|$ by
$|T|^{1/2}\max_T|\phi|$.  Applying the same statement on $T\times T$
separates $\mathcal R(a,b)$ as a finite sum of products of characters.
These expansions remain valid at noncoprime primary pairs because
$\mathcal R$ there is the fixed-ray bicharacter, not a symbolic quotient.
For example, for a good prime $p$ and every primary $n$ outside $S$,
\[
 \chi_n(p)\overline{\chi_p(n)}
       =\mathcal R(p,n)1_{(n,p)=1}.
\]
On coprime pairs this is sextic reciprocity, and on the remaining pairs both
sides vanish.  Thus cancellation of the moving local characters can leave
both a coprimality indicator and a fixed-ray phase; neither may be discarded.

\subsection{Smooth norm profiles}

The following conventions make the smooth dependence in character-polynomial
estimates quantitative.  They distinguish derivatives of a fixed test from powers of a
spectral height.  This distinction is needed when a Fourier tail is removed
only after the height range has been chosen.

For a profile $w$ on $(0,\infty)^d$ with logarithmic support in a fixed
compact set $\Omega\subset\mathbb R^d$, put
\[
 w_{\log}(\boldsymbol u)=w(e^{u_1},\ldots,e^{u_d}),\qquad
 p_j(w)=\sum_{|\alpha|\le j}\sup_{\boldsymbol u\in\mathbb R^d}
                  |\partial^\alpha w_{\log}(\boldsymbol u)|.
\]
We always understand that $w_{\log}$ is smooth and supported in $\Omega$.
Define
\[
 \widehat w(\boldsymbol t)=\int_{\mathbb R^d}w_{\log}(\boldsymbol u)
                    e^{-i\boldsymbol t\cdot\boldsymbol u}\,d\boldsymbol u,
 \qquad
 \|w\|_{J,\mathrm{sep}}
 =\int_{\mathbb R^d}|\widehat w(\boldsymbol t)|
                    (1+|\boldsymbol t|)^J\,d\boldsymbol t.
\]
For a fixed finite tuple $\boldsymbol w=(w_1,\ldots,w_r)$, possibly of
different fixed dimensions, we use
$p_j(\boldsymbol w)=1+\sum_{i=1}^r p_j(w_i)$.

\begin{lemma}[Smooth calculus]\label{lem:smooth-calculus}
For integers $J\ge0$,
\[
 \|w\|_{J,\mathrm{sep}}\ll_{J,d,\Omega}p_{J+d+2}(w).
\]
Logarithmic Fourier inversion separates any fixed norm monomial.  More
precisely, if $x_j=c_j\prod_{\ell=1}^r y_\ell^{a_{j\ell}}$ with $c_j,y_\ell>0$
and fixed real exponents $a_{j\ell}$, then
\[
 w(x_1,\ldots,x_d)=\frac1{(2\pi)^d}\int_{\mathbb R^d}
 \widehat w(\boldsymbol t)\prod_j c_j^{it_j}
       \prod_\ell y_\ell^{i\sum_j a_{j\ell}t_j}\,d\boldsymbol t.
\]
If the coefficient measure in this formula is common to a collection of
rows, Minkowski's inequality passes any separated row $\ell^2$ bound through
the integral using $\|w\|_{J,\mathrm{sep}}$ whenever the separated bound has
height growth at most $(1+|\boldsymbol t|)^J$.

For a unit box $I\subset\mathbb R^d$ and a smooth scalar function $F$ on
$I+[-1,1]^d$,
\begin{equation}\label{eq:parameter-sobolev}
 \sup_I|F|^2\ll_d
 \sum_{\alpha\in\{0,1\}^d}
 \int_{I+[-1,1]^d}|\partial^\alpha F|^2.
\end{equation}
Consequently, under the corresponding derivative bounds, rowwise choices
of scales in a polynomial range cost powers of $\log Z$, and rowwise choices
of norm-twist heights of absolute value at most $T_1$ cost a fixed power of
$1+T_1$.

For $T_1\ge1$ and integers $J,N\ge0$,
\begin{equation}\label{eq:late-fourier-tail}
 \int_{|\boldsymbol t|>T_1}|\widehat w(\boldsymbol t)|
             (1+|\boldsymbol t|)^J\,d\boldsymbol t
 \le T_1^{-N}\|w\|_{J+N,\mathrm{sep}}.
\end{equation}
With the Mellin convention
$\mathcal MW(s)=\int_0^\infty W(y)y^s\,dy/y$, a pure twist obeys
\[
 \mathcal M[W(y)y^{i\omega}](s)=\mathcal MW(s+i\omega).
\]
Let $D_y=y\partial_y$, let $I\subset\mathbb R$ be compact, and let $N\ge0$
be an integer.  Suppose the integrals below are finite and the boundary terms
in $N$ logarithmic integrations by parts vanish.  Then, uniformly for
$\sigma\in I$ and $T\in\mathbb R$,
\begin{equation}\label{eq:pointwise-mellin-tail}
 |\mathcal MW(\sigma+iT)|
 \ll_{I,N}(1+|T|)^{-N}
     \sum_{j=0}^N\int_0^\infty y^\sigma|D_y^jW(y)|\,\frac{dy}{y}.
\end{equation}
These hypotheses hold for annular $W$ on every such $I$.  They also hold
when $W$ is smooth at zero and Schwartz at infinity and $I$ is a compact
subset of $(0,\infty)$.  In particular a horizontal contour join, whose
imaginary coordinate is fixed, is estimated by this pointwise bound, not
solely by the integrated tail in Equation~\eqref{eq:late-fourier-tail}.

The following joint version will also be useful.  For measurable functions
$a(v),b(w)$ for which the right side is finite, and integers $J,N\ge0$,
\begin{equation}\label{eq:joint-gaussian-slice}
 \begin{split}
 &\int_{\mathbb R^2}e^{-(T+v)^2}|a(v)b(w)|
       (1+|T|+|v|+|w|)^J\,dv\,dw\\
 &\quad\ll_{J,N}(1+|T|)^{-N}
       \sup_v(1+|v|)^{J+N}|a(v)|
       \int_{\mathbb R}(1+|w|)^J|b(w)|\,dw.
 \end{split}
\end{equation}
Thus the product of a Gaussian in the sum of two heights and rapidly
decreasing Mellin transforms in the other two heights has rapid pointwise
decay on a fixed-height slice.  For integrated tails, the linear map
$(t_1,t_2,t_3)\mapsto(t_1+t_2,t_2,t_3)$ is invertible, so all fixed weighted
$L^1$ moments of this product also control complements of boxes in the
original heights.  Appending finitely many separating frequencies and fixed
linear translations of these three arguments gives the same conclusion by
a block triangular change of variables.

After translating a pure twist in the Mellin variable, a discarded
separated integrand bounded by $Z^B(1+|\boldsymbol t|)^J$ has absolute tail
at most
$Z^BT_1^{-N}\|w\|_{J+N,\mathrm{sep}}$.  Here $B$ and $J$ must be fixed
before $N$ is chosen.  Further derivatives of a uniformly smooth separating
profile change this last seminorm, not the previously fixed height order.
On a join of bounded real length at height comparable to $T_1$,
Equation~\eqref{eq:pointwise-mellin-tail} with order $N+\lceil J\rceil$
gives $Z^BT_1^{-N}$ times the corresponding finite weighted derivative
norm of the external test, provided the other factors have the stated bound
there.  This is $O(Z^BT_1^{-N})$ when those external norms are uniform.
\end{lemma}

\begin{proof}
For an integer $m$ with $2m>J+d$, integration by parts gives
\[
 (1+|\boldsymbol t|^2)^m\widehat w(\boldsymbol t)
 =\int_{\mathbb R^d}(1-\Delta)^m w_{\log}(\boldsymbol u)
                       e^{-i\boldsymbol t\cdot\boldsymbol u}\,d\boldsymbol u.
\]
Its absolute value is bounded by the volume of $\Omega$ times
$C_{m,d}p_{2m}(w)$.  The weighted integral is finite because $2m>J+d$;
one can choose $2m\le J+d+2$.  Fourier inversion gives the monomial formula.
For row vectors $B(\boldsymbol t)$ in $\ell^2$, the precise inequality used
there is
\[
 \left\|\frac1{(2\pi)^d}\int\widehat w(\boldsymbol t)
                 B(\boldsymbol t)\,d\boldsymbol t\right\|_{\ell^2}
 \le\frac1{(2\pi)^d}\int|\widehat w(\boldsymbol t)|
                 \|B(\boldsymbol t)\|_{\ell^2}\,d\boldsymbol t.
\]

In one dimension, the fundamental theorem of calculus, averaging a base
point over the enlarged interval, and Cauchy--Schwarz give
$\sup_I|F|^2\ll\int_{I+[-1,1]}(|F|^2+|F'|^2)$.  Applying this successively
in each coordinate proves Equation~\eqref{eq:parameter-sobolev}.  It can be
summed over rows before the derivative integrals, since all terms are
nonnegative.  A logarithmic scale range of length $O(\log Z)$ needs
$O(\log Z)$ unit intervals; a height range $[-T_1,T_1]$ needs
$O(1+T_1)$ intervals.  Derivatives with respect to a logarithmic scale insert
$yW'(y)$, together with the constant derivative of a central normalization.
Derivatives of a normalized twist $y^{it}$ insert powers of $\log y$.
These are again annular profiles with finite seminorm bounds.  For example,
if a fixed-parameter squared row bound is $O((1+|t|)^H)$ and there are
$d_t$ height parameters, the covering and integration cost at most a fixed
multiple of $(1+T_1)^{H+d_t}$.

On $|\boldsymbol t|>T_1$ one has
$(1+|\boldsymbol t|)^{-N}\le T_1^{-N}$, proving
Equation~\eqref{eq:late-fourier-tail}.  The Mellin shift is immediate from
its definition.  To prove Equation~\eqref{eq:pointwise-mellin-tail}, put
$F_\sigma(u)=e^{\sigma u}W(e^u)$.  For $|T|\ge1$, integration by parts gives
\[
 \mathcal MW(\sigma+iT)=(-iT)^{-N}
              \int_{\mathbb R}F_\sigma^{(N)}(u)e^{iTu}\,du,
 \qquad
 F_\sigma^{(N)}(u)=e^{\sigma u}\sum_{j=0}^N
       \binom Nj\sigma^{N-j}D_y^jW(e^u).
\]
Taking absolute values gives the bound on a compact real strip.  For
$|T|<1$, the absolute integral gives it after increasing the constant.
The two stated classes of $W$ have the required endpoint decay: compact
support suffices in the first case, while $e^{\sigma u}$ at $-\infty$ and
Schwartz decay at $+\infty$ suffice in the second.

For Equation~\eqref{eq:joint-gaussian-slice}, use
\[
 1+|T|\le(1+|T+v|)(1+|v|),\qquad
 1+|T|+|v|+|w|\le2(1+|T+v|)(1+|v|)(1+|w|).
\]
Move $(1+|T|)^N$ to the left, bound the weighted $a(v)$ pointwise, and
integrate the remaining polynomial weight against the Gaussian in $T+v$.
The remaining $w$ integral is the one displayed.  This proves the joint
bound and also the asserted join estimate.  For the integrated assertion,
both the map $(t_1,t_2,t_3)\mapsto(t_1+t_2,t_2,t_3)$ and its inverse have
fixed operator norm.  A fixed polynomial weight in the original variables
is therefore bounded by a fixed polynomial weight in the transformed ones.
On a complement of a box one inserts the additional inverse power of that
weight and integrates the Gaussian and the two Mellin factors separately.
The appended block triangular map has the same property because all of its
coefficients and those of its inverse are fixed.

A kernel depending on a common product, such as $y_1y_2$,
gives the same norm power on both variables in the monomial formula; this
preserves an equal-product-scale difference.  A nonsmooth arithmetic mask
has no such derivative bound and must instead be resolved before this lemma
is applied.
\end{proof}

\begin{lemma}[Gaussian annular decomposition]\label{lem:gaussian-annular}
Put
\[
 W_{\mathrm G}(y)=\frac1{2\sqrt\pi}
                         \exp\bigl(-\tfrac14(\log y)^2\bigr).
\]
This function is not annular.  Choose a fixed $\chi\in C_c^\infty(\mathbb R)$
such that $\sum_{k\in\mathbb Z}\chi(u-k)=1$, and define
\[
 W_{{\mathrm G},k}(y)=W_{\mathrm G}(y)\chi(\log y-k),\qquad
 w_k(x)=W_{\mathrm G}(e^k x)\chi(\log x).
\]
The profiles $w_k$ have one fixed compact logarithmic support and, for every
fixed $A\ge0$ and integer $j\ge0$,
\begin{equation}\label{eq:gaussian-annular-sum}
 \sum_{k\in\mathbb Z}e^{A|k|}p_j(w_k)<\infty.
\end{equation}
Consequently Lemma~\ref{lem:smooth-calculus} may be applied on each annulus
after $y=e^k x$, and the resulting weighted separation norms are summable.
For fixed $\kappa>0$, $A,B,C_0\ge0$, and $M>0$ one also has
\[
 Z^B\sum_{|k|>\kappa\log Z-C_0}e^{A|k|}p_j(w_k)\ll Z^{-M}
       \qquad(Z\to\infty).
\]
Directly, $\mathcal MW_{\mathrm G}(s)=e^{s^2}$, and the weighted logarithmic
derivatives in Equation~\eqref{eq:pointwise-mellin-tail} are finite uniformly
on every compact real strip.
\end{lemma}

\begin{proof}
Such a partition is obtained by normalizing the integer translates of a
nonnegative compactly supported smooth function positive on $[-1/2,1/2]$.
Let $\operatorname{supp}\chi\subset[-C,C]$.  Each logarithmic derivative of
$W_{\mathrm G}(e^k x)$ is a polynomial in $k+\log x$ times the same Gaussian.
The product rule therefore gives
\[
 p_j(w_k)\le C_j(1+|k|)^j
             \exp\bigl(-\tfrac14((|k|-C)_+)^2\bigr).
\]
This is summable after multiplication by $e^{A|k|}$ for every fixed $A$.
The first assertion follows, and the separation norms follow from
Lemma~\ref{lem:smooth-calculus}.  In the profile's logarithmic Fourier
transform, rescaling an annulus inserts only the unit phase $e^{-ikt}$, so it
does not change these norms; any real normalization of a surrounding
polynomial remains explicit.  On the indicated
tail, the logarithm of the last bound after multiplication by $Z^Be^{A|k|}$
is at most $-c\kappa^2(\log Z)^2+O(\log Z)$ for some fixed $c>0$; summing
the Gaussian tail proves the stated estimate for every $M$.  Finally, set
$u=\log y$ and complete the square in its Gaussian integral to obtain
$\mathcal MW_{\mathrm G}(s)=e^{s^2}$.  The same Gaussian bounds every
$e^{\sigma u}(d/du)^jW_{\mathrm G}(e^u)$ in $L^1(\mathbb R)$, uniformly
for $\sigma$ in a compact interval.
\end{proof}

We record explicit finite-order continuity statements for the two types of
kernels that accompany this separation.  They make no assertion about an
arithmetic transform; that transform must supply the stated kernel and its
available real lines.

\begin{lemma}[Finite seminorms for Fourier and Mellin kernels]
\label{lem:kernel-seminorms}
For a Schwartz function $f$ on $\mathbb R^d$, put
\[
 s_j(f)=\sum_{|\alpha|,|\beta|\le j}
             \|x^\alpha\partial^\beta f(x)\|_{L^1(\mathbb R^d)}.
\]
For every $A\ge0$ and multi-index $\beta$, the ordinary Fourier transform
satisfies
\[
 \sup_\xi(1+|\xi|)^A|\partial^\beta\widehat f(\xi)|
 \ll_{A,\beta,d}s_{\lceil A\rceil+|\beta|}(f).
\]
The same conclusion, with a fixed change in the constant, holds for any
fixed nondegenerate linear Fourier pairing.  For a radial transform written
as $\widehat f(\xi)=F(|\xi|^2)$, any prescribed bound
\[
 \sup_{r\ge0}(1+r)^A|(r\partial_r)^jF(r)|
\]
is therefore controlled by finitely many Schwartz seminorms of $f$.

For the Mellin assertion, let $m(s)$ be holomorphic on a fixed closed
vertical strip and suppose, uniformly there,
$|m(\sigma+it)|\le C(1+|t|)^h$ for a fixed $h\ge0$.  On an available line
$\Re s=\sigma$ in that strip define
\[
 K_W(x)=\frac1{2\pi i}\int_{(\sigma)}\mathcal MW(-s)m(s)x^{-s}\,ds,
\]
where $W$ has the finite weighted logarithmic derivatives needed to make the
integral absolutely convergent, with the boundary terms in the logarithmic
integrations by parts vanishing.  For every integer $j\ge0$,
\[
 |(x\partial_x)^jK_W(x)|
 \ll x^{-\sigma}\int_{\mathbb R}
       |\mathcal MW(-\sigma-it)|(1+|t|)^{h+j}\,dt.
\]
The integral on the right is bounded by finitely many integrals of
$y^{-\sigma}|(y\partial_y)^kW(y)|\,dy/y$, uniformly for $\sigma$ in the
fixed strip.  For $W(y)y^{i\omega}$ the same bound costs at most an additional
factor $(1+|\omega|)^{h+j}$.  After multiplication by a fixed annular cutoff,
these conclusions also bound every fixed logarithmic seminorm of
$y\mapsto K_W(Ry)$, with the factor $R^{-\sigma}$.
\end{lemma}

\begin{proof}
Differentiating $\widehat f$ inserts $x^\beta$, and multiplication by a
monomial in $\xi$ differentiates $x^\beta f$ before Fourier transformation.
The $L^1$ bound for the Fourier transform, the product rule, and the bound of
$(1+|\xi|)^A$ by a finite sum of monomials of degrees at most $\lceil A\rceil$
give the first assertion.  For a radial function,
$r\partial_r$ corresponds to $(\xi\cdot\partial_\xi)/2$.  Expanding its
$j$th power gives finitely many polynomial multiples of Fourier derivatives,
so the first assertion gives the radial one.  A fixed linear change of
coordinates changes only its constants.

For the Mellin assertion, differentiation under the absolutely convergent
integral inserts $(-s)^j$.  On the fixed strip this is bounded by a constant
times $(1+|t|)^j$.  Write $y=e^u$ and integrate the Fourier transform of
$e^{-\sigma u}W(e^u)$ by parts more than $h+j+1$ times.  The product rule
bounds the resulting $L^1$ derivatives by the stated weighted logarithmic
derivatives, because $\sigma$ ranges over a fixed set.  For a pure twist the
Mellin integrand is $\mathcal MW(-\sigma-it+i\omega)$.  Substitute
$v=t-\omega$ and use
$(1+|v+\omega|)^{h+j}\le(1+|v|)^{h+j}(1+|\omega|)^{h+j}$.
Finally the product rule for a fixed annular cutoff and the chain rule for
$Ry$ give the last statement.
\end{proof}

\subsection{Growth and logarithmic control for Hecke functions}

The disk estimate will be applied only after zeros have been excluded from
that disk.  The following elementary growth bound provides the input for the
complex-analytic argument and makes clear which constants are uniform in the
conductor.  When $\psi$ is a Hecke character, $L(s,\psi)$ abbreviates
$L_F(s,\psi)$.  In this subsection $\Gamma(s)$ denotes Euler's gamma function;
the finite quadratic function $\Gamma(c)$ was confined to the arithmetic
identities above.

\begin{lemma}[Hecke strip growth]\label{lem:hecke-strip-growth}
Let $\psi$ be a primitive nonprincipal finite-order Hecke character of $F$,
with conductor norm $Q$.  Then
\begin{equation}\label{eq:hecke-growth}
 |L(\sigma+it,\psi)|\ll Q^{3/5}(3+|t|)^2,
 \qquad-1/10\le\sigma\le11/10.
\end{equation}
For the principal character, the same bound with $Q=1$ holds for
$(s-1)\zeta_F(s)/(s+1)$, with the removable value used at $s=1$.
\end{lemma}

\begin{proof}
A finite-order character has trivial infinite type, since $\mathbb C^\times$
is connected.  The primitive Hecke functional equation in this case is
\[
 \Lambda(s,\psi)=(3Q)^{s/2}(2\pi)^{-s}\Gamma(s)L(s,\psi),\qquad
 \Lambda(s,\psi)=\varepsilon(\psi)\Lambda(1-s,\overline\psi),
 \quad |\varepsilon(\psi)|=1.
\]
This is the functional equation for primitive characters of trivial infinite
type stated in \cite[Equation (1.1)]{GZ}; a nonprincipal $L(s,\psi)$ is
entire.  Absolute Euler convergence bounds $L(11/10+it,\psi)$ uniformly in
$Q,t$.  Applying the functional equation and Stirling's formula on
$\Re s=-1/10$ gives
\[
 |L(-1/10+it,\psi)|\ll Q^{3/5}(3+|t|)^{6/5}.
\]
The estimate for bounded $t$ follows by continuity of the gamma quotient on
that line, so the constant is uniform there as well.

For completeness, the growth hypothesis needed to use the strip principle
can be obtained directly from a lattice theta integral.  Let
$\mathfrak f=(f)$ be the conductor and define the periodic residue character
$\phi(a)=\psi((a))$ when $(a,\mathfrak f)=1$, with value zero otherwise.
For the principal conductor $(1)$ put $\phi(a)=1$ for all $a$, including
$a=0$.  Since the ideal class group is trivial, conductor one has no
nonprincipal character.  Periodicity modulo $\mathfrak f$ follows from the ray character
property; $\phi$ is trivial on the six units because $(u)=(1)$ for a unit.
Thus, for $\Re s>1$,
\[
 6\pi^{-s}\Gamma(s)L(s,\psi)
 =\int_0^\infty\bigl(\Theta_\phi(v)-\phi(0)\bigr)v^{s-1}\,dv,
 \qquad
 \Theta_\phi(v)=\sum_{a\in\mathcal O}\phi(a)e^{-\pi vq_a}.
\]
The factor six counts the generators of each ideal.  If
$\widehat\phi(h)=Q^{-1}\sum_{a\bmod f}\phi(a)e(-ha/f)$,
finite Fourier inversion and Gaussian Poisson summation give
\[
 \Theta_\phi(v)=\frac{2}{\sqrt3v}
 \sum_{h\bmod f}\widehat\phi(h)
 \sum_{b\in\mathcal O}
    \exp\left(-\frac{4\pi|b-h/f|^2}{3v}\right).
\]
Changing the generator $f$ only reindexes the finite sum.  For each fixed
conductor, the nonzero dual vectors have a positive minimum length.  With
$A_\phi=2\widehat\phi(0)/\sqrt3$, this proves
\[
 \Theta_\phi(v)=A_\phi v^{-1}+O_\phi(v^{-1}e^{-c_\phi/v})\quad(0<v\le1),
 \qquad
 \Theta_\phi(v)=\phi(0)+O_\phi(e^{-c_\phi v})\quad(v\ge1)
\]
for some $c_\phi>0$.  Splitting the Mellin integral at one consequently gives
\[
 \begin{split}
 6\pi^{-s}\Gamma(s)L(s,\psi)
 ={}&\frac{A_\phi}{s-1}-\frac{\phi(0)}s
 +\int_0^1\bigl(\Theta_\phi(v)-A_\phi/v\bigr)v^{s-1}\,dv\\
 &+\int_1^\infty\bigl(\Theta_\phi(v)-\phi(0)\bigr)v^{s-1}\,dv.
 \end{split}
\]
Both integrals are entire and bounded in height on each bounded real strip,
with constants that may depend on the fixed conductor.  Stirling's formula
for $1/\Gamma(s)$ therefore gives at most exponential height growth for
$L(s,\psi)$ on that strip.  In the principal case the same conclusion holds
after multiplication by $(s-1)/(s+1)$.  Only this qualitative growth, for
each fixed conductor, is used in the strip principle.

To see explicitly that the fixed-conductor growth constants do not enter
the uniform strip bound, put
\[
 H_Q(s)=Q^{-3/5}\frac{L(s,\psi)}{(s+2)^2}.
\]
It is holomorphic in the closed strip and is bounded by one absolute
constant $C$ on both vertical boundary lines, by the bounds already proved.
For $\delta>0$, apply the maximum principle on the rectangle of height $T$
to $H_Q(s)\exp(\delta(s-1/2)^2)$.  On the vertical sides the exponential
has modulus at most $e^{9\delta/25}$, because
$|\Re s-1/2|\le3/5$.  On the horizontal sides its modulus is at most
$e^{\delta(9/25-T^2)}$, which tends to zero faster than the qualitative
fixed-$Q$ exponential growth as $T\to\infty$.  First let $T\to\infty$ for
this fixed $Q,\delta$, and then let $\delta\downarrow0$.  The result is
$|H_Q(s)|\le C$ throughout the strip, with the same constant for every $Q$.
Multiplication by $Q^{3/5}|s+2|^2$ proves
Equation~\eqref{eq:hecke-growth}.

For the principal function, the factor $(s-1)/(s+1)$ removes its only pole
in the strip and is bounded on the boundary.  Its functional equation gives
the same boundary bounds with $Q=1$.  This equation also follows from the
preceding Poisson formula with $\phi=1$: after writing $v=2t/\sqrt3$, the
theta relation is
$\Theta_\phi(2t/\sqrt3)=t^{-1}\Theta_\phi(2/(\sqrt3t))$, whose split Mellin
integral is invariant under $s\mapsto1-s$.  Apply the same damped-rectangle
argument to $(s-1)\zeta_F(s)/((s+1)(s+2)^2)$.
\end{proof}

\begin{lemma}[Logarithmic control]\label{lem:logarithmic-control}
Let $\psi$ be a primitive nonprincipal finite-order Hecke character of
conductor norm $Q$, and put $\mathcal L(s)=L(s,\psi)$.  For the principal
character put $Q=1$ and
$\mathcal L(s)=(s-1)\zeta_F(s)/(s+1)$, with its removable value at one.
Fix $a\in[1/2,1]$ and $0<e<10^{-3}$.  Suppose $\mathcal L$ has no zero in
the open disk of radius $2-a-2e$ centered at $2+it$.  Uniformly on the
closed concentric disk of radius $2-a-6e$,
\begin{equation}\label{eq:disk-control}
 |\mathcal L(s)|+|\mathcal L(s)^{-1}|
 \ll_{e,\epsilon}\{2Q(3+|t|)^2\}^{\epsilon}.
\end{equation}
On the disk of radius $2-a-8e$ one also has
\[
 |\mathcal L'(s)/\mathcal L(s)|\ll_e\log\{2Q(3+|t|)^2\}.
\]
In particular, for every fixed $b\ge\beta_*$ and $v>0$, these bounds hold
on $\Re s\ge b+v$, with constants depending on $v,\epsilon$ (and with the
height of $s$ in place of $t$).  In the principal case the reciprocal bound
passes to $1/\zeta_F$; the upper and logarithmic-derivative bounds are for
the regularized function $\mathcal L$.
\end{lemma}

\begin{proof}
Write $R_j=2-a-je$ and $\mathcal C=2Q(3+|t|)^2$.  All the disks used here
lie in $\Re s>1/2$, where the principal regularizer is holomorphic.  On the
zero-free disk choose a holomorphic logarithm $g=\log\mathcal L$ whose value
near the center is the Euler logarithm, together with the logarithm of the
regularizing factor in the principal case.  The value $g(2+it)$ is uniformly
bounded.  Lemma~\ref{lem:hecke-strip-growth} and Euler convergence to the
right of $11/10$ give
\[
 \Re g(s)=\log|\mathcal L(s)|\ll\log\mathcal C
       \qquad(|s-(2+it)|\le R_3).
\]
The possible heights differ from $t$ by at most $3/2$, which is included in
$\mathcal C$.  Borel--Carath\'eodory on radii $R_3,R_4$, whose difference is
$e$, now gives $|g|\ll_e\log\mathcal C$ on the disk of radius $R_4$.

On the disk of fixed radius $r_0=49/100$, one has $\Re s\ge151/100>1$.
The absolutely convergent Euler logarithm is uniformly bounded there.  For
the principal function the logarithm of $(s-1)/(s+1)$ is also bounded on
this disk, using its branch in $\Re s>1$.  Since
$r_0<R_6<R_4$, Hadamard's three-circles theorem applied to $g$ gives
\[
 \max_{|s-(2+it)|\le R_6}|g(s)|
 \ll_e(\log\mathcal C)^\theta,
 \qquad
 \theta=\frac{\log(R_6/r_0)}{\log(R_4/r_0)}<1.
\]
For fixed $e$, continuity on the compact interval $1/2\le a\le1$ makes
$\theta$ uniformly smaller than one.  For every $\epsilon>0$,
$\exp(C_e(\log\mathcal C)^\theta)\ll_{e,\epsilon}\mathcal C^\epsilon$.
Applying this to both $e^g$ and $e^{-g}$ proves
Equation~\eqref{eq:disk-control}.  Cauchy's estimate between radii $R_6$ and
$R_8$ gives the stated bound for $g'=\mathcal L'/\mathcal L$.

For the global assertion first suppose $\beta_*\le b\le1$.  Choose
$0<e<\min(10^{-3},v/8)$ and put $a=b$.  The disk of radius $R_2$ is contained
in $\Re s>b+2e>\beta_*$, so it is zero-free by the definition of $\beta_*$
and absolute Euler convergence beyond one.  The disk of radius $R_8$ covers
the points $b+v\le\Re s\le2$ at its center's height; Euler convergence
covers $\Re s\ge2$.  The preceding constants are uniform in $b$.  If $b>1$,
Euler convergence on $\Re s\ge1+v$ suffices.  Finally
\[
 \frac1{\zeta_F(s)}=\frac{s-1}{s+1}\mathcal L(s)^{-1},
 \qquad \left|\frac{s-1}{s+1}\right|\le1\quad(\Re s\ge0),
\]
which proves the principal reciprocal assertion.
\end{proof}

\begin{lemma}[Deleted Euler factors]\label{lem:deleted-euler-factors}
Let $R$ be a squarefree ideal and let $|a_p|\le1$ for $p\mid R$.  Put
$D_R(s)=\prod_{p\mid R}(1-a_pq_p^{-s})$.  For fixed $\sigma_0>0$ and every
$\epsilon>0$,
\[
 |D_R(s)|+|D_R(s)^{-1}|\ll_{\sigma_0,\epsilon}q_R^\epsilon
       \qquad(\Re s\ge\sigma_0).
\]
On a bounded real strip one has, uniformly in the height,
\[
 |D_R(s)|\ll_\epsilon q_R^{(-\Re s)_++\epsilon},
 \qquad
 \left|\frac{D_R'(s)}{D_R(s)}\right|
 \ll_{\sigma_0}\log(2q_R)\quad(\Re s\ge\sigma_0).
\]
In particular a support $q_R\le Z^B$ with bounded $B$ costs any prescribed
positive power of $Z$ on a fixed positive real half-plane.  For an
imprimitive function $L(s,\psi^*)D_R(s)$, both the primitive conductor and
the deletion radical $R$ must be included when applying logarithmic control.
\end{lemma}

\begin{proof}
Every factor is nonzero on $\Re s>0$.  For $\Re s\ge\sigma_0$, both its
absolute value and the absolute value of its inverse are at most
$(1-q_p^{-\sigma_0})^{-1}$.  For sufficiently large $q_p$, depending on
$\sigma_0,\epsilon$,
$-\log(1-q_p^{-\sigma_0})\le\epsilon\log q_p$.  The finitely many smaller
primes contribute a fixed constant.  This proves the first estimate.
For any real $\sigma$,
\[
 \prod_{p\mid R}(1+q_p^{-\sigma})
 \le q_R^{(-\sigma)_+}2^{\#\{p:p\mid R\}}
 \ll_\epsilon q_R^{(-\sigma)_++\epsilon}.
\]
The last inequality follows by separating the finitely many primes with
$q_p<2^{1/\epsilon}$.  Finally, logarithmic differentiation gives
\[
 \left|\frac{D_R'}{D_R}(s)\right|
 \le\sum_{p\mid R}\frac{\log q_p}{q_p^{\sigma_0}-1}
 \ll_{\sigma_0}\log(2q_R).
\]
If $q_R\le Z^B$, choose the exponent in the first estimate smaller than the
desired exponent of $Z$ divided by the bounded value of $B$; the case $B=0$
is immediate.  This proves the stated interpretation for deleted factors.
\end{proof}

\section{Completed cubic reflection and unmarked row energy}
\label{sec:completed-reflection}

This section first transforms a completed sum of Gauss coefficients while
retaining every zero extension in its character.  It then combines that
identity with the quadratic large sieve to bound its mean square over
arbitrary nonzero element rows.  At equal row and completed lengths, the
result is the bound $Z^{1+\epsilon}$ needed in the balanced argument.

The theta function and its Fourier coefficients at three fixed cusps are
the unconditional input from Dunn and Radziwi{\l}{\l}~\cite[Section~5 and
Appendix~A]{DR}.  The original coefficient and cusp calculations are in
Patterson~\cite[Theorem~8.1 and Sections~7--8]{Patterson}; the formal input
here remains the formulas in~\cite{DR}.  The finite transform below derives
the masked formula, including its local factors and phase.  The later
mean-square argument uses this phase only after fixing its finite cusp sectors.

\subsection{The completed reflection}

Write
\[
 \check e(z)=\exp\bigl(2\pi i(z+\bar z)\bigr),
 \qquad e(z)=\check e(z/\lambda)\qquad(z\in\mathbb C).
\]
On $F$, the exponent in $\check e$ is $2\pi i\operatorname{Tr}_{F/\mathbb Q}z$.
For every integer exponent, a power of a residue character is understood
to be zero at a nonunit.  In particular, $\chi_p(x)^0=1_{p\nmid x}$.

\begin{proposition}[Completed cubic reflection]
\label{prop:completed-reflection}
Fix a finite family $\mathcal X$ of finite-ray characters whose conductors
are supported on $S$.  Choose one fixed integral modulus $\mathfrak E$ with
prime support exactly $S$, divisible by every conductor in this family.
On primary elements extend each member of $\mathcal X$ by zero away from the
$\mathfrak E$-units.  Choose a single $L\in\mathcal O\setminus\{0\}$ with
prime support $S$, with $(18)\mid(L)$ and $\mathfrak E\mid(L)$, large enough
that, for every $\Psi_0\in\mathcal X$, the function on $\mathcal O$
\[
 \phi(x)=
 \begin{cases}
 \chi_x(\lambda)^2\Psi_0(x),&x\equiv1\pmod3,\ (x,\mathfrak E)=1,\\
 0,&\text{otherwise},
 \end{cases}
 \qquad
 \widehat\phi(h)=q_L^{-1}\sum_{x\bmod L}\phi(x)e(-hx/L)
\]
is $L$-periodic.  The zero branch is evaluated without evaluating either
character.  Such a choice of $L$ exists by the cubic supplementary law and
the ray periodicity, as verified below.  Both $\mathfrak E$ and $L$ are fixed
for the entire family before any moving prime is chosen.  Fix a member
$\Psi_0\in\mathcal X$ and put $\mathfrak L=(L)$.  Let $\mathcal P$ be any
finite set of distinct primary
primes not dividing $L$, choose $j_p\in\{0,1,\ldots,5\}$ for each
$p\in\mathcal P$, and set, on primary elements,
\[
 \Psi(n)=\Psi_0(n)\prod_{p\in\mathcal P}\chi_p(n)^{j_p}.
\]
No coprimality between the two variables in the following product is
imposed:
\[
 \begin{split}
 D(s,\Psi)&=\sum_{\substack{n\equiv1\ (3)\\(n,\mathfrak E)=1}}^{\rm sf}
       \gamma_2(n)\bar\alpha(n)\Psi(n)q_n^{-s},\\
 L_0(s,\Psi)&=\sum_{\substack{b\equiv1\ (3)\\(b,\mathfrak E)=1}}
       \bar\alpha(b)^3\Psi(b)^3q_b^{-3s+1/2},
       \qquad T(s,\Psi)=L_0(s,\Psi)D(s,\Psi).
 \end{split}
\]
Both series and their product converge absolutely for $\Re s>1$.

Let $X>0$.  Suppose that $V\in C^\infty(0,\infty)$ and that, for every
$C>0$ and every integer $j\ge0$, $(x\partial_x)^jV(x)$ is $O_{C,j}(x^C)$ as
$x\to0$ and $O_{C,j}(x^{-C})$ as $x\to\infty$.  Define
\[
 \widehat V(t)=\int_0^\infty V(x)x^t\,\frac{dx}{x}.
\]
The smoothed expression to be transformed is the completed sum
\[
 \begin{split}
 &\frac1{2\pi i}\int_{(a)}\widehat V(s-1/2)X^{s-1/2}T(s,\Psi)\,ds\\
 &\quad=\sum_{\substack{n\ {\rm sf},\ b\\n,b\equiv1\ (3)\\(nb,\mathfrak E)=1}}
 \frac{\gamma_2(n)\overline{\alpha(nb^3)}\Psi(nb^3)}
      {\sqrt{q_n}\,q_b}
 V\!\left(\frac{q_nq_b^3}{X}\right),\qquad a>1.
 \end{split}
\]
This equality follows by absolute convergence and Mellin inversion.
The character acts on the whole index $nb^3$, with its zero values retained;
$n$ and $b$ may share prime factors.

The reflected test is defined by
\[
 K=\frac{(2\pi)^4}{27},\qquad
 R(t)=\prod_{\pm}\frac{\Gamma(1+t\pm1/6)}{\Gamma(1-t\pm1/6)},
\]
\[
 V^\sharp(x)=\frac1{2\pi i}\int_{(\sigma)}
       \widehat V(-t)R(t)(Kx)^{-t}\,dt\qquad(\sigma\ge0).
\]
The integral defining $V^\sharp$ is absolutely convergent and is
independent of $\sigma\ge0$.

The product $T$ has an entire continuation.  On every fixed vertical
strip it has polynomial growth, with constants allowed to depend on
$L,\Psi_0,\mathcal P,(j_p)$ and the strip.  There is one fixed triple of
cusp coefficient functions from $\lambda^{-4}\mathcal O\setminus\{0\}$ to
$\mathbb C$, constructed from the theta expansions in the proof.  The same
triple is used for all choices of the arithmetic data, scale, and test profile.
For each fixed arithmetic choice above, there is a finite family of terms,
indexed as in (1), independent
of $X$, $V$, and the contour parameter $a>1$.  In each term, $d$ is selected
from this triple and $\vartheta$ is an additive character of $\mathcal O$
modulo a fixed modulus depending only on $L$; their sectorwise dependence is
exactly that stated in (3), and the remaining data have properties (1)--(3).
For these terms and every $a>1$, the identity is
\begin{equation}\label{eq:reflection}
 \begin{aligned}
 &\frac1{2\pi i}\int_{(a)}\widehat V(s-1/2)X^{s-1/2}T(s,\Psi)\,ds\\
 &\quad=\sum_{\text{terms}}\zeta
 \sum_{0\ne\mu\in\lambda^{-4}\mathcal O}
 \frac{d(\mu)\alpha(\mu)\vartheta(\lambda^4\mu)}{\sqrt{q_\mu}}
 \prod_{p\ {\rm active}}B_p(\lambda^4\mu)
 V^\sharp\!\left(\frac{q_\mu X}{q_c^2}\right).
 \end{aligned}
\end{equation}
Every inner sum is absolutely convergent.  The terms have the following
precise properties.

\begin{enumerate}
\item A term is specified by $h_0\bmod L$ and a subset of active primes.
Every $p$ with $j_p\ne0$ is active, while a prime with $j_p=0$ may be
active or inactive.  Write $r=\prod_{p\ {\rm active}}p$.  Then
$c=c_Fr$, where $c_F$ is the reduced denominator of
$\lambda^2h_0/L$, with a normalizing unit.  Thus $c_F$ ranges over a
fixed finite set and its prime divisors divide $L$.  There are
$O_{\mathfrak L}(2^{|\mathcal P|})$ terms, or
$O_{\mathfrak L}(4^{|\mathcal P|})$ after splitting each Ramanujan
factor in the next display into its two summands.

\item The local column factors are
\[
 B_p(x)=\begin{cases}
 \chi_p(x)^{-j_p-2},&j_p\ne0,4,\\
 q_p^{-1/2}(-1+q_p1_{p\mid x}),&j_p=4,\\
 q_p^{-1/2}\chi_p(x)^{-2},&j_p=0\text{ and }p\text{ is active}.
 \end{cases}
\]
The function $d$ is one of three fixed cusp coefficient functions.
Writing $u$ for an Eisenstein unit, their support and size satisfy
\begin{equation}
 \operatorname{supp}d\subset
 \{u\lambda^k n b^3:k\ge-4,\ n,b\equiv1\pmod3,\ n\ {\rm sf}\},
 \qquad |d(\mu)|\le27\,3^{k/6}|b|.
 \label{old-eq:4.3}
\end{equation}
The bound refers to the displayed representation of a supported index.
It permits common primes of $n$ and $b$, and it imposes no exclusion at
the primes of $L$ other than the displayed restriction at $\lambda$.
The character $\vartheta$ is an additive character of a quotient of
$\mathcal O$ by a fixed modulus depending only on $L$.

\item The scalar $\zeta$ is independent of $\mu$ and satisfies
$|\zeta|\le1/81$.  Its moving-prime dependence can be specified exactly.
For $m\not\equiv0\pmod6$, put
\[
 \tau_{p,m}^{\pm}=q_p^{-1/2}\sum_{y\bmod p}\chi_p(y)^m e(\pm y/p),
 \qquad |\tau_{p,m}^{\pm}|=1.
\]
For each active $p$, define units modulo $p$ by
\[
 \sigma_p=\lambda^2c/p,\qquad
 \epsilon_p=-\big((\lambda^3c/p)\sigma_p\big)^{-1},
\]
and put
\[
 \omega_{p,j}=\begin{cases}
 \tau_{p,j}^{-}\tau_{p,j+2}^{+}\chi_p(\epsilon_p)^{-j-2},&j\ne0,4,\\
 \tau_{p,4}^{-},&j=4,\\
 -\tau_{p,2}^{+}\chi_p(\epsilon_p)^{-2},&j=0.
 \end{cases}
\]
The indices of $\tau$ are read modulo six.  With
$M=\lambda^{12}L^4$, there is a number $\kappa_F$ of absolute value one,
depending only on $h_0$ and the class of $r$ modulo $M^2$, such that
\begin{equation}\label{eq:reflection-row-phase}
 \zeta=-\frac i{81}\bar\alpha(c)^2\widehat\phi(h_0)\bar\kappa_F
 \prod_{p\ {\rm inactive}}(1-q_p^{-1})
 \prod_{p\ {\rm active}}\chi_p(\sigma_p)^{-2}\omega_{p,j_p}.
\end{equation}
In each of the finitely many classes of $h_0$ and $r\bmod M^2$, both
$d$ and $\vartheta$ are fixed.  Consequently these are common column
factors within that sector.  Apart from the displayed scale $q_c^2$,
all other moving-prime dependence outside the $B_p$ is in the scalar
$\zeta$, which may depend on the whole active set.

The branch conventions are simultaneous for all local prime sets with
the same $L$ and $\Psi_0$.  In particular, if a good prime $p$ is added
to the local set with exponent zero and is declared inactive, the resulting
branch has the same $c_F,c,d,\vartheta,\kappa_F$, the same data at every
other active prime, and the same kernel as the branch in which $p$ is absent.
Its scalar is multiplied by $1-q_p^{-1}$.  This compatibility includes every
$h_0$, not only the unit classes modulo $L$.
\end{enumerate}

\emph{Bounds for the transformed test.}
For $A\ge0$ and an integer $B\ge0$, put
\[
 \mathfrak M_{A,B}(V)=
 \sup_{-A\le\eta\le1/4}\int_{\mathbb R}
       (1+|u|)^B|\widehat V(\eta+iu)|\,du.
\]
For every integer $j\ge0$,
\begin{equation}\label{eq:reflection-kernel}
 |(x\partial_x)^jV^\sharp(x)|
 \ll_{A,j}\min\{x^{1/4},(1+x)^{-A}\}
       \mathfrak M_{A,\lceil4A\rceil+j+2}(V).
\end{equation}
If $\chi\in C_c^\infty(0,\infty)$ is fixed, $Y>0$, and $J\ge0$ is an
integer, then
\begin{equation}\label{eq:reflection-annular-fourier}
 \int_{\mathbb R}(1+|u|)^J
 \left|\int_0^\infty\chi(y)V^\sharp(Yy)y^{-iu}\frac{dy}{y}\right|du
 \ll_{A,J,\chi}\min\{Y^{1/4},(1+Y)^{-A}\}
       \mathfrak M_{A,\lceil4A\rceil+J+4}(V).
\end{equation}
These are finite smooth seminorms.  More precisely,
\begin{equation}\label{eq:reflection-test-seminorm}
 \mathfrak M_{A,B}(V)\ll_{A,B}
 \int_0^\infty(1+x^{-A}+x^{1/4})
       \bigl(|V(x)|+|(x\partial_x)^{B+2}V(x)|\bigr)\frac{dx}{x}.
\end{equation}
Replacing $V(x)$ by $x^{iu_0}V(x)$ multiplies
$\mathfrak M_{A,B}$ by at most $(1+|u_0|)^B$.
\end{proposition}

\begin{proof}
We first express the character masks through finitely many translated theta
values.  We then transform their rational cusps and compute the resulting
automorphy multipliers and local Fourier factors; the final Mellin comparison
gives the reflection identity and the bounds for its transformed test.

\emph{The fixed theta input.}
Let $\Gamma=\mathrm{SL}_2(\mathcal O)$,
$\Gamma_1(3)=\{g\in\Gamma:g\equiv I\pmod3\}$, and
$\Gamma_2=\langle\mathrm{SL}_2(\mathbb Z),\Gamma_1(3)\rangle$.
The unconditional formulas of
\cite[Equations (5.4), (5.7), (5.9), and (5.12)--(5.17)]{DR}
give a smooth cubic theta function $\theta$ on
$\mathbb H^3=\mathbb C\times\mathbb R_{>0}$ with
\[
 \theta(gw)=\kappa(g)\theta(w)\quad(g\in\Gamma_2),\qquad
 \kappa\!\begin{pmatrix}a&b\\c&\delta'\end{pmatrix}
   =\left(\frac ca\right)_3\quad
   \left(\begin{pmatrix}a&b\\c&\delta'\end{pmatrix}\in\Gamma_1(3),\ c\ne0\right).
\]
Here $\kappa$ is one on $\mathrm{SL}_2(\mathbb Z)$ and is one on an
element of $\Gamma_1(3)$ with lower left entry zero.  The subscript $3$
denotes the ordinary cubic residue symbol; for a good prime $p\notin S$,
one has $(x/p)_3=\chi_p(x)^2$.
These automorphy and coefficient formulas have no hypothesis about zeros
of $L$-functions.

Put
\[
 \mathsf L(t)=\begin{pmatrix}1&0\\t&1\end{pmatrix},\qquad
 (t_0,t_1,t_2)=(0,\omega^2,\omega),\qquad
 \Theta_b(w)=\theta(\mathsf L(t_b)w).
\]
If $v\in\mathbb Z+3\mathcal O$, then $\mathsf L(v)$ is a product of
an element of $\mathrm{SL}_2(\mathbb Z)$ and a lower translation in
$\Gamma_1(3)$ with multiplier one.  The analogous assertion
holds for upper translations.  Therefore
\[
 \theta(\mathsf L(\lambda)w)=\Theta_1(w),\qquad
 \theta(\mathsf L(-\lambda)w)=\Theta_2(w),
\]
because $\lambda-\omega^2=2+3\omega$ and
$-\lambda-\omega=-1-3\omega$.  If $u_0=s+t\omega$ with
$0\le s,t<3$, and $E=\bigl(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\bigr)$,
then
\[
 \begin{pmatrix}u_0&-1\\1&0\end{pmatrix}=E\mathsf L(-u_0),
 \qquad -u_0-t_t\in\mathbb Z+3\mathcal O,
\]
and hence $\theta\bigl(\bigl(\begin{smallmatrix}u_0&-1\\1&0\end{smallmatrix}\bigr)w\bigr)
=\Theta_t(w)$.  These are the only cusp representatives needed below.

For any of them, define $d_H$ by
\begin{equation}\label{eq:reflection-cusp-expansion}
 \bar\theta(H(z,v))=C_H(v)+
 \sum_{\mu\ne0}d_H(\mu)vK_{1/3}(4\pi|\mu|v)\check e(\mu z).
\end{equation}
Here $C_H(v)$ is independent of $z$.
The three functions $\Theta_0,\Theta_1,\Theta_2$ are respectively
$F_1,F_{19},F_{10}$ in~\cite{DR}.  Its Appendix~A, rows $1,19,10$,
expresses their coefficients in terms of $\tau,\tau_1,\tau_2$ in its
Equations (5.7), (5.13), and (5.14).  In their notation the conjugate
expansion uses $\overline{d_j(-\mu)}$, as in their Equation (5.16).
Those formulas give the support inclusion in
Equation~\eqref{old-eq:4.3}.  For completeness, the two possible
magnitudes from $\tau$ at $\mu=u\lambda^k n b^3$ are
\[
 3^{k/6+8/3}|b|\quad\hbox{and}\quad3^{k/6+3}|b|,
\]
and $\tau_1,\tau_2$ have magnitude $9|b|$ and $k=-4$.
Here the Gauss sums use only the cubic symbol, including at primes in $S$.
To make their domain explicit, for every primary squarefree $n$ define,
as in~\cite[Equation (1.7)]{DR},
\begin{equation}\label{eq:reflection-cubic-source-gauss}
 \widetilde g_3(n)=q_n^{-1/2}\sum_{v\bmod n}
       \left(\frac vn\right)_3\check e(v/n),
 \qquad \widetilde g_3(1)=1.
\end{equation}
The ordinary cubic symbol is defined at every prime other than $\lambda$
and is extended by zero on nonunits and multiplicatively in the denominator.
Every primary $n$ is prime to $\lambda$.  At each prime divisor of $n$ the
cubic character is nontrivial, and $\check e(v/n)=e(\lambda v/n)$ is primitive
because $\lambda$ is a unit there.  The finite-field Gauss identity and the
Chinese remainder theorem therefore give $|\widetilde g_3(n)|=1$ for every
such squarefree $n$.  This includes the primary prime $-2$ of norm $4$;
no sextic character with that denominator is used.  Also, whenever $(t,n)=1$,
the change of variable $y=tv$ gives
\[
 q_n^{-1/2}\sum_{v\bmod n}\left(\frac vn\right)_3\check e(tv/n)
       =\left(\frac tn\right)_3^{-1}\widetilde g_3(n).
\]
This covers the twists $t=\lambda^2,\omega\lambda^2,\omega^2\lambda^2$
in the other source coefficient families.  Thus the unnormalized Gauss sums
occurring in all three cusp families have magnitude $|n|$.  Consequently
each coefficient magnitude displayed above is at most
$27\,3^{k/6}|b|$, including $k=-4$.  The formulas require only that
$n$ be squarefree and that $n,b$ be primary; they do not require
$(n,b)=1$.  Prime valuations of $nb^3$ also show that its representation
by such $n,b$, when it exists, is unique.  In particular the same bound
is valid when the variables share primes or contain primes excluded
from the primal sums.

At infinity the condition $x=\lambda^3\mu\equiv1\pmod3$ isolates
exactly $\mu=\lambda^{-3}nb^3$ with $n,b$ primary and $n$ squarefree.
The infinity support has $\lambda^3\mu\in\mathcal O$; every other ramified
exponent in Equation (5.7) of~\cite{DR}
makes $x$ divisible by $\lambda$, and the negative sign in the remaining
pair gives $x\equiv-1\pmod3$.  At the isolated index the coefficient in
$\bar\theta$ is
\begin{equation}\label{eq:reflection-infinity-coefficient}
 d_I(\lambda^{-3}nb^3)=3^{5/2}|b|\widetilde g_3(n).
\end{equation}
This is the unmasked source formula, valid for every primary squarefree $n$
and every primary $b$, with no coprimality condition between them.  Only when
$(n,S)=1$ may it be expressed in the global sextic notation: substituting
$y=\lambda v$ in $\gamma_2(n)$ gives
\begin{equation}\label{eq:reflection-cubic-sextic-conversion}
 \widetilde g_3(n)=\left(\frac\lambda n\right)_3^{-1}\gamma_2(n)
       =\chi_n(\lambda)^{-2}\gamma_2(n)\qquad((n,S)=1).
\end{equation}
The cubic supplementary laws in~\cite[Equation (1.5)]{DR} make
$(\lambda/x)_3$ periodic on all primary $x$ modulo $9$.  On primary
elements prime to $S$ it equals $\chi_x(\lambda)^2$.  A character in $\mathcal X$ is
periodic on primary elements prime to $\mathfrak E$ modulo its conductor, and the
zero condition $(x,\mathfrak E)\ne1$ is periodic modulo $\mathfrak E$.
Thus a sufficiently divisible $L$ with prime support $S$ works for every
member of the fixed finite family.  This verifies its choice before the
moving local primes are selected.

\emph{Finite Fourier inversion with the zero extensions.}
For $j\in\{0,\ldots,5\}$ define
\[
 C_{p,j}(h)=q_p^{-1}\sum_{x\bmod p}\chi_p(x)^j e(-hx/p).
\]
The character pairing furnished by $e$ is nondegenerate on
$\mathcal O/(p)$.  Changing variables in a Gauss sum, and using the
orthogonality of a nontrivial multiplicative character, gives
\begin{equation}\label{eq:reflection-local-fourier}
 C_{p,j}(h)=\begin{cases}
 q_p^{-1/2}\tau_{p,j}^{-}\chi_p(h)^{-j},&j\ne0,\ h\ne0,\\
 0,&j\ne0,\ h=0,\\
 -q_p^{-1},&j=0,\ h\ne0,\\
 1-q_p^{-1},&j=0,\ h=0.
 \end{cases}
\end{equation}
Here and below $h=0$ means $h\equiv0\pmod p$.  Each nontrivial power of
$\chi_p$ is primitive modulo the prime $p$, so the usual finite-field
Gauss-sum identity gives $|\tau_{p,m}^{\pm}|=1$.

Fourier inversion on $\mathcal O/(L)$ and on each $\mathcal O/(p)$
now gives
\[
 \phi(x)\prod_p\chi_p(x)^{j_p}
  =\sum_h C(h)e\!\left(x\left(h_0/L+\sum_p h_p/p\right)\right),
 \qquad C(h)=\widehat\phi(h_0)\prod_pC_{p,j_p}(h_p).
\]
This uses every additive frequency, including nonunit $h_0$ and zero local
frequencies.  We do not invoke the twisted formula in
\cite[Lemma~5.2 and Corollary~5.1]{DR}, whose unit-Fourier-support hypothesis
need not hold for these masks.  The theta function with this multiplier on
its infinity coefficient $x=\lambda^3\mu\in\mathcal O$ is the finite sum
\begin{equation}\label{eq:reflection-translated-theta}
 \mathcal F(z,v)=\sum_hC(h)\bar\theta(z+z_h,v),\qquad
 z_h=\lambda^2\left(h_0/L+\sum_p h_p/p\right).
\end{equation}
For an isolated infinity index $x=nb^3$, the piecewise definition of $\phi$
first discards every term for which $(nb,\mathfrak E)\ne1$.  This is a mask on
the whole completed index: $(nb^3,\mathfrak E)=1$ if and only if both $n$
and $b$ are $\mathfrak E$-units.  Only on that remaining domain do we use
complete multiplicativity, including $\chi_b(\lambda)^6=1$, to write
\[
 \phi(nb^3)\prod_p\chi_p(nb^3)^{j_p}
 =\chi_n(\lambda)^2\Psi(n)\Psi(b)^3
 \qquad((nb,\mathfrak E)=1).
\]
It remains true when $n,b$ share a moving prime, since the corresponding
local powers retain their zero values, including for $j_p=0$.  For the
discarded terms the completed coefficient is defined to be zero directly;
no value of $\chi_n(\lambda)$ is evaluated there.  Combining
Equations~\eqref{eq:reflection-infinity-coefficient} and
\eqref{eq:reflection-cubic-sextic-conversion} on the retained domain now
identifies the direct Mellin series with the product $T$.  The dual coefficient
functions $d_H$ keep the full source support in Equation~\eqref{old-eq:4.3};
no $S$-mask is placed on them.

Fix $h_0$ and declare $p$ active precisely when $h_p\ne0$.
Equation~\eqref{eq:reflection-local-fourier} forces all primes with
$j_p\ne0$ to be active.  Choose once for each $h_0\bmod L$ a representative
and a reduced expression
$\lambda^2h_0/L=a_F/c_F$.  Multiplying numerator and denominator by
one unit, normalize the numerator to be $1\pmod3$ if $\lambda\mid c_F$
and the denominator to be $1\pmod3$ otherwise.  Since every active
prime is primary, this unit depends only on $h_0$.
For $r=\prod_{p\ {\rm active}}p$, Equation~\eqref{eq:reflection-translated-theta}
then gives
\begin{equation}\label{eq:reflection-cusp-fraction}
 z_h=\frac ac,\qquad c=c_Fr,\qquad
 a=a_Fr+\lambda^2c_F\sum_{p\mid r}(r/p)h_p.
\end{equation}
At a prime dividing $c_F$, the numerator is congruent to $a_Fr$ and is
a unit.  At $p\mid r$, it is congruent to
$\lambda^2c_F(r/p)h_p$ and is again a unit.  Thus this fraction is
reduced, even when $h_0$ is a nonunit modulo $L$.  The empty product
$r=1$ is allowed.  Replacing any lift by another changes $z_h$ by an
element of $\lambda^2\mathcal O=3\mathcal O$, a period of $\theta$.

\emph{Choice of the cusp matrix and its multiplier.}
The next construction determines the automorphy multiplier and reflected
additive phase for each reduced cusp fraction just obtained.
Set $M=\lambda^{12}L^4$.  For every active $p$, the Chinese remainder
theorem permits the chosen nonzero class $h_p\bmod p$ to be lifted with
$h_p\equiv0\pmod{M^2}$.  Restrict $r$ to one class modulo $M^2$.
Equation~\eqref{eq:reflection-cusp-fraction} shows that $a$ is then
fixed modulo $Mc_F$; in fact its moving terms are divisible by
$M^2c_F$, and $a_Fr$ is fixed modulo $M^2$, which is divisible by
$Mc_F$ because $c_F$ divides $L$ up to a unit.
It also shows $a\equiv1\pmod3$ if $\lambda\mid c$, and
$c\equiv1\pmod3$ otherwise.

Choose $\delta'$ by the following compatible local congruences:
\[
 \begin{array}{ll}
 \delta'\equiv a^{-1}\pmod{Mc_F}
       &\text{at every prime dividing }c_F,\\
 \delta'\equiv0\pmod M
       &\text{at every prime dividing }M\text{ but not }c_F,\\
 \delta'\equiv a^{-1}\pmod p& (p\mid r).
 \end{array}
\]
The first line means the indicated prime-power parts of $Mc_F$.
All inverses exist because $a/c$ is reduced.  Put
$b=(a\delta'-1)/c$ and $g=\bigl(\begin{smallmatrix}a&b\\c&\delta'\end{smallmatrix}\bigr)$.
Then $g\in\Gamma$, and division of $a\delta'-1$ by $c$ gives
\[
 b\equiv0\pmod M\text{ at primes of }c_F,\qquad
 b\equiv-c^{-1}\pmod M\text{ at the other primes of }M.
\]
Consequently $a,b,c,\delta'$ have fixed residues at the needed powers
of every prime dividing $L$, independent of the classes $h_p$.
Whenever a zero entry would occur in a residue-symbol calculation,
one may first translate $z_h$ by $\lambda^2M^2\mathcal O$ and then
add a multiple of the combined congruence modulus to $\delta'$.
These changes preserve all displayed congruences and avoid the finitely
many values making an entry zero.  They do not change the translated
theta function.

There are three cases, distinguished by $v_\lambda(c)$.
If $3\mid c$, put $H=I$.  Then $G=g\in\Gamma_1(3)$, and cubic
reciprocity for the primary $a$ and the primary primes of $r$ gives
\begin{equation}\label{eq:reflection-cusp-case-zero}
 \kappa(G)=\left(\frac{c_F}{a}\right)_3
             \left(\frac ar\right)_3.
\end{equation}
If $v_\lambda(c)=1$, take the unique $u_0\in\{\lambda,-\lambda\}$
with $u_0\equiv c\pmod3$ and put $H=\mathsf L(u_0)$; these are the two
possibilities because $\mathcal O/(\lambda)\simeq\mathbb F_3$.
The congruences above show that $G=gH^{-1}\in\Gamma_1(3)$.
Write $A=a-u_0b$.  Both $A$ and $a$ are primary, and they are coprime:
$(a,b)=1$ and $(a,u_0)=1$.  The determinant equation gives
\[
 a(c-u_0\delta')=cA-u_0,
 \qquad bc\equiv-1\pmod a.
\]
It follows, by cubic reciprocity for $a,A$, that
\[
 \begin{split}
 \kappa(G)
 &=\left(\frac{-u_0}{A}\right)_3
          \left(\frac aA\right)_3^{-1}\\
 &=\left(\frac{-u_0}{A}\right)_3
          \left(\frac{c/u_0}{a}\right)_3
  =\left(\frac{-u_0}{A}\right)_3
          \left(\frac{c_F/u_0}{a}\right)_3
          \left(\frac ar\right)_3.
 \end{split}
\]
For the second equality, reduce $A$ modulo $a$ and use
$(A/a)_3=(-u_0b/a)_3=(u_0/a)_3(c/a)_3^{-1}$; the cubic symbol of
$-1$ is one.  Finally suppose $(\lambda,c)=1$.  Choose
$u_0=s+t\omega\equiv a\pmod3$ with $0\le s,t<3$ and put
$H=\bigl(\begin{smallmatrix}u_0&-1\\1&0\end{smallmatrix}\bigr)$.
Here $c\equiv1$, $\delta'\equiv0$, and $b\equiv-1\pmod3$, so
\[
 G=gH^{-1}=\begin{pmatrix}-b&a+bu_0\\-\delta'&c+\delta'u_0\end{pmatrix}
       \in\Gamma_1(3).
\]
Both $-b$ and $c$ are primary.  Moreover $-bc=1-a\delta'\equiv1\pmod9$.
Factor $\delta'=u\lambda^k\delta_0$ with $\delta_0$ primary.  The
supplementary laws make $(u\lambda^k/(1-a\delta'))_3=1$; cubic
reciprocity makes $(\delta_0/(1-a\delta'))_3=1$ because the numerator
after reciprocity is $1\pmod{\delta_0}$.  Thus
\[
 \left(\frac{\delta'}{-bc}\right)_3=1,
 \qquad
 \kappa(G)=\left(\frac{\delta'}{-b}\right)_3
          =\left(\frac{\delta'}c\right)_3^{-1}
          =\left(\frac ac\right)_3
          =\left(\frac a{c_F}\right)_3\left(\frac ar\right)_3.
\]
The last inverse uses $a\delta'\equiv1\pmod c$.

In all three cases we have proved
\begin{equation}\label{eq:reflection-multiplier}
 \kappa(G)=\kappa_F\prod_{p\mid r}\chi_p(a)^2,
 \qquad |\kappa_F|=1.
\end{equation}
The factor $\kappa_F$ is constant as the active $h_p$ vary in the
fixed sector.  If $3\mid c$, it is $(c_F/a)_3$: the unit and
$\lambda$-power factors are determined by $a\bmod9$, and reciprocity
determines the remaining fixed-prime factors from the residue of $a$ at
the primes of $c_F$.  In the middle case it is
$(-u_0/A)_3((c_F/u_0)/a)_3$, determined in the same way by the fixed
residues of $A=a-u_0b$ and $a$.  In the last case it is $(a/c_F)_3$,
which is determined directly by the fixed denominator $c_F$ and
$a\bmod c_F$.  The displayed congruences for $a,b,c,\delta'$ therefore
determine $\kappa_F$ using only $h_0$ and $r\bmod M^2$.
They also determine the chosen one of the three functions $\Theta_b$.

\emph{The reflected additive phase.}
Write $x=\lambda^4\mu$ and $D_F=\lambda^3c_F$.  At each $p\mid r$,
Equation~\eqref{eq:reflection-cusp-fraction} gives
$a\equiv\sigma_ph_p\pmod p$.  Hence
$\delta'\equiv\sigma_p^{-1}h_p^{-1}\pmod p$.  Additive Chinese
remaindering for the coprime factors of $D_Fr$ gives the exact identity
\begin{equation}\label{eq:reflection-additive-phase}
 \begin{split}
 \check e(-\delta'\mu/c)
 &=e(-\delta'x/(D_Fr))\\
 &=\vartheta(x)\prod_{p\mid r}e(\epsilon_ph_p^{-1}x/p),
 \qquad
 \vartheta(x)=e(-\delta'_F r^{-1}x/D_F).
 \end{split}
\end{equation}
Here $\delta'_F$ is the class of $\delta'$ modulo $D_F$, and
$r^{-1}$ in the formula for $\vartheta$ is taken modulo $D_F$.
For example, the local numerator at $p$ is
$-\delta'(D_Fr/p)^{-1}=\epsilon_ph_p^{-1}$; this verifies the signs
and the powers of $\lambda$.  The chosen congruences fix $\delta'_F$,
since $D_F$ divides the corresponding local moduli, and the sector fixes
$r^{-1}\bmod D_F$.  A common multiple of the finitely many $D_F$ is
therefore a fixed modulus for all the characters $\vartheta$.

Conjugating Equation~\eqref{eq:reflection-multiplier} contributes
$\bar\kappa_F\prod_{p\mid r}\chi_p(\sigma_p)^{-2}\chi_p(h_p)^{-2}$.
For $j\ne0$, multiplication by
Equation~\eqref{eq:reflection-local-fourier} leaves the local sum
\[
 q_p^{-1/2}\tau_{p,j}^{-}
 \sum_{h\ne0}\chi_p(h)^{-j-2}e(\epsilon_ph^{-1}x/p).
\]
Upon putting $y=h^{-1}$, the sum is a Gauss sum of exponent $j+2$.
If $j\ne4$, its value is
$q_p^{1/2}\tau_{p,j+2}^{+}\chi_p(\epsilon_px)^{-j-2}$, also at
$p\mid x$ because both sides then vanish.  If $j=4$, it is the
Ramanujan sum
\[
 \sum_{y\ne0}e(\epsilon_pxy/p)=-1+q_p1_{p\mid x}.
\]
For an active $j=0$, the same substitution gives
\[
 -q_p^{-1}\sum_{h\ne0}\chi_p(h)^{-2}e(\epsilon_ph^{-1}x/p)
 =-q_p^{-1/2}\tau_{p,2}^{+}\chi_p(\epsilon_px)^{-2}.
\]
These are exactly $\omega_{p,j}B_p(x)$; an inactive zero exponent
contributes $1-q_p^{-1}$.  This proves every local factor, including
its zero extension.  The choices of $h_0$ and the active subset give at
most $q_L2^{\#\{p:j_p=0\}}$ terms.  Splitting the $j_p=4$ factors
multiplies this by at most $2^{\#\{p:j_p=4\}}$, proving the stated
counts.

The denominator and the fixed-sector data constructed before these local
sums depend on the local prime set only through $h_0$ and the active radical
$r$; the remaining active-frequency dependence is exactly the local dependence
just summed.  We use the same representatives
and normalized fractions for each $h_0$, and the same fixed-sector choices
of cusp function, fixed additive character, and $\kappa_F$ whenever those
data recur.  An absent prime and an inactive zero-exponent prime have the
same active radical; their other active $\sigma_p,\epsilon_p$ are therefore
also identical.  The inactive coefficient $1-q_p^{-1}$ is the only change.
This proves the simultaneous branch compatibility in the statement.

The marked reflection in Section~\ref{par:reflection-marks} will use the
following dependence on pairs of active primes.  Since
\[
 \epsilon_p=-\lambda^{-5}(c/p)^{-2}\pmod p,
\]
there is a scalar $\xi_{p,j}$ of absolute value one, depending on $p,j$ but on no other
active prime such that
\begin{equation}\label{eq:reflection-cross-prime-phase}
 \chi_p(\sigma_p)^{-2}\omega_{p,j}
       =\xi_{p,j}\chi_p(c/p)^{2j+2}.
\end{equation}
For $j=4$ this uses $2j+2\equiv-2\pmod6$; for the other exponents it
follows immediately by inserting the expression for $\epsilon_p$.
Thus the contribution of another active prime to the phase at $p$ has
exponent exactly $2j+2$ modulo six.  In particular an active exponent
$j=1$ has column coupling $\chi_p(x)^{-3}=\chi_p(x)^3$.

\emph{Mellin normalization and continuation.}
Let
\[
 J(s)=\int_0^\infty
       \left.\partial_{\bar z}\mathcal F(z,v)\right|_{z=0}
       v^{2s-1}\,dv.
\]
The Bessel identity used in~\cite[Lemma~5.3]{DR} is
\[
 \int_0^\infty K_{1/3}(y)y^{w-1}\,dy
  =2^{w-2}\Gamma((w-1/3)/2)\Gamma((w+1/3)/2)
       \qquad(\Re w>1/3).
\]
On $\Re s>1$ the coefficient sums converge absolutely, so we may
differentiate and integrate their expansions term by term.
Since $\partial_{\bar z}\check e(\mu z)=2\pi i\bar\mu\check e(\mu z)$,
the identity with $w=2s+1$ gives
\[
 J(s)=\frac i4(2\pi)^{-2s}\Gamma(s+1/3)\Gamma(s+2/3)
       \sum_{0\ne\mu\in\lambda^{-3}\mathcal O}d_I(\mu)\phi(\lambda^3\mu)
          \prod_p\chi_p(\lambda^3\mu)^{j_p}
          \bar\alpha(\mu)q_\mu^{-s}.
\]
First use the whole-index mask to restrict to $(nb,\mathfrak E)=1$ as above,
then insert Equations~\eqref{eq:reflection-infinity-coefficient} and
\eqref{eq:reflection-cubic-sextic-conversion}.  Using
$q_{\lambda^{-3}}=27^{-1}$ and $\bar\alpha(\lambda^{-3})=-i$ gives
\begin{equation}\label{eq:reflection-direct-mellin}
 J(s)=\frac{3^{5/2}}4\left(\frac{27}{(2\pi)^2}\right)^s
       \Gamma(s+1/3)\Gamma(s+2/3)T(s,\Psi).
\end{equation}

For one translated term $z_h=a/c$, use the matrix $g=GH$ constructed
above.  The coordinate action of $g^{-1}$ on hyperbolic space, from
\cite[Equation (5.1)]{DR}, gives
\[
 g^{-1}(z+a/c,v)=
 \left(-\frac{\delta'}c-\frac{\bar z}{c^2(v^2+|z|^2)},
               \frac{v}{q_c(v^2+|z|^2)}\right).
\]
At $z=0$, the derivative of its first coordinate with respect to
$\bar z$ is $-(cv)^{-2}$; the derivatives of its conjugate coordinate
and of its vertical coordinate with respect to $\bar z$ are zero.
The chain rule and $\bar\theta(gw)=\bar\kappa(G)\bar\theta(Hw)$
therefore give
\[
 \left.\partial_{\bar z}\bar\theta(z+a/c,v)\right|_{z=0}
 =-\frac{\bar\kappa(G)}{c^2v^2}
   \left.\partial_z\bar\theta(H(z,1/(q_cv)))\right|_{z=-\delta'/c}.
\]
In particular the derivative removes $C_H(v)$ from every cusp expansion
in Equation~\eqref{eq:reflection-cusp-expansion}.
Substituting that expansion and then $v\mapsto1/(q_cv)$ in the integral
gives, for $\Re s<0$, the contribution
\begin{equation}\label{eq:reflection-reflected-mellin}
 \begin{split}
 -\frac i4(2\pi)^{2s-2}\bar\alpha(c)^2q_c^{1-2s}\bar\kappa(G)
 \prod_{\pm}\Gamma(3/2-s\pm1/6)\\
 {}
 \times\sum_{\mu\ne0}d_H(\mu)\alpha(\mu)
       \check e(-\delta'\mu/c)q_\mu^{-(1-s)}.
 \end{split}
\end{equation}
To verify the scalar directly, before applying the Bessel integral the
factor is $-2\pi i c^{-2}q_c^{2-2s}$ and the remaining integral is
$\int_0^\infty u^{2-2s}K_{1/3}(4\pi|\mu|u)\,du$.
The Bessel identity with $w=3-2s$, together with
$c^{-2}=\bar\alpha(c)^2/q_c$, gives exactly the display.

We justify both the continuation and the contour operations here.
The coefficient bound in Equation~\eqref{old-eq:4.3} implies absolute
convergence of the reflected series when $\Re(1-s)>1$: the separate
majorants are
\[
 \sum_n^{\rm sf}q_n^{-(1-\Re s)},\qquad
 \sum_b q_b^{1/2-3(1-\Re s)},\qquad
 \sum_{k\ge-4}3^{k(1/6-(1-\Re s))}.
\]
Each converges in that region.  The fixed support has a positive lower
bound for $|\mu|$.  The exponential decay of $K_{1/3}$ and of all its
derivatives, the coefficient bound, and the direct expansion imply
exponential decay at $v\to\infty$ for every logarithmic derivative of
each translated derivative.  The chain-rule formula just obtained
implies exponential decay at $v\to0$ as well, with constants depending
on the fixed translated term.  Repeated integration by parts in
$\log v$ now shows that $J$ is entire and decreases faster than every
power of $|\Im s|$ on each fixed strip.

Equation~\eqref{eq:reflection-direct-mellin}, with reciprocal gamma
functions on its right, continues $T$ to an entire function.  Its
direct series is bounded on any line to the right of $1$.
Equation~\eqref{eq:reflection-reflected-mellin} and Stirling's formula
give a polynomial bound on any line to the left of $0$; the exponential
parts of the two gamma products cancel.  On a strip between such lines,
the rapid bound for $J$ and the reciprocal gamma factors first give
$|T(\sigma+it)|\ll (1+|t|)^C e^{\pi|t|}$ for some $C$.
Divide $T$ by a sufficiently large power of $B+s$, with $B$ chosen so
that $B+s$ has no zero on the strip, and multiply by $e^{\varepsilon s^2}$.
The horizontal sides of a growing rectangle tend to zero for each
$\varepsilon>0$.  The maximum principle, followed by
$\varepsilon\to0$, transfers the polynomial bounds on the two vertical
sides to the strip.  This proves the stated polynomial strip growth.

Shift the integral on the left of Equation~\eqref{eq:reflection} from
$\Re s=a$ to $\Re s=1/2-\sigma<0$ with $\sigma>1/2$.
The entire continuation, polynomial strip bound, and rapid Mellin decay
justify the shift: Equation~\eqref{eq:pointwise-mellin-tail} applied to the
weighted logarithmic derivatives of $V$ makes the horizontal joins tend
to zero.  Inserting
Equation~\eqref{eq:reflection-reflected-mellin} divided by
Equation~\eqref{eq:reflection-direct-mellin}, and putting $t=1/2-s$,
gives the gamma quotient $R(t)$ and the factor
\[
 -\frac i{81}\bar\alpha(c)^2q_\mu^{-1/2}
       \left(\frac{Kq_\mu X}{q_c^2}\right)^{-t}.
\]
Indeed $(2\pi)^{4s-2}27^{-s}3^{-5/2}
=3^{-4}K^{-t}$ and $q_c^{1-2s}q_\mu^{s-1}
=q_\mu^{-1/2}(q_\mu/q_c^2)^{-t}$.
The reflected coefficient series is absolutely convergent on this
line, so it may be interchanged with the integral.  Summing its finite
local factors using Equation~\eqref{eq:reflection-additive-phase}
and the computed Gauss sums proves Equation~\eqref{eq:reflection} and
Equation~\eqref{eq:reflection-row-phase}.  Since $|\phi|\le1$, we have
$|\widehat\phi(h_0)|\le1$; every other factor in the scalar has
absolute value at most one.  This proves $|\zeta|\le1/81$.

\emph{Uniform estimates for the transformed test.}
The first negative pole of the numerator of $R(t)$ is $-5/6$.
The reciprocal denominator gamma functions are entire.  Hence the
integrand defining $V^\sharp$ is holomorphic for $\Re t>-5/6$.
On $-1/4\le\sigma\le A$, Stirling's formula gives
\[
 |R(\sigma+iu)|\ll_A(1+|u|)^{4\sigma}.
\]
The estimate is uniform also for bounded $u$, since this closed strip
has no numerator pole.  Equation~\eqref{eq:pointwise-mellin-tail} makes
$\widehat V$ rapidly decreasing pointwise on every vertical strip.  Rectangular
contour shifts are therefore valid within $-1/4\le\Re t\le A$,
including between any two nonnegative lines.  Applying
$(x\partial_x)^j$ contributes $(-t)^j$.  The three lines
$-1/4,0,A$ bound the resulting integral respectively by constant
multiples of
\[
 x^{1/4}\mathfrak M_{A,\lceil4A\rceil+j+2}(V),\quad
 \mathfrak M_{A,\lceil4A\rceil+j+2}(V),\quad
 x^{-A}\mathfrak M_{A,\lceil4A\rceil+j+2}(V).
\]
Combining them proves Equation~\eqref{eq:reflection-kernel}.
The rapid large-$x$ bound, with $A$ arbitrarily large, also makes every
dual sum in Equation~\eqref{eq:reflection} absolutely convergent when
$V^\sharp$ is represented on the zero line.  Thus the shift of the
kernel contour does not require a termwise shift of a conditionally
convergent Dirichlet series.

For Equation~\eqref{eq:reflection-annular-fourier}, set
$f_Y(v)=\chi(e^v)V^\sharp(Ye^v)$.  This is supported on a fixed compact
interval.  Integrating its Fourier transform by parts $J+2$ times
and using the trivial bound for $|u|\le1$ gives
\[
 \int_{\mathbb R}(1+|u|)^J|\widehat f_Y(u)|\,du
 \ll_J\sum_{j=0}^{J+2}\|f_Y^{(j)}\|_{L^1(\mathbb R)}.
\]
Leibniz's rule and Equation~\eqref{eq:reflection-kernel} bound each
norm by the right side of Equation~\eqref{eq:reflection-annular-fourier},
because $e^v$ stays in a fixed compact subinterval of $(0,\infty)$.
Finally, for $|u|\ge1$, $B+2$ integrations by parts give
\[
 |\widehat V(\eta+iu)|
 \le |\eta+iu|^{-B-2}
       \int_0^\infty|(x\partial_x)^{B+2}V(x)|x^\eta\frac{dx}{x}.
\]
For $|u|\le1$ use the corresponding undifferentiated integral.
On $-A\le\eta\le1/4$,
$x^\eta\le1+x^{-A}+x^{1/4}$, proving
Equation~\eqref{eq:reflection-test-seminorm}.  The identity
$\widehat{x^{iu_0}V}(\eta+iu)=\widehat V(\eta+i(u+u_0))$
and $1+|u|\le(1+|u+u_0|)(1+|u_0|)$ give the asserted norm-twist
bound.  In particular the Gaussian test
$V(x)=(2\sqrt\pi)^{-1}\exp(- (\log x)^2/4)$, whose Mellin transform is
$e^{t^2}$, satisfies the hypotheses of this Proposition directly.
\end{proof}

\subsection{Element rows and fixed sectors}

To apply the reflection to a character $A\mapsto\chi_A(m)$, we separate
the fixed supplementary phases from the good prime divisors of the row.
The separation retains the zero at every shared good prime.

\begin{lemma}[Fixed ray sectors for nonzero rows]
\label{lem:reflection-row-sectors}
Let $\Psi_{\rm base}$ range over a fixed finite family of finite-ray
characters with conductors supported on $S$, extended by zero off the primary
elements prime to $S$.  Use the fixed generators $\pi_{\mathfrak l}$ for $\mathfrak l\in S$
from Lemma~\ref{lem:fixed-numerator-ray}.
For $0\ne m\in\mathcal O$, write uniquely
\[
 m=u m_Sm_{\rm good},\qquad
 m_S=\prod_{\mathfrak l\in S}\pi_{\mathfrak l}^{v_{\mathfrak l}(m)},\qquad
 m_{\rm good}=\prod_{p\notin S}p^{v_p(m)},
\]
where the good prime generators are primary and $u$ is a unit.  On every
primary element $A$ prime to $S$ one has
\begin{equation}\label{eq:reflection-row-reduction}
 \chi_A(m)=\chi_A(u m_S)\mathcal R(A,m_{\rm good})
       \prod_{p\mid m_{\rm good}}\chi_p(A)^{v_p(m)}.
\end{equation}
All powers on the right retain their zero values, including when
$v_p(m)\equiv0\pmod6$.  Fix $u$, the valuations $v_{\mathfrak l}(m)$ modulo six,
and the class of $m_{\rm good}$ in the fixed ray group through which
$\mathcal R$ factors.  In such a sector define
\begin{equation}\label{eq:reflection-sector-character}
 \Psi_0^{[m]}(A)=
 \begin{cases}
 \Psi_{\rm base}(A)\chi_A(u m_S)\mathcal R(A,m_{\rm good}),
       &A\equiv1\pmod3,\ (A,S)=1,\\
 0,&\text{otherwise}.
 \end{cases}
\end{equation}
The nonzero branch is a member of a fixed finite family of finite-ray
characters with conductors supported on $S$.  Therefore a single
$\mathfrak E,L$ in Proposition~\ref{prop:completed-reflection} works for all
these sectors.  The local prime set contains every $p\mid m_{\rm good}$,
with exponent $v_p(m)$ modulo six, even when this exponent is zero.
No moving good prime is absorbed into $L$.
\end{lemma}

\begin{proof}
If $(A,m_{\rm good})=1$, multiplicativity and the symmetric reciprocity
factor in Lemma~\ref{lem:fixed-gauss-phase} give
Equation~\eqref{eq:reflection-row-reduction}.  If a good prime is shared,
both sides are zero: the left side is the zero extension of $\chi_A(m)$,
and its local factor on the right is zero even for a six-divisible exponent.
This proves the identity on the entire stated domain.

On this domain $\chi_A(u m_S)$ depends only on $u$ and the $S$-valuations
modulo six, because every numerator factor is a unit modulo $A$.  For each
of the finitely many representatives
$d=u\prod_{\mathfrak l\in S}\pi_{\mathfrak l}^{e_{\mathfrak l}}$,
$0\le e_{\mathfrak l}<6$, Lemma~\ref{lem:fixed-numerator-ray} identifies
$A\mapsto\chi_A(d)$ with a finite-ray character whose conductor is supported
at primes over $6d$, all in $S$.  Once the good ray class is fixed,
$A\mapsto\mathcal R(A,m_{\rm good})$ is one of a fixed finite family of
characters supported at the primes over $2,3$.  Their products with the
finite family of base characters give the asserted fixed family.  Choosing
a common multiple of its conductors and the $S$-mask gives the common
$\mathfrak E$ and then $L$.  Additional good local residue-symbol factors in
a completed coefficient are placed in $\mathcal P$ and combined at a shared
prime with the same zero convention; they do not alter this fixed modulus.
The argument requires $m\ne0$ and makes no local-prime assertion for the row
$m=0$.
\end{proof}

\subsection{Common profiles and lattice tails}

The reflected series has a scale depending on its row.  We keep that
dependence inside one joint smooth profile until after Fourier inversion;
this gives one coefficient measure for the whole row norm.

\begin{lemma}[Common annular kernel profile]
\label{lem:reflection-common-profile}
Let $V$ satisfy the hypotheses of Proposition~\ref{prop:completed-reflection}.
Fix an integer $d\ge1$, a compact set $\Omega\subset\mathbb R^d$, and real
exponents $a_1,\ldots,a_d$.  Let $W$ be one smooth profile with logarithmic support
in $\Omega$, common to all rows in a given block, and let $Y>0$ be fixed
within that block.  Put
\[
 h(\boldsymbol y)=\prod_{i=1}^d y_i^{a_i},\qquad
 F_Y(\boldsymbol y)=W(\boldsymbol y)V^\sharp(Yh(\boldsymbol y)),\qquad
 \mathfrak m_A(Y)=\min\{Y^{1/4},(1+Y)^{-A}\}.
\]
For $A\ge0$ and integers $j,J\ge0$,
\begin{align}
 p_j(F_Y)&\ll_{A,j,d,\Omega,\boldsymbol a}
 \mathfrak m_A(Y)p_j(W)
 \mathfrak M_{A,\lceil4A\rceil+j+2}(V),
 \label{eq:reflection-common-profile}\\
 \|F_Y\|_{J,\mathrm{sep}}&\ll_{A,J,d,\Omega,\boldsymbol a}
 \mathfrak m_A(Y)p_{J+d+2}(W)
 \mathfrak M_{A,\lceil4A\rceil+J+d+4}(V).
 \label{eq:reflection-common-measure}
\end{align}
Here $p_j$ is the homogeneous seminorm of a single profile, not the
inhomogeneous seminorm of a tuple.  The constants are independent of $Y$
and of the actual norm labels within the block.
\end{lemma}

\begin{proof}
On the support of $W$, the monomial $h$ is bounded above and below by
positive constants depending only on $\Omega$ and the exponents.  Each
operator $y_i\partial_{y_i}$ acting on $V^\sharp(Yh(\boldsymbol y))$ is
$a_i$ times the Euler derivative of $V^\sharp$ at $Yh(\boldsymbol y)$.
It produces no additional power of $Y$.  Leibniz's rule and
Equation~\eqref{eq:reflection-kernel} prove
Equation~\eqref{eq:reflection-common-profile}.  Applying
Lemma~\ref{lem:smooth-calculus} with $j=J+d+2$ gives
Equation~\eqref{eq:reflection-common-measure}.
\end{proof}

Here is the norm consequence, including its order of operations.  Let
$\mathcal I$ be the original row set, and suppose a row vector $\mathcal U$
is linear in the whole profile $F_Y$.  The profile includes the cutoffs for
the full fixed logarithmic boxes of its row and column norm variables.
Assume that simultaneous logarithmic Fourier inversion expresses the vector
using one density common to every row.  After the relevant arithmetic
coefficient independences have been verified, write
$F_Y=\mathfrak m_A(Y)\widetilde F_Y$.  Minkowski gives
\begin{equation}\label{eq:reflection-common-minkowski}
 \|\mathcal U\|_{\ell^2(\mathcal I)}
 \le \frac{\mathfrak m_A(Y)}{(2\pi)^d}
       \int_{\mathbb R^d}|\widehat{\widetilde F_Y}(\boldsymbol t)|
       \|B(\boldsymbol t)\|_{\ell^2(\mathcal I)}\,d\boldsymbol t.
\end{equation}
Here $B(\boldsymbol t)$ is the separated row vector, whose norm powers all
have absolute value one.  The identity underlying this inequality is a
Bochner integral in $\ell^2(\mathcal I)$; the finite separation norm above
ensures its absolute integrability whenever the separated norm has the stated
polynomial height growth.  Uniform bounds for densities chosen separately
for individual rows would not give this identity.
Only inside the nonnegative norm on the right may the row set subsequently
be enlarged, using the same separated formula for $B$ on the added rows.
No comparison of the kernel argument with $Y$ is asserted
on the added rows.  Squaring this inequality gives the factor
$\mathfrak m_A(Y)^2$.  If only the small-argument bound is being used, the
same argument permits the weaker scalar $\min\{1,Y^{1/4}\}$, whose square is
$\min\{1,Y^{1/2}\}$.  The normalized profile has bounded tuple seminorms;
they are not claimed to be small, since the tuple seminorm includes a constant
one.  A kernel occurring once in an already expanded quadratic expression
instead contributes its scalar once, not automatically its square.

The Gaussian test in Proposition~\ref{prop:completed-reflection} is directly
admissible there and in Lemma~\ref{lem:reflection-common-profile}: only the
joint factor $W$ in the latter lemma needs compact logarithmic support.  If
the compact-support calculus is instead applied to the Gaussian itself,
Lemma~\ref{lem:gaussian-annular} supplies the required absolutely summable
annular decomposition.

\begin{lemma}[Lattice kernel tails]\label{lem:lattice-kernel-tail}
Let $\Lambda$ be either $\mathcal O$ or $\lambda^{-4}\mathcal O$.  Suppose
$K:(0,\infty)\to\mathbb C$ satisfies $|K(x)|\le C_Ax^{-A}$ for $x\ge1$,
where $A>1$.  Then, for $a>0$ and $U\ge1$,
\begin{equation}\label{eq:lattice-kernel-tail}
 \sum_{\substack{0\ne\mu\in\Lambda\\a q_\mu>U}}|K(aq_\mu)|
 \ll_A C_A(1+a^{-1})U^{1-A}.
\end{equation}
For $K=V^\sharp$ one may take
$C_A\ll_A\mathfrak M_{A,\lceil4A\rceil+2}(V)$.
\end{lemma}

\begin{proof}
The shell $2^jU<a q_\mu\le2^{j+1}U$ contains
$O(1+2^jU/a)$ points of either fixed lattice.  Its contribution is at most
$C_A(1+2^jU/a)(2^jU)^{-A}$.  Summing the two geometric series gives
$O_A(C_A(U^{-A}+a^{-1}U^{1-A}))$, which implies the display because $U\ge1$.
The last assertion follows from Equation~\eqref{eq:reflection-kernel}.
\end{proof}

\subsection{A quadratic norm for completed indices}

The next estimate bounds the quadratic character left by reflection on a
squarefree factor times a cube.  We begin with the imported sieve.  For
squarefree primary ideals outside a fixed set containing the
primes over \(6\), the quadratic kernel \(\chi_a(b)^3\) satisfies
\begin{equation}\label{eq:quadratic-large-sieve}
 \sum_{\substack{a\ {\rm sf}\\q_a\le U}}
 \left|\sum_{\substack{b\ {\rm sf}\\q_b\le V}}
             c_b\chi_a(b)^3\right|^2
 \ll_\epsilon (UV)^\epsilon(U+V)\sum_b|c_b|^2.
\end{equation}
This is Goldmakher--Louvel's quadratic large sieve
\cite[Definition 1 and Theorem 1.1]{GL}.  To verify its family hypotheses,
let \(\vartheta_a((x))\) denote the quadratic residue symbol evaluated at
the primary generator of \((x)\).  The adjustment of \(x\) to its primary
generator contributes
\(\varepsilon_\lambda(x)^{e(a)}\), where \(\varepsilon_\lambda\) is the
nontrivial character modulo \(\lambda\) and
\(e(a)=(q_a-1)/2\bmod2\).  The powers of \(\omega\) contribute nothing
because they are squares.  CRT therefore gives exact primitive conductor
\(a\lambda^{e(a)}\) and trivial infinite type.  We use the primitive
inducing character: the displayed
formula first specifies its restriction to the \(\lambda\)-units, and the
factor at \(\lambda\) is absent when \(e(a)=0\).  This does not change
any value on the indices in Equation~\eqref{eq:quadratic-large-sieve}.
In a fixed primary square
class modulo \(4\), two
indices have the same \(e(a)\); for coprime such indices the primitive
product has conductor exactly their product.  The reciprocity factor is
the fixed bicharacter \(\mathcal R\) of
Equation~\eqref{eq:reciprocity-four-class}.  These are precisely the
family conditions of the cited theorem.  A finite ray partition and
transpose duality give Equation~\eqref{eq:quadratic-large-sieve} in the
displayed orientation.

A fixed restriction on the row set decreases the positive outer sum.
A fixed restriction on the coefficient support is implemented by setting
the omitted coefficients to zero.  Neither observation permits an arbitrary
mask depending simultaneously on a row and a column.  The next reduction
resolves the collision mask produced by square factors of a completed index.

\begin{lemma}[Quadratic reduction for completed indices]
\label{lem:completed-quadratic-reduction}
Let $K,N,B\ge1$.  Let $\mathcal I$ be any set of squarefree primary good
ideals $k$ with $q_k\ll K$.  Let $n$ range over squarefree primary ideals
with $q_n\asymp N$, and let $b$ range over primary ideals with
$q_b\asymp B$.  The ideals $n,b$ may contain primes of $S$ other than
$\lambda$.  Let $\beta(n,b)$ be arbitrary complex coefficients independent
of $k$.  Fixed restrictions on $\mathcal I$ and fixed restrictions on the
$(n,b)$-support are allowed.

Write uniquely $b=gt^2$ with $g$ squarefree, and set
$c=(n,g)$, $n=cm$, $g=ch$.  Thus $c,m,h$ are pairwise coprime and
squarefree.  Fix one set of dyadic ranges
\[
 q_c\asymp C,\qquad q_g\asymp G,\qquad q_t\asymp T,
 \qquad B\asymp GT^2,
\]
where all comparison constants are fixed.  Let $\mathcal Q_{C,G,T}(\beta)$
be the sum over $k\in\mathcal I$ of the squared absolute value of the
$(n,b)$-sum restricted to this set of ranges, with summand
$\beta(n,b)\chi_k(nb)^3$.  For a squarefree good ideal $r$ and a fixed $t$,
let $\mathcal D_r(t)$ be the set of triples $(c',m,h)$ for which
\[
 c=rc',\qquad n=rc'm,\qquad b=rc'ht^2
\]
belongs to the original support and to these ranges, with the stated
squarefreeness and pairwise coprimality of $c,m,h$.  Then
\begin{equation}\label{eq:completed-quadratic-reduction}
 \begin{split}
 \mathcal Q_{C,G,T}(\beta)
 \ll_\epsilon{}&(KNB)^\epsilon T
 \sum_{q_t\asymp T}\ 
 \sum_{\substack{r\ {\rm sf},\ (r,S)=1\\q_r\ll\min(K,C)}}
 \left(\frac K{q_r}+\frac{NG}{C^2}\right)\frac C{q_r}
 \sum_{(c',m,h)\in\mathcal D_r(t)}
       |\beta(rc'm,rc'ht^2)|^2.
 \end{split}
\end{equation}
Empty quotient ranges contribute zero; all bounded ranges, including the
unit ideal, are included by the fixed comparison constants.  Fixed ray
sectors and bounded row scalars are permitted.  In particular, if
$|\beta(n,b)|\le1$, then
\begin{equation}\label{eq:completed-quadratic-norm}
 \sum_{k\in\mathcal I}
 \left|\sum_{n,b}\beta(n,b)\chi_k(nb)^3\right|^2
 \ll_\epsilon (KNB)^\epsilon(K+NB)NB.
\end{equation}
The constants may depend on the fixed support comparisons and on $S$, but
are uniform in the coefficients and in the permitted support restrictions.
\end{lemma}

\begin{proof}
The factorization $b=gt^2$ permits $g$ and $t$ to share primes.  Since
$nb=c^2mht^2$, the zero convention gives the exact identity
\begin{equation}\label{eq:completed-quadratic-zero-mask}
 \chi_k(nb)^3=\chi_k(mh)^3\,1_{(k,ct)=1}.
\end{equation}
Indeed a square has sixth power under the displayed quadratic character;
this is one on units and zero at a collision.  The identity therefore also
holds when $n$ or $g$ shares a prime with $t$.

For the chosen ranges there are $O(T)$ possible $t$.  Hilbert-space Cauchy
gives
\[
 \left\|\sum_{q_t\asymp T}F_t\right\|_2^2
 \ll T\sum_{q_t\asymp T}\|F_t\|_2^2.
\]
Fix $t$ in the positive sum.  The condition $(k,t)=1$ is now a fixed row
restriction.  For the remaining condition use the full identity
\[
 1_{(k,c)=1}=\sum_{r\mid(k,c)}\mu(r).
\]
For each $k$ the number of possible $r$ is at most $d_{\mathcal O}(k)$.
Rowwise Cauchy followed by summation over $k$ costs $(KNB)^\epsilon$
and gives a positive sum over fixed squarefree good $r$.  Write
$k=rk'$ and $c=rc'$.  Squarefreeness gives the fixed row restriction
$(k',r)=1$ and the coefficient restriction $(c',r)=1$; terms with
$(r,t)>1$ vanish, and otherwise $(k',t)=1$ is another fixed row restriction.
There is no remaining $(k',c')$ condition in an individual $r$-summand:
the full collision mask was already expanded.  A fixed ray sector for $k$
becomes a fixed sector for $k'$ once $r$ is fixed.

The factor $\chi_r(mh)^3$ is a bounded column factor.  Split the squarefree
parts of $m,h$ supported on $S$ from their good parts:
$m=m_Sm_{\rm g}$ and $h=h_Sh_{\rm g}$.  There are only finitely many
choices of $m_S,h_S$.  Their symbols are bounded row factors, while
$j=m_{\rm g}h_{\rm g}$ is squarefree and good, of norm $O(NG/C^2)$.
For fixed $r,t,m_S,h_S$, group the remaining columns by $j$.  If $C_j$
is the resulting coefficient, the quadratic large sieve gives
\[
 \sum_{k'}\left|\sum_j C_j\chi_{k'}(j)^3\right|^2
 \ll_\epsilon (KNB)^\epsilon
       \left(\frac K{q_r}+\frac{NG}{C^2}\right)\sum_j|C_j|^2.
\]
The outer sum may retain all its fixed restrictions.  On a nonempty range,
both quotient lengths in the last display are bounded below by a fixed
positive constant; replacing them by their maxima with one changes only
the comparison constant.

For each $j$, the number of factorizations into $m_{\rm g},h_{\rm g}$ is
divisor-bounded.  There are $O(C/q_r)$ possible $c'$ in its range.  Cauchy
over these two choices therefore gives
\[
 \sum_j|C_j|^2
 \ll_\epsilon (KNB)^\epsilon\frac C{q_r}
 \sum_{(c',m,h)\in\mathcal D_r(t)}
          |\beta(rc'm,rc'ht^2)|^2.
\]
The bounded column factor from $\chi_r$ has been discarded only in this
positive sum.  The fixed bad-part symbols were row factors of unit modulus
and were removed from the outer modulus before applying the sieve.
The ideals $m,h$ here are the original ideals,
reconstructed from their fixed $S$-parts; the $S$-part of $c'$ is untouched.
Combining the displays proves Equation~\eqref{eq:completed-quadratic-reduction}.

Suppose now $|\beta|\le1$.  Ideal counting gives
\[
 \#\mathcal D_r(t)\ll
       \frac C{q_r}\frac NC\frac GC.
\]
For $q_r\asymp r_0$, its product with the preceding factor $C/q_r$ is
$O(NG/r_0^2)$.  There are $O(r_0)$ possible $r$ in this range and $O(T)$
possible $t$.  Since $B\asymp GT^2$, their contribution to
Equation~\eqref{eq:completed-quadratic-reduction} is at most
\[
 (KNB)^\epsilon\frac{NB}{r_0}
       \left(\frac K{r_0}+\frac{NG}{C^2}\right)
 \ll (KNB)^\epsilon NB(K+NB).
\]
Here $r_0,C,T$ are bounded below on a nonempty range and $G\ll B$.
The number of dyadic choices for $C,G,T,r_0$ is logarithmic in the norm
ranges.  Hilbert-space triangle over the $C,G,T$ blocks, followed by a
rescaling of $\epsilon$, proves Equation~\eqref{eq:completed-quadratic-norm}.
\end{proof}

\subsection{The unmarked reflected energy}

For a fixed finite-ray character $\nu$, extended by zero away from primary
elements prime to $S$, and for $m\ne0$, define the central completed sum
\begin{equation}\label{eq:unmarked-central-completion}
 \mathcal C_V(X;m,\nu)=
 \sum_{\substack{c\ {\rm sf},\ n\\c,n\equiv1\ (3)\\(cn,S)=1}}
 \frac{\gamma_2(c)\overline{\alpha(cn^3)}\nu(cn^3)\chi_{cn^3}(m)}
      {\sqrt{q_c}\,q_n}
 V\!\left(\frac{q_cq_n^3}{X}\right).
\end{equation}
Here $c,n$ may share primes.  The defining series is absolutely convergent
for every $X>0$ when $V$ satisfies the hypotheses of
Proposition~\ref{prop:completed-reflection}.  Complete multiplicativity,
with the zeros retained, gives
$\chi_{cn^3}(m)=\chi_c(m)\chi_n(m)^3$.  Thus for every $a>1$,
\begin{equation}\label{eq:unmarked-central-mellin}
 \mathcal C_V(X;m,\nu)=\frac1{2\pi i}\int_{(a)}
 \widehat V(s-1/2)X^{s-1/2}T(s,\nu\chi_\bullet(m))\,ds.
\end{equation}
Indeed, absolute convergence permits termwise Mellin inversion on that line.


We bound the mean square of this completed sum over nonzero element rows.
At equal completed and row norm scales, the goal is
\[
 \sum_{0<q_m\le C_mZ}|\mathcal C_V(Z;m,\nu)|^2
 \ll Z^{1+\epsilon}\mathfrak M_{A,J}(V)^2
\]
for every fixed $C_m\ge1$ and $\epsilon>0$, with suitable finite orders
$A,J$.  Lemma~\ref{lem:unmarked-completed-moment} below gives the more
general estimate that tracks the repeated prime factors of the row.

The reason for the quadratic sieve is already visible for squarefree rows.
Take $m=R$ primary, squarefree and good, with $q_R\asymp Z$, and set $X=Z$.
Fix a row sector, a cusp sector, a dual unit $u$, and one integer $k\ge-4$
in the dual support.  There are no frozen good local primes in this case.
Every prime of $R$ enters reflection with exponent one, so for
$\mu=u\lambda^k n b^3$ its column factor gives
\[
 \chi_R(\lambda^4\mu)^3
 =\chi_R(u\lambda^{k+4})^3\chi_R(nb)^3.
\]
This identity retains all zeros, and $n,b$ may share primes.  The first
factor depends only on $R$; the cusp coefficient and additive character are
common throughout the sector.  Their coefficient bound leaves the column
normalization $q_n^{-1/2}q_b^{-1}$, up to a constant depending on the fixed
$k$.  Thus the only arithmetic interaction between the row and the two
column indices is the kernel of Lemma~\ref{lem:completed-quadratic-reduction}.

On dual dyads $q_n\asymp U$, $q_b\asymp B$, the reflected kernel has argument
comparable to $UB^3/Z$.  Its rapid decay permits discarding whole blocks
with $UB^3>Z^{1+\tau}$ for any fixed $\tau>0$; the tail argument below makes
this restriction uniform when the ramified exponent and the other parameters
vary.  On the retained blocks, Lemma~\ref{lem:reflection-common-profile}
separates the kernel with one common coefficient measure while retaining
the actual row annulus.  Write
$\mathcal U_{U,B,k}(R)$ for this dyadic piece of the reflected sum.  For a
retained block, the quadratic norm and the squared column normalization give
\[
 \sum_R|\mathcal U_{U,B,k}(R)|^2
 \ll_{V,k,\tau,\epsilon}Z^\epsilon
       \frac{(Z+UB)UB}{UB^2}
 =Z^\epsilon\frac{Z+UB}{B}
 \ll_{V,k,\tau,\epsilon}Z^{1+\tau+\epsilon}.
\]
Here $B\ge1$ and $UB^3\le Z^{1+\tau}$ give the last inequality.  This explains
the balanced energy scale.  The general argument must also sum the ramified
exponents, retain the kernel saving at unequal scales, and treat repeated
prime factors of the row.  We freeze the prime powers of the row whose exponents are at least two
and apply the same quadratic norm to its squarefree residual factor.  The next definition
records the resulting local terms; the completed-row lemma then sums them.

\begin{definition}[An unmarked reflected block]
\label{def:unmarked-reflected-block}
Fix $Z\ge2$, $X=Z^N$, one of the fixed characters $\Psi_0$ and its common
modulus $L$ from Proposition~\ref{prop:completed-reflection}, and put
$M_{\rm ref}=\lambda^{12}L^4$.  For the current norm over $R$, freeze a finite set $\mathcal F$ of
good primes with exponents $j_p\in\{0,\ldots,5\}$.  Let $R$ range over
squarefree primary good ideals coprime to every prime of $\mathcal F$, in
one fixed class modulo $M_{\rm ref}^2$ and with any further fixed row
restrictions.  These restrictions are independent of the dual variables.
On primary elements prime to $S$ set
\[
 \Psi_R(A)=\Psi_0(A)\prod_{p\in\mathcal F}\chi_p(A)^{j_p}
                         \prod_{p\mid R}\chi_p(A).
\]
Every local power retains its zero.  In the reflection for $\Psi_R$, fix
$h_0$, the active or inactive decision at every $j_p=0$ prime of
$\mathcal F$, and one of the two summands of every $j_p=4$ Ramanujan factor.
All primes of $R$ are active and have exponent one.  Let $F_{\rm act}$ be
the product of the active primes in $\mathcal F$.

For each chosen divisibility summand at $j_p=4$, use the exact partition
\[
 1_{p\mid n_0b_0^3}=1_{p\mid n_0}+1_{p\nmid n_0}1_{p\mid b_0}.
\]
Choose one term.  Let $N_F$ be the product of the primes assigned to the
squarefree ideal $n_0$, and let $B_F$ be the product assigned to $b_0$ but
not $n_0$.  Write $n_0=N_Fn$ and $b_0=B_Fb$.  In the first assignment only
the occurrence in $n_0$ is extracted, even if $b_0$ contains that prime;
in the second exactly one occurrence is extracted from $b_0$.  All remaining
source restrictions are retained on $n,b$.  Let $F_{\rm sm}$ be the product
of the small $j_p=4$ summands and the active $j_p=0$ primes, and put
\[
 A_0=\log_Zq_{F_{\rm act}},\qquad S_0=\log_Zq_{F_{\rm sm}},\qquad
 N_0=\log_Zq_{N_F},\qquad B_0=\log_Zq_{B_F}.
\]
Empty products have log-norm zero.

Fix a unit $u$, an integer $k\ge-4$, and dual ranges
\[
 \mu=u\lambda^kN_Fn(B_Fb)^3,\qquad
 q_n\asymp Z^v,\quad q_b\asymp Z^{\ell_b},\quad
 q_{\lambda^k}=Z^{e_\lambda},
\]
where $v,\ell_b\ge0$.  The squarefree $n$ and the primary $b$ retain the
full source support of Equation~\eqref{old-eq:4.3}, including permitted
primes of $S$ and shared primes.  Let $H\ge0$, and choose a common smooth
profile $W_1(y_R,y_n,y_b)$ on a fixed compact logarithmic box, where
\[
 y_R=q_R/Z^H,\qquad y_n=q_n/Z^v,\qquad y_b=q_b/Z^{\ell_b}.
\]
Choose fixed smooth annular cutoffs $\chi_{\rm row},\chi_n,\chi_b$ for these
three ranges, and put
\[
 W_0(\boldsymbol y)=\chi_{\rm row}(y_R)\chi_n(y_n)\chi_b(y_b)W_1(\boldsymbol y).
\]
The supports and invoked seminorms are fixed uniformly over the block.
Denote by $\mathcal U_{v,\ell_b,e_\lambda}(R)$ the specified
term of the right side of Equation~\eqref{eq:reflection}, restricted to this
dual representation and multiplied by $W_0(y_R,y_n,y_b)$.  The original row
set $\mathcal I_H$ consists of the allowed $R$ for which
$\chi_{\rm row}(q_R/Z^H)\ne0$.  This definition is made separately for
every fixed choice above.
\end{definition}

The frozen primes in this definition may vary between invocations; they are
not added to the fixed arithmetic modulus $L$.  The definition fixes them
only before taking the current norm over $R$.

\begin{lemma}[Unmarked reflected block]
\label{lem:unmarked-reflected-block}
For an unmarked reflected block, set
\[
 T_0=2H+2A_0-N-N_0-3B_0,\qquad
 \nu_0=\frac v2+\ell_b+\frac{e_\lambda}{3}+\frac{S_0+B_0}{2}.
\]
Let all these log-lengths range over fixed bounded sets.  Put
\[
 Y=\frac{q_{\lambda^kN_FB_F^3}X Z^vZ^{3\ell_b}}
          {q_{c_FF_{\rm act}}^2Z^{2H}},\qquad
 W(\boldsymbol y)=y_n^{-1/2}y_b^{-1}W_0(\boldsymbol y),\qquad
 F_Y(\boldsymbol y)=W(\boldsymbol y)V^\sharp(Yy_ny_b^3y_R^{-2}).
\]
There are functions $\rho(R)$ and $\beta(n,b)$, with absolute values at
most one, such that $\beta$ is independent of $R$ and
\begin{equation}\label{eq:unmarked-structural-block}
 \mathcal U_{v,\ell_b,e_\lambda}(R)
 =\frac13 Z^{-\nu_0}\rho(R)
       \sum_{n,b}\beta(n,b)\chi_R(nb)^3F_Y(y_R,y_n,y_b).
\end{equation}
The sum retains all the fixed dual restrictions from the definition.  The
only arithmetic factor here that depends simultaneously on $R$ and the
dual variables is the displayed zero-extended quadratic symbol.  Moreover,
\begin{equation}\label{eq:unmarked-actual-annuli}
 \frac{q_\mu X}{q_c^2}=Yy_ny_b^3y_R^{-2},\qquad
 c=c_FF_{\rm act}R.
\end{equation}
This is an exact identity on the original support, including when $N_F$
shares primes with $b$.

For every $\epsilon>0$ and every fixed $A\ge0$,
\begin{equation}\label{eq:unmarked-reflected-energy}
 \begin{split}
 \sum_{R\in\mathcal I_H}|\mathcal U_{v,\ell_b,e_\lambda}(R)|^2
 &\ll_{\epsilon,A} Z^{E_0+\epsilon}p_5(W)^2
       \mathfrak M_{A,\lceil4A\rceil+7}(V)^2,\\
 E_0={}&\max(H,v+\ell_b)-S_0-B_0-\ell_b-\frac{2e_\lambda}{3}
       -\frac12(T_0-v-3\ell_b-e_\lambda)_+.
 \end{split}
\end{equation}
The constants may depend on the fixed kernel order $A$, the fixed arithmetic
data, the bounded log-length ranges, and the fixed support box, but not on the
moving frozen local set $\mathcal F$.  Norm twists in $W_0$ have
the polynomial cost supplied by its finite seminorms.
\end{lemma}

\begin{proof}
For the fixed $h_0$ and the fixed class of $R$ modulo $M_{\rm ref}^2$, the
class of $F_{\rm act}R$ is fixed.  Proposition~\ref{prop:completed-reflection}
therefore gives one common $d$ and one common additive character
$\vartheta$ for the whole row set.  Write $x_F=u\lambda^{k+4}N_FB_F^3$,
so that $\lambda^4\mu=x_Fnb^3$.  The residual local factors are exactly
\[
 \prod_{p\mid R}B_p(\lambda^4\mu)
 =\chi_R(x_F)^3\chi_R(nb^3)^3
 =\chi_R(x_F)^3\chi_R(nb)^3.
\]
Both equalities include zeros; no unit symbol has been divided.  In
particular, any collision with $x_F$ is a restriction depending only on $R$.
Every remaining local factor is fixed and depends only on the dual index.
The scalar $\zeta_R$ of the reflection depends on $R$ but not on $\mu$.

Define on the specified dual support
\[
 \delta(n,b)=
 \frac{d(\mu)\alpha(\mu)\vartheta(\lambda^4\mu)}
      {27\,3^{k/6}|B_Fb|}.
\]
The coefficient bound gives $|\delta(n,b)|\le1$.  Its denominator is
nonzero, and the definition remains valid when the numerator is zero.
Using $q_\mu=3^kq_{N_Fn}q_{B_Fb}^3$ gives the exact factorization
\[
 \frac{d(\mu)\alpha(\mu)\vartheta(\lambda^4\mu)}{\sqrt{q_\mu}}
 =27\delta(n,b)q_{\lambda^k}^{-1/3}
                  q_{N_Fn}^{-1/2}q_{B_Fb}^{-1}.
\]
A small Ramanujan factor or an active zero-mask factor contributes
$q_p^{-1/2}$.  A divisibility factor contributes $q_p^{1/2}$; for a prime
assigned to $N_F$ it cancels $q_p^{-1/2}$ from the squarefree denominator,
and for a prime assigned to $B_F$ it leaves $q_p^{-1/2}$ after extraction.
When a prime assigned to $N_F$ also divides $b$, its entire occurrence in
$b$ remains in $q_b^{-1}$ and in $\delta(n,b)$.  These facts give exactly
the common power $Z^{-\nu_0}y_n^{-1/2}y_b^{-1}$.

Absorb $\delta$, the signs of small Ramanujan terms, the bounded fixed local
characters, and the fixed dual indicators into $\beta(n,b)$.  This function
is bounded by one and is independent of $R$.  Put
$\rho(R)=81\zeta_R\chi_R(x_F)^3$; it is bounded by one because
$|\zeta_R|\le1/81$.  The factor $27/81=1/3$ now proves
Equation~\eqref{eq:unmarked-structural-block}.  Multiplicativity of the norm
proves Equation~\eqref{eq:unmarked-actual-annuli}.  It does not use any
coprimality between $N_F$ and $b$.

By the definitions of the log-norms,
\[
 Y=q_{c_F}^{-2}Z^{v+3\ell_b+e_\lambda-T_0}.
\]
The finite set of possible $c_F$ depends only on $L$.  Consequently
$\mathfrak m_A(Y)\ll_L Z^{-(T_0-v-3\ell_b-e_\lambda)_+/4}$.
Apply Lemma~\ref{lem:reflection-common-profile} in dimension three and
Equation~\eqref{eq:reflection-common-minkowski} to the structural identity.
The single density is common to all $R$ because $W$ uses the row and both
dual norms as coordinates.  At each Fourier mode the norm powers have
absolute value one and can be absorbed into the bounded row scalar and the
common dual coefficient.  Only now enlarge the positive row norm from
$\mathcal I_H$ to squarefree good $R$ with $q_R\ll Z^H$, retaining any
fixed restrictions.  No kernel estimate is used on the added rows.

Equation~\eqref{eq:completed-quadratic-norm}, with
$K=Z^H$, $N_{\rm col}=Z^v$, and $B_{\rm col}=Z^{\ell_b}$, bounds the
separated squared norm by
\[
 Z^{\epsilon+\max(H,v+\ell_b)+v+\ell_b}.
\]
The squared common coefficient in the structural identity has exponent
$-v-2\ell_b-2e_\lambda/3-S_0-B_0$.  The square of the extracted profile
scalar contributes $-(T_0-v-3\ell_b-e_\lambda)_+/2$.  Combining these
exponents and the dimension-three separation norm proves
Equation~\eqref{eq:unmarked-reflected-energy}.
\end{proof}

For later summation, call a dual block retained when
\[
 v+3\ell_b+e_\lambda\le T_0+\tau_{\rm ref},
 \qquad \tau_{\rm ref}>0.
\]
On the fixed support box, Equation~\eqref{eq:unmarked-actual-annuli} places
the kernel argument in $[C_{\rm ker}^{-1},C_{\rm ker}]$ times
$Z^{v+3\ell_b+e_\lambda-T_0}$, for one fixed $C_{\rm ker}$.  If
$\log Z\ge2\log C_{\rm ker}/\tau_{\rm ref}$, every unretained whole
block has actual argument greater than $Z^{\tau_{\rm ref}/2}$ throughout
its support.  These blocks can be removed before Fourier inversion with
arbitrary power saving when the other log-lengths lie in bounded ranges.
Here are the details needed for that uniform assertion.

On the full source support,
\[
 \frac{|d(\mu)|}{\sqrt{q_\mu}}
 \le27q_{\lambda^k}^{-1/3}q_{n_0}^{-1/2}q_{b_0}^{-1}
 \le27q_\lambda^{4/3}.
\]
After its indicator is discarded, each local factor is bounded by
$q_p^{1/2}$, uniformly in $\mu$.  The product of these bounds has a fixed
polynomial size when the active log-norm is bounded.  For a fixed row put
$a=X/q_c^2$.  Its factor $1+a^{-1}$ also has a fixed polynomial bound on
the stated row and conductor ranges.  The finite local choices and bounded
row counts have a further fixed polynomial cost; call the sum of these
norm-scale exponents $B_{\rm tail}$.  The fixed profile seminorms and their
polynomial height costs remain multiplicative factors outside this exponent.
It is chosen before the kernel order and does not count discarded dual indices.
The remaining dual sum is the unrestricted lattice sum in
Lemma~\ref{lem:lattice-kernel-tail}, with $U=Z^{\tau_{\rm ref}/2}$.
For any $D>0$, choosing
\[
 A>1+\frac{2(B_{\rm tail}+D)}{\tau_{\rm ref}}
\]
makes the discarded contribution $O(Z^{-D})$ times a finite seminorm of
$V$ and the fixed profile bounds.  The assertion holds either for the
absolute total or, after increasing $B_{\rm tail}$ to include the bounded
row count, for the row norm.  The dyadic cutoffs have bounded overlap, so
this lattice estimate controls their whole sum.  In particular no bound on
the discarded dual lengths has been assumed.

We finish by summing the reflected blocks for an element row.  The following
norm observation records the actual support that governs this summation.
Suppose an element row has an ideal factorization
$(m)=m_{\rm pow}m_{\rm sup}R$, with integral factors, and the fixed support
constants give
\[
 q_m\le C_mZ^M,\qquad q_{m_{\rm pow}}\ge Z^O/C_O,
 \qquad q_R\ge Z^H/C_H.
\]
Since $q_{m_{\rm sup}}\ge1$, multiplicativity gives
\begin{equation}\label{eq:actual-residual-row-dyad}
 H\le M-O+\frac{\log C_{\rm res}}{\log Z},
 \qquad C_{\rm res}=C_HC_mC_O.
\end{equation}
This inequality does not require $m_{\rm sup}$ to have bounded norm, and it
need not be an equality.  We will use it with the three factors on disjoint
prime supports and with the unit residual ideal in the bounded $H=0$ dyad.

\begin{lemma}[Unmarked completed-row moment]
\label{lem:unmarked-completed-moment}
Let $\nu$ range over a fixed finite family of multiplicative finite-ray
characters with conductors supported on $S$, all extended by zero away from
the primary elements prime to $S$.  Fix bounded ranges for $M\ge0$ and $N\in\mathbb R$,
a constant $C_m\ge1$, and fixed dyadic support comparisons.  For a nonzero
element $m$, let
\[
 m_{\rm pow}=\prod_{v_{\mathfrak p}(m)\ge2}
                 \mathfrak p^{v_{\mathfrak p}(m)}
\]
be its maximal powerful ideal factor.  Use nonnegative centers $O$ for its
dyadic ranges, with center zero for the unit range.

For every $\epsilon>0$ there exist a finite $A\ge0$ and an integer $J\ge0$
such that, for every test $V$ satisfying
Proposition~\ref{prop:completed-reflection}, every actual $O$-dyad, and all
sufficiently large $Z$,
\begin{equation}\label{eq:unmarked-completed-moment}
 \sum_{\substack{0<q_m\le C_mZ^M\\q_{m_{\rm pow}}\asymp Z^O}}
       |\mathcal C_V(Z^N;m,\nu)|^2
 \ll Z^{O/2+\max\{M-O,\,2M-O-N\}+\epsilon}
       \mathfrak M_{A,J}(V)^2.
\end{equation}
The finite orders, constant, and lower threshold may depend on the fixed
arithmetic data, the parameter ranges, $C_m$, the dyadic support comparisons,
and $\epsilon$.  They are uniform in the element rows and in $\nu$ in the
fixed family.  The estimate directly includes the Gaussian test; no annular
support assumption is made on $V$.  Replacing $V(x)$ by $x^{it_0}V(x)$ has
at most a fixed polynomial cost in $1+|t_0|$.
\end{lemma}

\begin{proof}
Fix one character $\nu$.  Lemma~\ref{lem:reflection-row-sectors}, with
$\Psi_{\rm base}=\nu$, supplies a fixed finite family of literal characters
for the reflection of $\nu(A)\chi_A(m)$.  Choose their common $\mathfrak E,L$
before any good row prime varies.  Partition the rows into the finitely many
unit, $S$-valuation residue, and good-ray sectors from that lemma.  This keeps every
good local prime, including those whose row valuation is divisible by six.

For a row in the prescribed $O$-dyad, split its valuation-one primes into
$m_{\rm sup}$ supported on $S$ and the squarefree good product $R$.  Thus
$(m)=m_{\rm pow}m_{\rm sup}R$ on disjoint prime supports.  Freeze the unit,
the actual ideals $m_{\rm pow},m_{\rm sup}$, and then partition $R$ into
actual smooth annuli $q_R\asymp Z^H$, with $H\ge0$ and the unit in the
bounded zero dyad.  Keep this row cutoff in each reflected vector.  Equation
\eqref{eq:actual-residual-row-dyad} gives
\begin{equation}\label{eq:unmarked-row-width}
 H\le M-O+O_{\rm fixed}(1/\log Z).
\end{equation}
The fixed row restrictions, including coprimality with the frozen factors,
are independent of the dual variables.

After the row-sector conversion, take $\mathcal F$ in
Definition~\ref{def:unmarked-reflected-block} to be the good primes of
$m_{\rm pow}$, with their valuations reduced modulo six.  The primes of
$R$ have exponent one.  Fix $h_0$, a class of $R$ modulo $M_{\rm ref}^2$,
and the local active and Ramanujan choices at the frozen primes.  Every
active frozen good prime divides $m_{\rm pow}$ to exponent at least two.
Consequently
\begin{equation}\label{eq:unmarked-powerful-radical}
 q_{F_{\rm act}}^2\le q_{m_{\rm pow}},\qquad
 2A_0\le O+O_{\rm fixed}(1/\log Z).
\end{equation}
This uses the actual powerful dyad's upper comparison.  The possible
$S$-prime factors of $m_{\rm pow}$ only increase its norm.

Apply Equation~\eqref{eq:reflection} at $X=Z^N$ to the Mellin representation
in Equation~\eqref{eq:unmarked-central-mellin}.  Its dual sum is absolutely
convergent.  Insert fixed smooth dyadic partitions in the residual squarefree
and cube norms, and fix the unit and ramified exponent of the dual index.
These operations give the blocks of Definition~\ref{def:unmarked-reflected-block}
with uniformly bounded seminorms for $W$.  No partition of the primal test
$V$ is needed: the compact profile here is the product of the dual and row
cutoffs with the normalized inverse roots, and $V^\sharp$ remains the kernel.

Fix a small $\tau_{\rm ref}>0$.  The row lengths $H,O$, the active length
$A_0$, and $N$ lie in fixed bounded ranges.  The same is true of
$S_0,N_0,B_0$, since their prime products divide $F_{\rm act}$.  The tail
argument following Lemma~\ref{lem:unmarked-reflected-block} therefore removes
all unretained whole dual blocks with any prescribed power saving, with a
finite seminorm of $V$.  Its polynomial cost includes the row and local
counts below but not the discarded dual indices.  Choose its kernel order
first, and increase $A,J$ in the statement so that
$\mathfrak M_{A,J}(V)$ dominates this tail seminorm and the block seminorm.
This is possible because increasing $A$ widens the supremum defining
$\mathfrak M_{A,J}$ and increasing $J$ increases its weight.

Write $e=e_\lambda$.  The source restriction $k\ge-4$ gives
$e\ge-4\log 3/\log Z$.  For a retained block,
\[
 v+3\ell_b+e\le T_0+\tau_{\rm ref}.
\]
Since $v,\ell_b\ge0$ and $T_0$ is bounded, this also bounds every retained
dual length.  There are only logarithmically many retained choices for the
dual annuli and for $k$.  Choose a nonnegative
$\epsilon_Z=O_{\rm fixed}(1/\log Z)$ large enough to cover the support
offsets in Equations~\eqref{eq:unmarked-row-width} and
\eqref{eq:unmarked-powerful-radical}, and to ensure $e\ge-\epsilon_Z$.

The exponent in Equation~\eqref{eq:unmarked-reflected-energy} satisfies
\[
 E_0\le\max\{H,T_0-S_0-B_0\}+\frac53\epsilon_Z+\tau_{\rm ref}.
\]
To see this, use $\max(H,v+\ell_b)-\ell_b=\max(H-\ell_b,v)$ and discard
the nonpositive kernel term.  The first branch is at most
$H+2\epsilon_Z/3$, since $\ell_b,S_0,B_0\ge0$.  In the second branch,
retention gives $v\le T_0-3\ell_b-e+\tau_{\rm ref}$, so it is at most
$T_0-S_0-B_0+5\epsilon_Z/3+\tau_{\rm ref}$.  Moreover,
\[
 \begin{split}
 T_0-S_0-B_0
 &=2H+2A_0-N-N_0-S_0-4B_0\\
 &\le 2M-O-N+O(\epsilon_Z),
 \end{split}
\]
by Equations~\eqref{eq:unmarked-row-width} and
\eqref{eq:unmarked-powerful-radical}.  Thus every retained block, with its
frozen row parts fixed, has exponent at most
\[
 \max\{M-O,2M-O-N\}+\tau_{\rm ref}+O(\epsilon_Z).
\]

It remains to count the frozen parts and the finite expansions.  Every
powerful ideal is uniquely $x^2y^3$ with $y$ squarefree: at a prime, the
exponent of $y$ is the parity of the powerful exponent, and an odd positive
exponent is at least three.  Ideal counting and
$\sum_yq_y^{-3/2}<\infty$ therefore give
\[
 \#\{m_{\rm pow}:q_{m_{\rm pow}}\ll Z^O\}\ll Z^{O/2}.
\]
The squarefree $m_{\rm sup}$ is a divisor of the fixed radical of $S$ and
has only finitely many choices.  The local branches at the good primes of
$m_{\rm pow}$ have divisor-bounded multiplicity, as do their Ramanujan
assignments.  Rowwise divisor Cauchy and then summation of the local choices
cost an arbitrarily small power of $Z$.  The choices of $h_0$, fixed ray
sectors, and units are finite, and the actual row and retained dual annuli
cost only powers of $\log Z$.  This counts the powerful contribution $O/2$
exactly once.

Choose $\tau_{\rm ref}$ and the local small-power losses within the prescribed
$\epsilon$ allowance.  Then increase the fixed-data threshold for $Z$ so that
the $O(\epsilon_Z)$ terms lie within that allowance.  The block bound, the
preceding counts, and the already chosen tail saving prove
Equation~\eqref{eq:unmarked-completed-moment}.  The twist assertion follows
from the norm-twist bound for $\mathfrak M_{A,J}$ in
Proposition~\ref{prop:completed-reflection}.
\end{proof}

For clarity about the test used at equal lengths, put
\[
 V_{\rm G}(y)=\frac1{2\sqrt\pi}
           \exp\!\left(-\frac{(\log y)^2}{4}\right).
\]
It satisfies the hypotheses of Proposition~\ref{prop:completed-reflection}
directly, has $\widehat V_{\rm G}(t)=e^{t^2}$, and has finite
$\mathfrak M_{A,J}$ for every finite $A,J$.  Thus the preceding lemma applies
to it without an additional annular extension.  At $X=Z$ its exact Mellin form
is
\[
 \mathcal C_{V_{\rm G}}(Z;m,\nu)=\frac1{2\pi i}\int_{(4)}
       Z^t e^{t^2}T(t+1/2,\nu\chi_\bullet(m))\,dt.
\]
In the defining sum the coefficients remain normalized by
$q_c^{-1/2}q_n^{-1}$; the scale occurs in $V_{\rm G}(q_cq_n^3/Z)$.

Finally take $M=N=1$.  The exponent in an actual $O$-dyad is
$1-O/2+\epsilon$.  The nonnegative $O$-dyads are logarithmically many, and
their unit dyad is included.  Summing them, with a smaller preliminary loss,
gives the equal-length conclusion for every admissible $V$:
\begin{equation}\label{eq:unmarked-balanced-completed-moment}
 \sum_{0<q_m\le C_mZ}|\mathcal C_V(Z;m,\nu)|^2
 \ll Z^{1+\epsilon}\mathfrak M_{A,J}(V)^2.
\end{equation}

\section{The base probe and its balanced low estimate}\label{sec:probe}

We now place the completed sums of the preceding section inside one
average $I_\eta(X,Y,Z)$. Its original representation separates into an
additive polynomial and a completed theta row. At the balanced scales
$X=Y=Z^{1/2}$, their mean-square estimates give the direct bound
$I_\eta\ll_\eta Z^{1/4+\epsilon}$. We first define the average for
independent positive scales $X,Y,Z$; only the final proposition in this
section specializes them. The next section applies Poisson summation to
this same average and identifies the reciprocal of the target
$L$-function in its principal row.

\subsection{Definition and separation of the two factors}

Choose a finite ray group $T$, independently of the target character,
through which the functions $\mathcal R$, $G$, and the fixed primary
and supplementary phases supported over $2,3$ in
Lemma~\ref{lem:fixed-gauss-phase} factor.  The target-dependent
fixed-numerator characters below need not factor through $T$.  For a
finite-order Hecke character $\eta$, write
\[
 \Theta=\langle\eta,\widehat T\rangle.
\]
Use one excluded set $S$ for the finite family
$\{\eta\theta:\theta\in\widehat T\}$.  It contains the primes over
$6$, the prime supports of the defining moduli of the fixed
zero-extended presentations of $\eta$ and the characters in
$\widehat T$, and every prime of norm at most a fixed $P_0$.  These
full presentations belong to the fixed arithmetic data.  They are fixed
before $X,Y,Z$ and the varying row and prime parameters are chosen.
Enlarging $S$ will not change $T$.

The finite transform should exclude frequencies meeting $S$, including
zero, and should leave coefficients that factor prime by prime. The
following character and phase corrections arrange these two properties.

Let $b_*$ generate the squarefree product of the primes in $S$.
Choose a residue character $\xi$ modulo $b_*$ whose restriction to
each prime factor is nonprincipal and whose order divides six.  Such
a choice exists: at a prime of odd residue characteristic one may
use the quadratic character, and the residue field at the prime over
$2$ has order four and has a character of order three.  The Chinese
remainder theorem makes $\xi$ primitive modulo $b_*$.  All residue
characters below are extended by zero on nonunits.  Put
\[
 \begin{gathered}
 g_\psi(c,k)=\sum_{d\bmod c}\psi(d)e(kd/c),\qquad
 \tau=q_{b_*}^{-1/2}g_\xi(b_*,1),\qquad
 \Xi(a)=\xi(a)\chi_a(b_*),\\
 \Psi_{m,s,\eta}(a)=
       \eta(a)\Xi(a)^{-1}\mathcal R(a,s)\chi_a(m)
       \qquad ((a,S)=1).
 \end{gathered}
\]
Finite character orthogonality at each prime gives $|\tau|=1$.
By Lemma~\ref{lem:fixed-numerator-ray}, the fixed-numerator symbol
$a\mapsto\chi_a(b_*)$ is a finite ray character whose conductor is
supported on $S$.  Thus $\Xi$ is a fixed finite character for this
target, although it need not factor through $T$.

The primitive character $\xi$ at the primes in $S$ will force the
Poisson frequency to be prime to $S$. The accompanying normalizations
cancel the unit factors introduced by the auxiliary modulus $b_*$.
For the completed index $A=cn^3$, the factor $\mathcal R(A,s)$
cancels the cross-prime reciprocity phases between $A$ and $s$, while
$\overline{G(A)}$ cancels the remaining pair phases within $A$.
We verify these cancellations coefficientwise in
Section~\ref{sec:local-euler}.

Define the corrected row at spectral parameter $t+1/2$ by
\[
 T_{m,s,\eta}(t)=
 \sum_{\substack{c\ {\rm sf}\\n}}
 \gamma_2(c)\overline{\alpha(cn^3)}\Psi_{m,s,\eta}(cn^3)
 \overline{G(cn^3)}q_c^{-1/2-t}q_n^{-1-3t}.
\]
Here and below the ideal variables avoid $S$.  In particular every
factor in $\Xi(a)^{-1}$ is evaluated on a unit.  The value
$G(cn^3)$ is the finite-ray extension in
Equation~\eqref{eq:G-quadratic-refinement}, also when $cn^3$ is not
squarefree.

Fix nonnegative, nonzero functions $W_0,W_1\in C_c^\infty(0,\infty)$
and put $\Phi(t)=e^{t^2}$.  Define
\begin{equation}\label{eq:general-probe}
\begin{aligned}
 I_\eta(X,Y,Z)={}&\frac1Y\sum_s
 \frac{W_1(q_s/Y)\chi_s(b_*)}
      {\tau\xi(s)\sqrt{q_{b_*}q_sX}}\\
 &\quad\cdot\sum_{m\in\mathcal O}
       \xi(m)q_s^{-1/2}g_{\chi_s}(s,-m)
       W_0\!\left(\frac{q_m}{q_{b_*}q_sX}\right)
       \frac1{2\pi i}\int_{(4)}
             Z^t\Phi(t)T_{m,s,\eta}(t)\,dt.
\end{aligned}
\end{equation}
The masks in this definition remove nonunits at $S$.  All sums in
Equation~\eqref{eq:general-probe} are absolutely convergent on the
displayed line; the $m$ and $s$ sums are finite because the weights
are annular.

The fixed finite Fourier expansion of the correction is
\[
 \overline{G(A)}=\sum_{\theta\in\widehat T}a_\theta\theta(A),
 \qquad
 a_\theta=\frac1{|T|}\sum_{v\in T}
                 \overline{G(v)}\,\overline{\theta(v)}.
\]
Parseval and Cauchy--Schwarz give
$\sum_\theta|a_\theta|\le |T|^{1/2}$.  For a ray class
$\sigma\in T$ choose a representative $s_\sigma$ and put
\[
 \begin{aligned}
 \nu_\sigma(a)&=\eta(a)\Xi(a)^{-1}\mathcal R(a,s_\sigma)
       \quad ((a,S)=1),\\
 B_{m,\sigma}(Z)&=\sum_{\theta\in\widehat T}a_\theta
 \frac1{2\pi i}\int_{(4)}Z^t\Phi(t)
       T(t+1/2,\nu_\sigma\theta\chi_\bullet(m))\,dt.
 \end{aligned}
\]
Extend $\nu_\sigma$ by zero away from the primary elements prime to $S$, without
evaluating $\Xi(a)^{-1}$ there.  The function $T$ here is the completed
product in
Equation~\eqref{eq:reflection}, with its original zero masks.
Since $\mathcal R(A,s)=\mathcal R(A,s_\sigma)$ for $s\in\sigma$,
the displayed Mellin integral of the corrected row equals
$B_{m,\sigma}(Z)$ throughout that class.

For $m\ne0$, the central Mellin identity
\eqref{eq:unmarked-central-mellin} gives the direct specialization
\[
 B_{m,\sigma}(Z)=\sum_{\theta\in\widehat T}a_\theta
       \mathcal C_{V_{\rm G}}(Z;m,\nu_\sigma\theta).
\]
The characters $\nu_\sigma\theta$ form one fixed finite family whose
conductor and defining-modulus supports are contained in $S$. They need
not factor through $T$. Their original zero extensions are those in the
completed-row moment, so that estimate applies directly to this family
of row sums. The triangle inequality in the row Hilbert space costs only
the fixed factor $\sum_\theta|a_\theta|$. The row-sector conversion used
by that estimate retains the zeros at shared good primes, including when
a row valuation is divisible by six.

Write $Q=q_{b_*}XY$.  Choose a smooth compactly supported function
$\Omega$ on $(0,\infty)$ equal to one on every value of $q_m/Q$
that can occur on the support of
$W_1(q_s/Y)W_0(q_m/(q_{b_*}q_sX))$.  For $v\in\mathbb R$, set
\[
 A_{m,\sigma,v}(Y)=\frac1Y\sum_{s\in\sigma}
 \frac{W_1(q_s/Y)\chi_s(b_*)}{\tau\xi(s)}
 (q_s/Y)^{-1/2+iv}q_s^{-1/2}g_{\chi_s}(s,-m).
\]
Our Mellin convention for $W_0$ is
$\widehat W_0(iv)=\int_0^\infty W_0(r)r^{iv}\,dr/r$.
Since
$q_m/(q_{b_*}q_sX)=(q_m/Q)/(q_s/Y)$, Mellin inversion gives the
exact identity
\begin{equation}\label{eq:low-separated}
 I_\eta(X,Y,Z)=\frac{Q^{-1/2}}{2\pi}
 \sum_{\sigma\in T}\int_{\mathbb R}\widehat W_0(iv)
 \sum_{m\ne0}\Omega(q_m/Q)\xi(m)(q_m/Q)^{-iv}
                 A_{m,\sigma,v}(Y)B_{m,\sigma}(Z)\,dv.
\end{equation}
The omitted $m=0$ term vanishes by the annular support of $W_0$.
The factor $(q_m/Q)^{-iv}$ has absolute value one.  In $A$, the
arithmetic coefficient is fixed and bounded independently of $m,v$;
all dependence on $v$ occurs in a norm power of the annular variable.

The Gaussian $V_{\rm G}$ has Mellin transform $e^{t^2}$ and satisfies
the reflection hypotheses directly. Thus
Lemma~\ref{lem:unmarked-completed-moment} applies without an annular
decomposition of this test.

\subsection{The balanced low estimate}

For the balanced estimate, the completed-row moment already bounds the
mean square of $B_{m,\sigma}(Z)$. It remains to bound the additive factor
$A_{m,\sigma,v}(Y)$. Its Gauss expansion gives distinct reduced fractions
in $\mathbb C/\mathcal O$; their separation and coefficient mass give the
required mean square. The planar additive large sieve is classical;
compare Huxley's multivariable and number-field inequality \cite{Huxley}
and the Poisson proof in \cite[Section~3, Theorem~3]{BaierBansal}. We
include the lattice proof to record the normalization used here.

\begin{lemma}[Planar additive large sieve]
\label{lem:planar-additive-sieve}
Let $0<\delta\le1$, let $Q\ge1$, and let $z_1,\ldots,z_J$ be a finite set
of points in $\mathbb C/\mathcal O$ satisfying
\[
 \inf_{n\in\mathcal O}|z_j-z_k-n|\ge\delta
 \qquad(j\ne k).
\]
Then, for arbitrary complex numbers $a_1,\ldots,a_J$,
\[
 \sum_{\substack{m\in\mathcal O\\q_m\le Q}}
 \left|\sum_{j=1}^J a_j e(mz_j)\right|^2
 \ll (Q+\delta^{-2})\sum_{j=1}^J|a_j|^2.
\]
The implied constant is absolute for the lattice, additive character, and
self-dual measure fixed above.  A fixed multiple of $Q$ in the row ball is
allowed by changing this constant.
\end{lemma}

\begin{proof}
The assertion is immediate when $J=0$.  The characters $e(mz_j)$ are
well defined on $\mathbb C/\mathcal O$ by self-duality.  Choose
representatives in $\mathbb C$ for the other cases.  We use the Fourier transforms
\[
 \widehat F(y)=\int_{\mathbb C}F(x)e(-xy)\,d\mu(x),\qquad
 \check\phi(x)=\int_{\mathbb C}\phi(y)e(xy)\,d\mu(y).
\]
Choose a nonnegative $\phi\in C_c^\infty(\mathbb C)$ of integral one,
supported in a sufficiently small disk about zero.  Since
$|e(xy)-1|\le (4\pi/\sqrt3)|x||y|$, its support can be chosen so that
\[
 |\check\phi(x)-1|\le\frac12\qquad(|x|\le1).
\]
Thus $F(x)=4|\check\phi(x)|^2$ is a nonnegative Schwartz function and
$F(x)\ge1$ on the unit disk.  If
$\widetilde\phi(y)=\overline{\phi(-y)}$, Fourier inversion and the product
formula, with convolution taken with respect to $d\mu$, give
\[
 \widehat F=4\phi*\widetilde\phi.
\]
In particular, $\widehat F$ is bounded and supported in a disk of some fixed
radius $L$.  This is the bandlimited majorant we need.

Put $R=\sqrt Q$.  Positivity of $F$ first gives
\[
 \sum_{q_m\le Q}\left|\sum_j a_je(mz_j)\right|^2
 \le \sum_{m\in\mathcal O}F(m/R)
                    \left|\sum_j a_je(mz_j)\right|^2.
\]
All sums on the right converge absolutely.  For any $z\in\mathbb C$,
the Fourier transform of $x\mapsto F(x/R)e(xz)$ at $y$ is
$R^2\widehat F(R(y-z))$.  The factor $R^2$ is the Jacobian of the real
two-dimensional dilation; no lattice-volume factor occurs because $d\mu$
has covolume one on $\mathcal O$.  Poisson summation therefore expands the
right side as
\[
 R^2\sum_{j,k}a_j\overline{a_k}
       \sum_{n\in\mathcal O}
       \widehat F\bigl(R(n-z_j+z_k)\bigr).
\]

For each fixed $j$, a term in the inner sum can be nonzero only if the lift
$z_k+n$ lies within distance $L/R$ of $z_j$.  The set of all lifts
$\{z_k+n:1\le k\le J,\ n\in\mathcal O\}$ is $\delta$-separated in
$\mathbb C$.  Distinct classes have this property by hypothesis, and two
distinct lifts of one class differ by a nonzero Eisenstein integer, whose
absolute value is at least one and hence at least $\delta$.
The disks of radius $\delta/3$ about lifts in a disk of radius $L/R$ are
disjoint and lie in the concentric disk of radius $L/R+\delta/3$.
Comparing their Euclidean areas bounds the number of these lifts by
\[
 \left(1+\frac{3L}{R\delta}\right)^2
 \ll 1+(R\delta)^{-2}.
\]
The same bound holds with $j$ and $k$ interchanged.  Taking absolute values
in the Poisson expansion, using the fixed bound for $\widehat F$, and then
using $2|a_ja_k|\le |a_j|^2+|a_k|^2$, we obtain
\[
 \sum_{q_m\le Q}\left|\sum_j a_je(mz_j)\right|^2
 \ll R^2\bigl(1+(R\delta)^{-2}\bigr)\sum_j|a_j|^2.
\]
Since $R^2=Q$, this is the asserted estimate.  Replacing $Q$ by a fixed
multiple proves the final statement.
\end{proof}

We apply this estimate to the full residue classes in the Gauss sums.
The zero extension of the character is important here: it makes the
fractions reduced with respect to the displayed modulus, not merely with
respect to an inducing conductor.

\begin{lemma}[The balanced additive norm]
\label{lem:balanced-additive-norm}
Fix the arithmetic data of the probe, the annular weight $W_1$, a ray class
$\sigma\in T$, and a row-ball constant $C>0$.  For $Y,Q\ge1$ and every
real $v$, the polynomial $A_{m,\sigma,v}(Y)$ satisfies
\[
 \sum_{\substack{m\in\mathcal O\\q_m\le CQ}}
       |A_{m,\sigma,v}(Y)|^2
 \ll_{\mathcal A,W_1,C}\frac{Q+Y^2}{Y}.
\]
The bound is uniform in $v$.  In particular, when $Q=q_{b_*}Y^2$,
\[
 \sum_{q_m\le CQ}|A_{m,\sigma,v}(Y)|^2
 \ll_{\mathcal A,W_1,C}\frac QY.
\]
\end{lemma}

\begin{proof}
For a primary $s$ outside $S$, put $r_s=q_s/Y$ and
\[
 c_\sigma(s)=1_{s\in\sigma}\frac{\chi_s(b_*)}{\tau\xi(s)}.
\]
The inverse is evaluated only on these elements prime to $S$.  Here $|\tau|=1$,
$|\xi(s)|=1$, and $|\chi_s(b_*)|=1$, because $b_*$ is supported on $S$.
Thus $|c_\sigma(s)|\le1$.  The definition of $A$ and the Gauss expansion
give the exact identity
\[
 A_{m,\sigma,v}(Y)
 =Y^{-3/2}\sum_s c_\sigma(s)W_1(r_s)r_s^{-1+iv}
       \sum_{d\bmod s}\chi_s(d)e(-md/s).
\]
The sum is over primary ideals outside $S$.  For every such $s$, the
character $\chi_s(d)$ is zero exactly when $(d,s)>1$, and has absolute value
one otherwise.  This includes nonsquarefree $s$: a local exponent divisible
by six is the indicator of the units at that prime, not the constant
function one.  For $s=1$ the single residue class is included and
$\chi_1=1$.

We verify that the fractions $d/s$ for the contributing pairs are distinct
modulo $\mathcal O$.  Suppose that $(d,s)=(d',s')=1$ and
$d/s-d'/s'\in\mathcal O$.  Then $ds'-d's$ is divisible by $ss'$.
Reducing this divisibility modulo $s$ and using the inverse of $d$ modulo
$s$ gives $s\mid s'$.  The symmetric argument gives $s'\mid s$.
The primary generator of an ideal outside $S$ is unique, so $s=s'$;
the original congruence then gives $d=d'\pmod s$.  The same conclusion
includes the unit modulus.  Changing a residue representative only
translates its fraction by an element of $\mathcal O$.

Let $0<r_-<r_+$ be fixed with $\operatorname{supp}W_1\subset[r_-,r_+]$.
For two distinct contributing fractions and every $n\in\mathcal O$,
the Eisenstein integer $ds'-d's-nss'$ is nonzero.  Hence
\[
 \left|\frac d s-\frac{d'}{s'}-n\right|
 =\frac{|ds'-d's-nss'|}{|ss'|}
 \ge\frac1{|ss'|}\ge\frac1{r_+Y}.
\]
The last inequality uses $q_s,q_{s'}\le r_+Y$.  Thus these fractions are
$\delta_Y$-separated in $\mathbb C/\mathcal O$, where
$\delta_Y=\min(1,r_+^{-1})/Y\le1$.

Writing the expanded polynomial as
$\sum_{s,d}a_{s,d}e(-md/s)$, with only the unit residue classes retained,
its coefficient square mass is exactly
\[
 \begin{split}
 \sum_{s,d}|a_{s,d}|^2
 &=Y^{-3}\sum_s |c_\sigma(s)W_1(r_s)|^2r_s^{-2}
       \varphi_{\mathcal O}(s),\\
 \varphi_{\mathcal O}(s)
 &:=\#(\mathcal O/s\mathcal O)^\times,\qquad
 \varphi_{\mathcal O}(1):=1.
 \end{split}
\]
There is no dependence on $v$ in this expression.  Since
$\varphi_{\mathcal O}(s)\le q_s\le r_+Y$, $r_s\ge r_-$ on the support,
and there are $O(Y)$ ideals with $q_s\le r_+Y$, this mass is $O(Y^{-1})$.
Lemma~\ref{lem:planar-additive-sieve}, applied to the points $-d/s$ with
separation $\delta_Y$, now proves the first bound.  Finally $q_{b_*}\ge1$,
so $Q=q_{b_*}Y^2$ implies $Q+Y^2\le2Q$, proving the second.
\end{proof}

The additive norm now has precisely the scale needed to pair with the
unmarked completed-row norm.  The following proposition performs that
pairing for the original probe.

\begin{proposition}[Balanced low estimate]
\label{prop:balanced-low}
Fix the arithmetic data and smooth weights used to define $I_\eta$.
For every $\epsilon>0$, as $Z\to\infty$,
\[
 \bigl|I_\eta(Z^{1/2},Z^{1/2},Z)\bigr|
 \ll_{\mathcal A,\epsilon} Z^{1/4+\epsilon}.
\]
The exponent is independent of the target character; the implied constant
and lower threshold may depend on its fixed arithmetic data.
\end{proposition}

\begin{proof}
Set $X=Y=Z^{1/2}$ and $Q=q_{b_*}XY=q_{b_*}Z$.  The function $\Omega$ in
Equation~\eqref{eq:low-separated} is bounded and supported in a fixed
compact subinterval of $(0,\infty)$.  Thus all rows in that identity lie
in $0<q_m\le C_\Omega Q$ for a fixed constant $C_\Omega$.

The specialization $M=N=1$ of
Lemma~\ref{lem:unmarked-completed-moment}, followed by the finite ray
decomposition defining $B_{m,\sigma}$, gives for every $\epsilon_1>0$
\[
 \sum_{0<q_m\le C_\Omega Q}|B_{m,\sigma}(Z)|^2
 \ll_{\mathcal A,\epsilon_1} Z^{1+\epsilon_1}.
\]
Indeed, $q_{b_*}$ is fixed, so the row ball is a fixed multiple of the
$M=1$ ball.  The completed scale is $Z$, which is $N=1$ in that lemma.
The lemma applies directly to the Gaussian profile in the
completed integrals defining $B$;
the fixed finite Fourier sum over ray characters costs only a fixed factor
by the triangle inequality in the row Hilbert space.  In particular this
application retains the original row zero masks.

Apply Cauchy--Schwarz to the row sum in
Equation~\eqref{eq:low-separated}.  The factor $(q_m/Q)^{-iv}$ has absolute
value one, $|\xi(m)|\le1$ with its zero extension, and $\Omega$ is bounded.
Lemma~\ref{lem:balanced-additive-norm} is uniform in $v$, while $B$ is
independent of $v$.  Since $W_0$ is smooth and annular,
$\int_{\mathbb R}|\widehat W_0(iv)|\,dv<\infty$.  The finite sum over
$\sigma\in T$ therefore gives
\[
 \begin{split}
 |I_\eta(X,Y,Z)|
 &\ll_{\mathcal A,\epsilon_1}
 Q^{-1/2}\left(\frac QY\right)^{1/2}Z^{1/2+\epsilon_1/2}\\
 &=Z^{1/4+\epsilon_1/2}.
 \end{split}
\]
Taking $\epsilon_1=2\epsilon$ proves the proposition.
\end{proof}

\section{The Poisson representation and its Euler factors}
\label{sec:poisson-representation}

The direct estimate is now available. To compare the probe with the
Mellin signal in Proposition~\ref{lem:continuation-criterion}, we return
to independent positive scales $X,Y,Z$ and apply Poisson summation in
the element variable $m$. The resulting rows are indexed by the
sixth-power-free part of the frequency. We will factor each row into
Hecke $L$-functions and a holomorphic Euler product; the row $u=1$
will contain the reciprocal of the target function.

\subsection{The exact Poisson series}

The correction $\overline{G(A)}$ is independent of $m$, and the
varying character in the completed row is precisely $\chi_A(m)$,
including its zeros.  Consequently Poisson summation uses the full
modulus $sb_*A$, even if $s$ and $A$ share primes.  Expand the
$s$-Gauss sum and first sum the lifts of a residue modulo $b_*A$.
The lifts impose $H\equiv b_*Ad\pmod s$ and give the coefficient
\begin{equation}\label{eq:local-F}
 F(s,A,H)=\sum_{\substack{d\bmod s\\H\equiv b_*Ad\pmod s}}
 \chi_s(d)g_{\xi\chi_A}\!\left(b_*A,\frac{H-b_*Ad}{s}\right).
\end{equation}
Indeed, the finite Fourier transform of
$\xi(m)\chi_A(m)q_s^{-1/2}g_{\chi_s}(s,-m)$ modulo $sb_*A$ is
$q_s^{1/2}F(s,A,H)$.  The Poisson prefactor for the row scale
$q_{b_*}q_sX$ is $X/q_A$.  After the outer square-root normalization in
Equation~\eqref{eq:general-probe}, leaving its separate $Y^{-1}$ outside,
their product is
$X^{1/2}/(q_{b_*}^{1/2}q_A)$.  This also verifies all factors of
$q_s$ in the transformation.

At the primes of $b_*$ the primitive Gauss sum vanishes unless
$(H,S)=1$.  It therefore also removes $H=0$.  Every remaining
element has a unique expression
$H=ua^6$, where $u$ is a sixth-power-free element, including its
unit factor, and $a$ is the primary generator of an ideal.  Write
$\sum_u^{(6)}$ for the sum over such nonzero $u$ with $(u,S)=1$.

Let $\widetilde W_0(|y|^2)$ be the planar Fourier transform of
$W_0(|y|^2)$ for the self-dual measure and character fixed in
Section~\ref{sec:arithmetic}.  The radial Fourier argument after
Poisson is $Xq_H/q_A$.  Put
\[
 M(z)=\int_0^\infty\widetilde W_0(r)r^{z-1}\,dr,
 \qquad
 \widehat W_1(w)=\int_0^\infty W_1(r)r^{w-1}\,dr.
\]
Smoothness at zero and Schwartz decay show that $M$ is holomorphic
for $\Re z>0$.  On every compact positive real strip it has
arbitrary polynomial decay in $\Im z$: every derivative of
$e^{(\Re z)v}\widetilde W_0(e^v)$ is integrable in $v$, uniformly
on that strip, so repeated integration by parts applies.  The same
argument gives arbitrary polynomial decay for $\widehat W_1$ on
every fixed real strip.

We will need the strict positivity
\begin{equation}\label{eq:radial-mellin-positive}
 M(1/6)>0.
\end{equation}
Here is a proof that does not require the Fourier transform itself
to be nonnegative.  For $0<\sigma<1$,
\[
 |y|^{2\sigma-2}=\frac1{\Gamma(1-\sigma)}
             \int_0^\infty t^{-\sigma}e^{-t|y|^2}\,dt.
\]
The pairing of the left side with $\widetilde W_0(|y|^2)$ is
absolutely integrable.  Fubini and Fourier duality express it as
\[
 \frac1{\Gamma(1-\sigma)}\int_0^\infty t^{-\sigma}
 \int_{\mathbb C}W_0(|x|^2)
       \frac{2\pi}{\sqrt3\,t}
       \exp\!\left(-\frac{4\pi^2|x|^2}{3t}\right)
                  d\mu(x)\,dt>0.
\]
The inner integral is strictly positive for every $t>0$, since
$W_0$ is nonnegative and nonzero.  Absolute integrability follows
either from the displayed representation on the Fourier side or
from the annular support of $W_0$ on this side.  Polar integration
identifies the original pairing with $(2\pi/\sqrt3)M(\sigma)$,
proving Equation~\eqref{eq:radial-mellin-positive}.

Set $x=t+1-z$ after Mellin inversion of the two weights.  For a
nonzero sixth-power-free $u$ with $(u,S)=1$, define
\begin{equation}\label{eq:local-full-series}
 \mathcal F_{\eta,u}(x,w,z)=
 \sum_{\substack{c\ {\rm sf}\\n,s,a}}
 K_\eta(c,n,s,a;u)\,
 q_c^{-1/2-x}q_n^{-1-3x}q_s^{-w}q_a^{-6z},
\end{equation}
where $A=cn^3$ and the coefficient, with all normalizations retained,
is
\begin{equation}\label{eq:local-full-coefficient}
 K_\eta(c,n,s,a;u)=
 \frac{F(s,A,ua^6)\chi_s(b_*)}
 {\sqrt{q_{b_*}}\tau\xi(s)\overline{\xi(u)}}
 \gamma_2(c)\overline{\alpha(A)G(A)}
 \eta(A)\Xi(A)^{-1}\mathcal R(A,s).
\end{equation}
For convenience define the common Mellin weight
\[
 \mathcal W(X,Y,Z;x,w,z)=
 X^{1/2-z}Z^{x+z-1}Y^{w-1}
 \Phi(x+z-1)M(z)\widehat W_1(w).
\]
For example, the lines $\Re x=3$, $\Re w=3$, $\Re z=2$ correspond
to the original line $\Re t=4$ and lie in absolute convergence.
The complete high identity is
\begin{equation}\label{eq:probe-poisson-identity}
 \begin{split}
 I_\eta(X,Y,Z)=\frac1{(2\pi i)^3}
 \int_{(3)}\int_{(3)}\int_{(2)}
 &\mathcal W(X,Y,Z;x,w,z)\\
 &\cdot\sum_u^{(6)}q_u^{-z}\overline{\xi(u)}
             \mathcal F_{\eta,u}(x,w,z)\,dz\,dw\,dx.
 \end{split}
\end{equation}
One may justify all interchanges on these lines by the elementary
bound $|F(s,A,H)|\le q_sq_{b_*A}$ and by the annular weights.
In particular, Equations~\eqref{eq:low-separated} and
\eqref{eq:probe-poisson-identity} are identities for the same
probe, not estimates for separately chosen test expressions.


\subsection{The scalar Euler identity}
\label{sec:local-euler}

We next evaluate the coefficient in
Equation~\eqref{eq:local-full-coefficient}.  Our objective is to prove
that the high series is a scalar Euler product for the target $\eta$.
The calculation also records separately the terms with positive valuation
of the completed index $A$ at a given prime.  We keep every ray phase until
it has been cancelled or assigned to a local factor.

Fix a nonzero sixth-power-free row $u$ with $(u,S)=1$ and a prime
$p\notin S$.  Set
\begin{equation}\label{eq:local-parameters}
 \begin{gathered}
 Q=q_p,\qquad j=v_p(u)\in\{0,\ldots,5\},\qquad
 \rho=\chi_p(u/p^j),\qquad \omega_p=\chi_p(-1),\\
 a_p=\overline{\alpha(p)}^3\eta(p)^3,\qquad
 b_p=\overline{\alpha(p)}^2\eta(p)^2\overline{\chi_p(4)},
 \qquad b_p^3=a_p^2,\\
 V=Q^{-6z},\qquad R=a_p^2Q^{4-6x-6z},\qquad
 W_{\rm loc}=\rho Q^{-w}.
 \end{gathered}
\end{equation}
The values $\rho,a_p,b_p$ have absolute value one, and
$\omega_p\in\{1,-1\}$.  No ray-class restriction on $p$ is imposed.

\paragraph{The finite scalar and its unit factors.}
Write $G_r=g_{\chi_p^r}(p,1)$, with $G_0=-1$, and
$\gamma_r=Q^{-1/2}G_r$ for $1\le r\le5$.  For $t\ge1$ and
$j'\ge0$, summing the lifts of a residue modulo $p$ gives
\[
 g_{\chi_p^r}(p^t,p^{j'})=
 \begin{cases}
 Q^{t-1}G_r1_{j'=t-1},&6\nmid r,\\
 Q^t1_{j'\ge t}-Q^{t-1}1_{j'\ge t-1},&6\mid r.
 \end{cases}
\]
The second line is the Ramanujan sum for the principal character
extended by zero.  Orthogonality also gives
$G_1G_{-1}=\omega_pQ$.

Let $e_0=v_p(c)\in\{0,1\}$, $l=v_p(n)$, $t=e_0+3l$,
$k=v_p(s)$, and $m=v_p(a)$, so $v_p(H)=j'=j+6m$.
The part of Equation~\eqref{eq:local-F} at $p$, before its unit
factors are restored, is
\begin{equation}\label{eq:local-scalar-definition}
 C_p(t,k,j')=
 \sum_{\substack{d\bmod p^k\\p^{j'}\equiv p^td\pmod{p^k}}}
 \chi_p^k(d)
 g_{\chi_p^t}\!\left(p^t,\frac{p^{j'}-p^td}{p^k}\right).
\end{equation}
At modulus one both factors in this formula mean one.  Its complete
evaluation is
\begin{equation}\label{eq:local-scalar-values}
 C_p(t,k,j')=
 \begin{cases}
 1,&t=k=0,\\
 1_{j'=0},&t=0,\ k>0,\\
 g_{\chi_p^t}(p^t,p^{j'}),&t>0,\ k=0,\\
 \omega_p^{-1}G_1g_{\chi_p^{t-1}}(p^t,p^{j'-1}),
                      &t>0,\ k=1,\ j'\ge1,\\
 0,&\text{otherwise}.
 \end{cases}
\end{equation}
For $t=0$ and $k>0$, the congruence fixes $d=p^{j'}$ modulo
$p^k$, whose zero-extended character is nonzero exactly when
$j'=0$.  For the fourth line the congruence first forces $j'\ge1$.
Expanding the inner Gauss sum, the sum over $d\bmod p$ is
\[
 \sum_{d\bmod p}\chi_p(d)e(-vd/p)
       =\omega_p^{-1}G_1\chi_p(v)^{-1}
       \qquad(p\nmid v),
\]
which changes the Gauss character from $\chi_p^t$ to
$\chi_p^{t-1}$.  If $t>0$ and $k\ge2$, the permitted lifts
$d=d_0+p^{k-1}v$ do not change $\chi_p^k(d)$, but their Gauss phase
is $e(-yv/p)$ with the Gauss variable $y$ a unit.  Their sum is
zero.  This proves Equation~\eqref{eq:local-scalar-values}, including
the cases where the Gauss character is principal and zero-extended.

For the original units write $h_p=H/p^{j'}$, $s_p=s/p^k$, and
$A_p^\circ=A/p^t$.  Chinese remaindering and the substitution
$d=h_p(b_*A_p^\circ)^{-1}d'$ give the additional unit factor
\begin{equation}\label{eq:local-unit-factor}
 U_p=\chi_p^{k-t}(h_p)\chi_p^t(s_p)
       \chi_p^{t-k}(A_p^\circ)\chi_p^{t-k}(b_*).
\end{equation}
More explicitly, the outer residue character gives
$\chi_p^k(h_p)\chi_p^{-k}(b_*A_p^\circ)$, while the Chinese
remainder factor and the rescaling of the Gauss argument give
$\chi_p^t(b_*A_p^\circ)\chi_p^{-t}(h_p/s_p)$.  Their product is
Equation~\eqref{eq:local-unit-factor}.  Since
$h_p=(u/p^j)(a/p^m)^6$, one has $\chi_p(h_p)=\rho$.

At the primes of $b_*$, one has
$(H-b_*Ad)/s\equiv H/s\pmod {b_*}$.  The local factor in the
complete Gauss sum is
$\sqrt{q_{b_*}}\tau\xi(A)\overline{\xi(H/s)}$.
Here $\overline{\xi(H/s)}=\overline{\xi(u)}\xi(s)$ because
$\xi(a)^6=1$.  This cancels every displayed $\xi$ factor in
Equation~\eqref{eq:local-full-coefficient}.  The product of the last
factors in Equation~\eqref{eq:local-unit-factor} cancels
$\chi_s(b_*)\chi_A(b_*)^{-1}$.  This cancellation is coefficientwise.
For any fixed enlargement of $S$, the same calculation uses the same
ray group $T$.

\paragraph{Cancellation of the remaining pair phases.}
With $t_p=v_p(A)$ and $k_p=v_p(s)$, reciprocity at each pair of
distinct primes gives
\[
 \mathcal R(A,s)\prod_p
       \chi_p^{t_p}(s_p)\chi_p^{-k_p}(A_p^\circ)
       =\prod_p\mathcal R(p,p)^{t_pk_p}.
\]
Indeed, for $p\ne r$ the two orientations have opposite symbol
exponents, and their quotient is $\mathcal R(p,r)$; this cancels
the corresponding pair in the bicharacter $\mathcal R(A,s)$.
Only its diagonal remains.

Equation~\eqref{eq:signal} gives
$\gamma_2(c)=\mu(c)\alpha(c)G(c)/\gamma_1(c)$.
The Chinese remainder formula
\[
 \gamma_1(c)=\prod_{p\mid c}\gamma_1(p)
             \prod_{p<r,\ p,r\mid c}\chi_p(r)\chi_r(p)
\]
removes the squarefree part of the remaining pair product.  If
$e_p=v_p(c)$ and $l_p=v_p(n)$, its exponent at a pair $p<r$ is
$t_pt_r-e_pe_r=3(e_pl_r+e_rl_p+3l_pl_r)$.  The cube of
$\chi_p(r)\chi_r(p)$ is $\mathcal R(p,r)$, whose square is one.
Thus the pair product left after this division is
\[
 \prod_{p<r}\mathcal R(p,r)^{e_pl_r+e_rl_p+l_pl_r}
 =\mathcal R(c,n)G(n^3)
   \prod_p\overline{G(p^3)}^{\,l_p}
            \mathcal R(p,p)^{-e_pl_p-l_p(l_p-1)/2}.
\]
The equality follows by expanding the quadratic refinement
Equation~\eqref{eq:G-quadratic-refinement} on the prime factors of
$n^3$.  The same refinement gives
$G(c)\mathcal R(c,n)G(n^3)=G(cn^3)$, which is cancelled by the
inserted $\overline{G(cn^3)}$.  Finally
Equation~\eqref{eq:fixed-gauss-phases} gives
$G(p^3)=\gamma_3(p)$ and $\mathcal R(p,p)=\omega_p$.
These identities account for all pair phases, with no restriction
on the ray class of $p$.

The coefficient in Equation~\eqref{eq:local-full-coefficient}
has therefore separated into prime factors.  In absolute convergence,
its factor at $p$ is the series
\begin{equation}\label{eq:local-P}
\begin{aligned}
 P_p={}&\sum_{\substack{e_0=0,1\\l,k,m\ge0}}
 (-1)^{e_0}\gamma_1^{-e_0}\eta(p)^{e_0}
 (a_p\overline{\gamma_3})^l\rho^{k-t}
 \omega_p^{tk-e_0l-l(l-1)/2}C_p(t,k,j+6m)\\
 &\hspace{33mm}\cdot
 Q^{-(x+1/2)e_0-(1+3x)l-wk-6zm},\qquad t=e_0+3l.
\end{aligned}
\end{equation}
Let $P_p^*$ be the same sum restricted to $t>0$, equivalently to
positive valuation of the completed index $A$.

\begin{lemma}[The complete local identity]\label{lem:local-euler}
For every nonzero sixth-power-free $u$ with $(u,S)=1$ and every
prime $p\notin S$, the factors just defined are
\begin{equation}\label{old-eq:5.13a}
\begin{aligned}
 P_p^*&=\frac1{1-R}\left\{
 \frac{R(1-Q^{-1})-\eta(p)(Q-1)
 Q^{-x-w}V^{1_{j\le1}}}{1-V}+J_j\right\},\\
 P_p&=\frac1{1-V}+1_{j=0}\frac{W_{\rm loc}}{1-W_{\rm loc}}+P_p^*,
\end{aligned}
\end{equation}
where the six values of $J_j$ are
\[
\begin{array}{c|l}
 j&J_j\\\hline
 0&-\eta(p)\rho^{-1}Q^{-x}+W_{\rm loc}R\\
 1&\eta(p)Q^{-x-w}\\
 2&a_p\rho^{-3}Q^{3/2-3x}\\
 3&-\eta(p)b_p\rho^{-2}Q^{2-3x-w}
       +b_p^2\rho^{-4}Q^{2-4x}\\
 4&-\eta(p)a_p\rho^{-3}Q^{5/2-4x-w}\\
 5&-a_p^2Q^{3-6x}.
\end{array}
\]
Put $W=\chi_p(u)Q^{-w}$ and
$D=\eta(p)\overline{\chi_p(u)}Q^{-x}$, with the original zero
extension at $p\mid u$, and define
\begin{equation}\label{eq:local-H-definition}
 H_p=P_p\frac{(1-V)(1-W)}{1-D},\qquad
 \mathcal H_{\eta,u}=\prod_{p\notin S}H_p.
\end{equation}
Then the series in Equation~\eqref{eq:local-full-series} has the
scalar factorization
\begin{equation}\label{eq:probe-high}
 \mathcal F_{\eta,u}(x,w,z)=
 \frac{\zeta_F^S(6z)L^S(w,\chi_\bullet(u))}
 {L^S(x,\eta\overline{\chi_\bullet(u)})}
 \mathcal H_{\eta,u}(x,w,z).
\end{equation}
Write $x_r=\Re x$, $w_r=\Re w$, and $z_r=\Re z$.  For every
fixed $\epsilon_0>0$, the correction product converges normally
on a neighborhood of every point in each of the following regions,
and hence defines a holomorphic function there:
\begin{align}
 x_r\ge51/100,\quad z_r\ge17/50,\quad w_r\ge-1/100,\quad
 x_r+w_r\ge1+\epsilon_0;\label{eq:local-region-one}\\
 x_r\ge7/8,\quad z_r\ge33/200,\quad w_r\ge19/20.
 \label{eq:local-region-two}
\end{align}
Uniformly in imaginary parts and unit phases,
$\mathcal H_{\eta,u}\ll_\epsilon q_u^\epsilon$, with the constant
also depending on $\epsilon_0$ in the first region.  For $u=1$ in the
second region,
$\mathcal H_{\eta,1}=1+O(P_0^{-c_H})$ with, for example,
$c_H=4/5$.  Equation~\eqref{eq:probe-high} begins in absolute
convergence and supplies a meromorphic continuation through these
regions.
\end{lemma}

The first region will contain the buffered contours for nonprincipal
rows, including the reflected numerator line. The second contains the
principal residue point $w=1$, $z=1/6$ and the contours used to reach it.

\begin{proof}
There are four families with $t>0$:
$(e_0,l)=(0,2r+2),(1,2r),(0,2r+1),(1,2r+1)$, where $r\ge0$.
In each family, increasing $r$ by one and the first permitted $m$
by one multiplies the summand of Equation~\eqref{eq:local-P} by
$R$.  In fact the phase ratio is $a_p^2$: the remaining ratio is
one because $\gamma_3^2=\omega_p$, $\omega_p^2=1$, and $\rho^6=1$.
Increasing $m$ further multiplies a summand by $V$.

For transparency, the following table lists every nonzero summand
after division by $R^r$.  In its first row $j$ is arbitrary; all
other conditions are as displayed.  Every omitted case is zero by
Equation~\eqref{eq:local-scalar-values}.
\begin{equation}\label{eq:local-family-table}
\begin{array}{c|c|l|l}
 (e_0,l)&k&\text{condition on }(j,m)&R^{-r}\text{ times summand}\\\hline
 (0,2r+2)&0&m\ge r+1&R(1-Q^{-1})V^{m-r-1}\\
 (0,2r+2)&0&j=5,\ m=r&-a_p^2Q^{3-6x}\\
 (0,2r+2)&1&j=0,\ m=r+1&W_{\rm loc}R\\
 (1,2r)&0&j=0,\ m=r&-\eta(p)\rho^{-1}Q^{-x}\\
 (1,2r)&1&j=1,\ m=r&\eta(p)Q^{-x-w}\\
 (1,2r)&1&m\ge r+1_{j\le1}&-\eta(p)(Q-1)Q^{-x-w}V^{m-r}\\
 (0,2r+1)&0&j=2,\ m=r&a_p\rho^{-3}Q^{3/2-3x}\\
 (0,2r+1)&1&j=3,\ m=r&-\eta(p)b_p\rho^{-2}Q^{2-3x-w}\\
 (1,2r+1)&0&j=3,\ m=r&b_p^2\rho^{-4}Q^{2-4x}\\
 (1,2r+1)&1&j=4,\ m=r&-\eta(p)a_p\rho^{-3}Q^{5/2-4x-w}.
\end{array}
\end{equation}
Here is a direct check of its entries.  In the first family $t=6r+6$.
For $k=0$, the principal Gauss lift is
$Q^t(1-Q^{-1})$ for $m\ge r+1$; its extra boundary value at
$(j,m)=(5,r)$ is $-Q^{t-1}$.  For $k=1$, the product
$\omega_p^{-1}G_1G_{-1}=Q$ allows only $(j,m)=(0,r+1)$.
These give the first three rows.  In the second family $t=6r+1$.
For $k=0$ the nonprincipal equality $j'=t-1$ gives the fourth row.
For $k=1$, the principal Gauss lift has its negative boundary at
$j'=t$, giving the fifth row, and its value $Q^{t-1}(Q-1)$ for
$j'>t$, giving the sixth row.  The latter condition is exactly
$m\ge r+1_{j\le1}$.

For the two odd families $t=6r+3$ and $t=6r+4$.  Their $k=0$
lines require $j'=t-1$ and their $k=1$ lines require $j'=t$.
The phase reductions are
\[
 \gamma_1\gamma_2
       =-\alpha(p)\overline{\chi_p(4)}\gamma_3,
 \qquad
 \frac{\gamma_4}{\gamma_1}
       =-\frac{\overline{\alpha(p)}}{G(p)}.
\]
The first is Equation~\eqref{eq:signal} at $p$; the second follows
from the first and $\gamma_2\gamma_4=1$.  Substitution gives the
last four rows, including both $j=3$ terms.  Thus the table covers
all five ramified valuations as well as $j=0$.

Summing $r\ge0$ gives $(1-R)^{-1}$, and the two unbounded $m$
ranges give $(1-V)^{-1}$.  These sums are precisely the formula for
$P_p^*$ in Equation~\eqref{old-eq:5.13a}.  If $t=0$, then
$e_0=l=0$.  The $k=0$ terms give $(1-V)^{-1}$, while the $k>0$
terms require $j=m=0$ and give $W_{\rm loc}/(1-W_{\rm loc})$.
This proves the
formula for $P_p$ as well.

It remains to justify the analytic assertions, especially where
$w_r$ is negative.  Put $\mathcal E_p=P_p^*+D$.  The contribution
from $t=0$ is $(1-V)^{-1}+W/(1-W)$, so an exact simplification gives
\begin{equation}\label{eq:local-H-defect}
 H_p-1=
 \frac{D(V+W-VW)-VW+(1-V)(1-W)\mathcal E_p}{1-D}.
\end{equation}
This expression has no $1-W$ denominator.  At $p\mid u$, where
$D=W=0$, it reduces to $(1-V)\mathcal E_p$.  In both stated regions
$|R|<1$, $|V|<1$, and $|D|<1$, uniformly away from one, so these
are holomorphic local expressions.

For $p\nmid u$ and $\vartheta=(-w_r)_+$, the $j=0$ row gives
\[
 |\mathcal E_p|\ll
 Q^{4-6x_r-6z_r+\vartheta}+Q^{1-x_r-w_r-6z_r}.
\]
In the first region $4-6x_r-6z_r\le-11/10$ and
$\vartheta\le1/100$.  The extra factor $1-W$ in
Equation~\eqref{eq:local-H-defect} therefore leaves this contribution
at most $Q^{-27/25}$.  The term $DW$ is at most
$Q^{-1-\epsilon_0}$; the other terms in that equation are smaller.
For $p\mid u$, the strict second-family term has exponent
$1-x_r-w_r\le-\epsilon_0$.  The closest other exponents in the
table are
\[
 3/2-3x_r\le-3/100,\qquad
 2-3x_r-w_r\le-1/50-\epsilon_0,
\]
and all remaining ones are no larger than $-3/100$.
Consequently, with $\epsilon_H=\min(\epsilon_0,1/50)$,
\[
 H_p-1=O(Q^{-1-\epsilon_H})\quad(p\nmid u),\qquad
 H_p-1=O(Q^{-\epsilon_H})\quad(p\mid u).
\]
In the second region the same comparison gives the stronger bounds
\[
 H_p-1=O(Q^{-363/200})\quad(p\nmid u),\qquad
 H_p-1=O(Q^{-33/40})\quad(p\mid u).
\]
The closest good-prime term here is
$Q^{1-x_r-w_r-6z_r}$, and the closest ramified term is
$Q^{1-x_r-w_r}$.

The sum over primes not dividing $u$ converges normally, and the
finite product over primes of $u$ is $O_\epsilon(q_u^\epsilon)$
by the divisor-product bound.  The estimates are uniform in all
imaginary parts and unit phases.  When $u=1$, the good-prime tail
above $P_0$ is $O(P_0^{-163/200})$ by the ideal count, which implies
the asserted $O(P_0^{-4/5})$ estimate for the product.
Finally extracting $(1-V)^{-1}(1-W)^{-1}(1-D)$ at every prime
gives Equation~\eqref{eq:probe-high}.  The normal convergence of
the remaining product proves its claimed continuation.
\end{proof}

For $u=1$, Equation~\eqref{eq:probe-high} contains
$1/L^S(x,\eta)$, and its numerator has the two principal factors
$\zeta_F^S(6z)$ and $\zeta_F^S(w)$. Their residues will produce the
target Mellin signal. For intermediate row norms, the next section
assigns one zero-free rectangle to the finite family of twists associated
with each row and produces simultaneous polynomial witnesses from a
selected zero. The principal and bounded rows, together with the
outer norm ranges, are treated in
Section~\ref{sec:shared-analytic-estimates}.

\section{A zero detector with saturated witnesses}\label{sec:detector}

The high expansion contains a sum over sixth-power-free rows \(u\).
We associate each nonprincipal row with a buffered zero-free rectangle
for the finite family of character presentations that it determines.
Whenever the resulting bin lies above a fixed floor, a zero produces
an inverse polynomial and a plain polynomial with simultaneous lower
bounds.  These witnesses can then be counted using different mean
estimates.  The analytic inputs here are Lemma~\ref{lem:logarithmic-control},
the smooth calculus of Lemma~\ref{lem:smooth-calculus}, and the global
growth estimate in Equation~\eqref{eq:hecke-growth}, together with the
deleted Euler-factor bounds of Lemma~\ref{lem:deleted-euler-factors}.

Throughout this section, assume \(\beta_*>51/100\), where \(\beta_*\)
is the global supremum in Equation~\eqref{old-eq:1.1b}.  This setting
does not select either of the two boundaries in the continuation
criterion.

\subsection{Buffered rectangles and pointwise bounds}

Fix the arithmetic data \(\mathcal A\) independently of \(Z\), as in
Section~\ref{sec:arithmetic}.  In particular, the finite group
\[
 \Theta=\langle\eta,\widehat T\rangle
\]
contains the target and the fixed ray twists used in the high-row
family, and every conductor prime of a member of \(\Theta\) belongs
to \(S\).  For a
sixth-power-free element \(u\), define the finite collection of
presentations
\[
 \mathcal X_u=
 \bigl\{\psi_{u,\nu,\varsigma}(n)=\nu(n)\chi_n(u)^\varsigma:
       \nu\in\Theta,\ \varsigma\in\{-1,1\}\bigr\}.
\]
At a nonunit the value of \(\chi_n(u)^{-1}\), like every other
power of the symbol, is zero.  The group property of \(\Theta\)
shows that \(\mathcal X_u\) is closed under conjugation.  It contains
the numerator character \(\chi_\bullet(u)\), the denominator
character \(\eta\overline{\chi_\bullet(u)}\), and every zero-extended
presentation obtained by multiplying either orientation of the sextic
symbol by a member of \(\Theta\).

Let \(\psi^*\) be the primitive character inducing a presentation
\(\psi\in\mathcal X_u\), and let \(Q_\psi\) be its conductor norm.
Reciprocity and the fixed conductor primes give
\[
 Q_\psi\ll_{\mathcal A}q_u,\qquad
 L_{\rm orig}(s,\psi)
   =L(s,\psi^*)\prod_{p\in E_u}(1-\psi^*(p)q_p^{-s}),
 \qquad
 E_u\subset S\cup\{p:p\mid u\}.
\]
The radical of the deleted product has norm \(O_S(q_u)\).
These identities retain the original zero extensions; in particular,
they do not replace a row-dependent mask by an independently chosen
mask.

\phantomsection\label{par:physical-row-ramification}
If \(v_p(u)=j\in\{1,\ldots,5\}\) at a prime \(p\notin S\), the
local character of \(\chi_\bullet(u)\) on units at \(p\) has order
\(6/\gcd(6,j)>1\).  A character in \(\Theta\) is unramified there,
so it cannot cancel this local character.  Consequently a
presentation in \(\mathcal X_u\) can induce the principal character
only when \(u\) is supported on \(S\).  There are finitely many such
sixth-power-free rows, including unit factors.  Remove them for the
present detector; each application must estimate these bounded physical
rows separately.

Fix \(0<d_{\min}<d_{\max}<\infty\) and write \(U=Z^d\), where
\(d_{\min}\le d\le d_{\max}\).  The rows in the present dyad satisfy
\(q_u\asymp U\), with fixed comparison constants.  Let
\[
 0<\tau\le d_{\min}/100,\qquad T_1=Z^\tau>2,\qquad
 0<e<10^{-3},\qquad I=\lceil1/e\rceil+2.
\]
The value of \(\tau\) will be chosen after the fixed height orders
are known.  For now it is fixed independently of \(Z\).  In
particular \(T_1\le U^{1/100}\).

\begin{lemma}[Buffered zero-free bins]\label{lem:buffered-bins}
For every retained row \(u\), there are an index
\(i\in\{1,\ldots,I-1\}\) and a grid point
\[
 a\in(51/100+e\mathbb Z_{\ge0})\cap[51/100,1]
\]
with the following properties.  For \(j=1,\ldots,I\), set
\[
 M_j(u)=\max\left(\{51/100\}\cup
 \left\{\Re\rho:
 \begin{array}{l}
  \psi\in\mathcal X_u,\quad L(\rho,\psi^*)=0,\\
  \Re\rho\ge51/100,\quad |\Im\rho|\le3jT_1
 \end{array}\right\}\right).
\]
Then
\[
 a\le M_i(u)<a+e,\qquad M_{i+1}(u)<a+2e.
\]
If \(a>51/100\), some \(\psi\in\mathcal X_u\) has a zero
\(\rho=\sigma+i\gamma\) with
\[
 a\le\sigma<a+e,\qquad |\gamma|\le3iT_1.
\]
For every \(\epsilon_1>0\), every \(\psi\in\mathcal X_u\), and
sufficiently large \(Z\), uniformly in the retained row,
\[
 |L_{\rm orig}(s,\psi)|+|L_{\rm orig}(s,\psi)^{-1}|
 \ll_{\mathcal A,e,\epsilon_1} U^{\epsilon_1}
 \quad
 \left(\Re s\ge a+6e,\ |\Im s|\le(3i+2)T_1\right).
\]
On the reflected line in the same height range,
\[
 |L_{\rm orig}(1-a-6e+it,\psi)|
 \ll_{\mathcal A,e,\epsilon_1}
 U^{a-1/2+12e+\epsilon_1}(3+T_1)^C,
\]
where \(C\) depends only on the fixed real strip and the field.
There are \(O_e(1)\) possible pairs \((i,a)\), independently of
the row and its conductor.
\end{lemma}

\begin{proof}
Each maximum exists: the collection is finite, the zeros of each
nonprincipal primitive \(L\)-function are discrete in compact
rectangles, and absolute Euler convergence excludes zeros with real
part greater than one.  The sequence \(M_j(u)\) is nondecreasing
and lies in \([51/100,1]\).  If every one of its \(I-1\) increments
exceeded \(e\), then
\[
 M_I(u)-M_1(u)>(I-1)e>1,
\]
which is impossible.  Choose an index with
\(M_{i+1}(u)-M_i(u)\le e\), and round \(M_i(u)\) down on the stated
grid.  This gives the two inequalities.  When \(a>51/100\), the
maximum \(M_i(u)\) is attained by an actual zero, giving \(\rho\).
This reasoning allows a zero on the line one.

For \(|t|\le(3i+2)T_1\), the closed disk centered at \(2+it\)
with radius \(2-a-2e\) has real part at least \(a+2e\).
Its radius is less than \(3/2<T_1\), so every point in it has
imaginary part of absolute value less than \(3(i+1)T_1\).
The bound \(M_{i+1}(u)<a+2e\) therefore excludes every zero of every
primitive function in this disk.  Lemma~\ref{lem:logarithmic-control}
controls the concentric disk of radius \(2-a-6e\).  Apply it with
center \(2+i\Im s\), and use absolute Euler convergence farther
right.  Since \(Q_\psi\ll_{\mathcal A}U\) and
\((3+|t|)^2\ll_e U^{1/50}\), the arbitrarily small power in that
lemma can be chosen so that its contribution is \(U^{\epsilon_1/2}\).
The estimates for polynomial-size deleted Euler products following
that lemma supply the other \(U^{\epsilon_1/2}\).

For the last assertion, the primitive functional equation recalled
in the proof of Lemma~\ref{lem:hecke-strip-growth}
\cite[Equation (1.1)]{GZ} relates the value at \(1-a-6e+it\) to
the conjugate primitive function at \(a+6e-it\).  The conjugate
presentation is in \(\mathcal X_u\).  The conductor and gamma
quotient contribute at most
\[
 Q_\psi^{a-1/2+6e}(3+|t|)^C.
\]
The reflected primitive value has an arbitrarily small \(U\)-power
by the preceding disk bound.  Finally, \(1-a-6e\ge-6e\), so the
upper bound for the deleted product is \(U^{6e+\epsilon_1}\), after
reducing the preliminary losses.  This proves the reflected
estimate.  The ranges of \(i\) and \(a\) give \(O_e(1)\) pairs.
\end{proof}

We call \((i,a)\) the bin of the row and put
\[
 \delta=2a-1,\qquad 1/50\le\delta\le1.
\]
Every primitive character inducing a presentation in \(\mathcal X_u\)
is a finite-order Hecke character.  The definition of \(\beta_*\),
together with \(\beta_*>51/100\), therefore gives \(M_i(u)\le\beta_*\).
In particular, the bin satisfies the exact inequalities
\[
 a\le\beta_*,\qquad \delta\le2\beta_*-1.
\]
The bin \(a=51/100\) will be bounded by the trivial row count.
For \(a>51/100\), the zero in Lemma~\ref{lem:buffered-bins} produces
large polynomials to which the moments can be applied.

\begin{lemma}[Pointwise dyadic estimates]\label{lem:detector-dyads}
Fix bounded nonnegative ranges for \(r,m\), and let
\[
 M_r=M_\psi(r;W_M),\qquad S_m=S_\psi(m;W_S)
\]
denote the polynomials of Equations~\eqref{old-eq:1.1a} and
\eqref{old-eq:1.1}, now at base \(U\).  Suppose
\(\psi\in\mathcal X_u\), the untwisted profiles have uniformly
bounded smooth seminorms on a fixed annulus, and any pure norm twist
has height at most \((3i+1)T_1\).  All additional Mellin frequencies
entering the same \(L\)-argument are required to have total absolute
value at most \(T_1/2\).

For every \(\epsilon>0\), there are \(e_0>0\), depending only on
\(\epsilon\) and the bounded length ranges, and a finite height
order \(A_{\mathcal A}\), uniform in the moving rows and profiles,
such that, when
\[
 0<e<e_0,\qquad (1+T_1)^{A_{\mathcal A}}\le U^{\epsilon/10},
\]
the following estimates hold for sufficiently large \(Z\):
\begin{equation}
 |M_r|^2\ll_{\mathcal A,\epsilon}U^{\delta r+\epsilon},
 \label{old-eq:6.2}
\end{equation}
\begin{equation}
 |S_m|^2\ll_{\mathcal A,\epsilon}
       U^{\delta\min(m,1-m)+\epsilon}.
 \label{old-eq:6.3}
\end{equation}
The external tail order may be chosen after the positive number
\(\tau\).  It does not change \(A_{\mathcal A}\).
\end{lemma}

\begin{proof}
We spell out the height restriction because a full infinite contour
shift would not be justified by the bin.  For a fixed annular \(W\),
a bounded real number \(\sigma\), and a pure twist \(y^{i\omega}\),
Mellin inversion on a line \(c+\sigma>1\) gives
\[
 \begin{split}
 &U^{-r/2}\sum_n\mu(n)\psi(n)
       W(q_n/U^r)(q_n/U^r)^{-\sigma+i\omega}\\
 &\quad=\frac1{2\pi i}\int_{(c)}
  \mathcal MW(s)\,
  U^{r(s+\sigma-i\omega-1/2)}
  L_{\rm orig}(s+\sigma-i\omega,\psi)^{-1}\,ds.
 \end{split}
\]
Here \(\mathcal MW\) is the Mellin transform defined in
Lemma~\ref{lem:smooth-calculus}.  The twist occurs in the
\(L\)-argument, not in the transform whose tails are estimated.
Shift only the portion with added frequency at most the assigned
fraction of \(T_1\) to \(\Re(s+\sigma)=a+6e\).
The horizontal joins and the shifted portion remain in the
zero-free rectangle of Lemma~\ref{lem:buffered-bins}.  Its bound
gives \(U^{(a-1/2+6e)r+\epsilon_1}\) for the central portion.

Leave the remaining tails on the absolute line, which we may take
to be \(\Re(s+\sigma)=2\).  There the reciprocal Euler product is
absolutely bounded.  On the inverse joins the reciprocal remains
inside the buffered rectangle.  Since the real join interval and
the length range are bounded, the remaining arithmetic and scale
factors on a join are \(O(U^B(1+|\Im s|)^J)\) for fixed \(B,J\)
chosen before the external tail order.  The pointwise Mellin bound
in Equation~\eqref{eq:pointwise-mellin-tail}, applied on that compact
real interval with derivative order \(N+\lceil J\rceil+1\), bounds
each horizontal join by \(O(U^B T_1^{-N})\).  The separate vertical
tails have the same bound by Equation~\eqref{eq:late-fourier-tail}.
Both estimates apply to the untwisted profile, uniformly in the
stated family.  Because \(T_1=Z^\tau\), \(\tau>0\), and all lengths
lie in a fixed bounded interval, a sufficiently large fixed \(N\)
makes these errors smaller than the claimed bound.
Taking \(e\) and \(\epsilon_1\) small in the prescribed
\(\epsilon\) proves Equation~\eqref{old-eq:6.2}.

For the plain polynomial the same calculation uses
\(L_{\rm orig}\) instead of its reciprocal.  Shifting to
\(\Re(s+\sigma)=a+6e\) gives
\[
 |S_m|\ll U^{(a-1/2+6e)m+\epsilon_1}
          +O(U^B T_1^{-N}).
\]
Alternatively shift its central portion to
\(\Re(s+\sigma)=1-a-6e\).  The function is entire because the
row is nonprincipal.  The reflected estimate of
Lemma~\ref{lem:buffered-bins} gives
\[
 |S_m|\ll
 U^{a-1/2+12e+\epsilon_1}
 U^{(1/2-a-6e)m}(3+T_1)^C
 +O(U^B T_1^{-N}).
\]
The joins here need only the global upper strip bound and the
negative-strip bound for the deleted product, since no reciprocal
is present.  These give fixed \(B,J\) as above, so the pointwise
Mellin estimate bounds the joins and the integrated Fourier estimate
bounds the absolute-line tails.  Taking the better central bound,
choosing \(N\) to
dominate also the possibly negative exponent when \(m>1\), and
using the displayed height hypothesis proves
Equation~\eqref{old-eq:6.3}.  A primitive conductor smaller than
\(U\) reduces the reflected conductor factor.  The original deleted
Euler factors have already been included in
Lemma~\ref{lem:buffered-bins}.

There are only finitely many separated variables in any one
\(L\)-argument.  Assign each a fixed fraction of the single
\(T_1/2\) allowance; do not assign that allowance anew at successive
shifts.  The profile windows and the finite height orders used in the
row estimates are fixed before \(N\).  Lemma~\ref{lem:smooth-calculus} differentiates only the
untwisted separating profiles when increasing \(N\), which proves
the last assertion.
\end{proof}

\subsection{Simultaneous saturated witnesses}

At a selected zero, the truncated-inverse construction below produces
a product with squared size at least $U^{\delta(r+m)-\epsilon}$.
The pointwise estimates bound the same product by
$U^{\delta(r+\min(m,1-m))+\epsilon}$. Since $\delta\ge1/50$,
these two bounds force $m\le1/2+O(\epsilon)$. Each factor then
attains its own pointwise exponent up to the prescribed loss; this
is the saturation asserted in the next proposition.

\begin{proposition}[Two saturated witnesses]\label{prop:detector-witness}
Let \(u\) have a bin \((i,a)\) with \(a>51/100\), and assume the
loss and height hypotheses of Lemma~\ref{lem:detector-dyads}.
For every \(t\in[1,3/2]\),
and every prescribed \(\epsilon>0\),
after reducing its preliminary losses, there are
\(\psi\in\mathcal X_u\), a zero \(\rho=\sigma+i\gamma\) as in
Lemma~\ref{lem:buffered-bins}, dyadic lengths \(D=U^r\), \(N=U^m\),
and two polynomials \(M_r,S_m\) for the common row character
\(\psi\) such that
\begin{equation}\label{eq:detector-note}
 r\le t+O(1/\log U),\qquad
 r+m\ge t-O(1/\log U),\qquad
 |M_rS_m|^2\gg_{\mathcal A,\epsilon}
 U^{\delta(r+m)-\epsilon}.
\end{equation}
Their lengths and individual values also satisfy
\begin{equation}\label{eq:saturated-witness}
 \begin{gathered}
 t-\tfrac12-O(\epsilon)\le r\le t+O(1/\log U),
 \qquad 0\le m\le\tfrac12+O(\epsilon),\\
 |M_r|^2\gg_{\mathcal A,\epsilon}U^{\delta r-\epsilon},
 \qquad
 |S_m|^2\gg_{\mathcal A,\epsilon}U^{\delta m-\epsilon}.
 \end{gathered}
\end{equation}
The constants in the \(O(\epsilon)\) terms are absolute on the
stated parameter ranges.  Both polynomials have the same twist
height \(\gamma-\nu\), where \(|\nu|\le cT_1\) for a fixed
\(c>0\) chosen within the cumulative frequency allowance.
Their untwisted profiles form a uniformly smooth annular family.
For each row, both witnesses use one presentation
\(\psi_{u,\vartheta,\varsigma}\), selected by a label
\((\vartheta,\varsigma)\) in the fixed finite set
\(\Theta\times\{-1,1\}\).  The presentation itself varies with \(u\).
The dyadic pair can likewise
be selected separately for each row from \(O((\log U)^2)\) possibilities.
\end{proposition}

\begin{proof}
The truncated-inverse and Gamma-integral construction is a form of the
classical zero detector; compare \cite[Appendix C]{MP}
and \cite[Section~13.1]{GM}.  The buffered row-dependent rectangle and
the simultaneous lower bounds needed here are established below.
Choose the zero \(\rho\) supplied by the bin and set
\[
 D_*=U^t,\qquad Y_*=U^{20}.
\]
Let \(V_{\le}\) be a fixed smooth function equal to one on
\([0,1]\) and zero on \([2,\infty)\).  Define
\[
 C_\psi(s)=\sum_l\mu(l)\psi(l)V_{\le}(q_l/D_*)q_l^{-s},
 \qquad
 J=\frac1{2\pi i}\int_{(2)}
       Y_*^z\Gamma(z)L_{\rm orig}(\rho+z,\psi)
       C_\psi(\rho+z)\,dz.
\]
The row character is nonprincipal, so \(L_{\rm orig}\) is entire.
Moving the line to \(\Re z=-1/4\) crosses only the pole of
\(\Gamma(z)\) at zero, and its residue vanishes because
\(L_{\rm orig}(\rho,\psi)=0\).  The global strip bound
in Equation~\eqref{eq:hecke-growth}, together with the deleted
Euler factors, bounds the new line by
\[
 U^{-5}U^2D_*^{\,1-\sigma+1/4+o(1)}T_1^2.
\]
Indeed \(\Re(\rho-1/4)\ge26/100>0\), so the deleted product has
an arbitrarily small \(U\)-power; the deliberately weaker
\(U^2\) includes the primitive conductor bound.  The finite
polynomial is bounded by the displayed power of \(D_*\) using the
ideal count.  Finally, the fixed polynomial in the added height is
integrable against \(\Gamma(-1/4+iv)\), since
\((1+|\gamma+v|)^2\le(1+|\gamma|)^2(1+|v|)^2\).
Uniformly for \(1\le t\le3/2\),
\[
 -3+t(5/4-\sigma)
 \le-3+\frac32\left(\frac54-\frac{51}{100}\right)
 =-\frac{189}{100}.
\]
Thus \(J=o(1)\).  This Gamma integration uses only global upper
bounds and consumes no reciprocal height allowance.

On the original line, absolute convergence and the Mellin formula
for the exponential give
\[
 J=\sum_{l,k}\mu(l)\psi(lk)V_{\le}(q_l/D_*)
       (q_lq_k)^{-\rho}e^{-q_lq_k/Y_*}.
\]
For \(1<q_n\le D_*\), the coefficient of \(\psi(n)q_n^{-\rho}\)
is \(\sum_{l\mid n}\mu(l)=0\).  For \(n=1\) it is one.
When \(q_n>D_*\), \(V_{\le}(2q_n/D_*)=0\).
Consequently insertion of \(1-V_{\le}(2q_lq_k/D_*)\) removes
exactly the unit contribution \(e^{-1/Y_*}=1+o(1)\).
The resulting tail has absolute value \(1+o(1)\).
Multiplying it by a fixed smooth terminal cutoff equal to one for
\(q_lq_k\le U^{21}\) and zero for \(q_lq_k\ge2U^{21}\) changes it
by \(o(1)\); this follows from \(Y_*=U^{20}\), exponential decay,
and the ideal count.

Partition \(q_l\asymp D\), \(q_k\asymp N\) by a fixed smooth
dyadic partition.  There are \(O((\log U)^2)\) relevant pairs,
and their support satisfies
\[
 D\ll D_*,\qquad DN\gg D_*,\qquad DN\ll U^{21}.
\]
With \(x=q_l/D\), \(y=q_k/N\), the profile for a pair is
\[
 W_1(x)V_{\le}(Dx/D_*)\,W_2(y)\,H_{D,N}(xy),
\]
where \(\Omega\) is a fixed annular cutoff equal to one on the
product of the supports of \(W_1,W_2\), and
\[
 H_{D,N}(s)=\Omega(s)
   [1-V_{\le}(2DNs/D_*)]e^{-DNs/Y_*}
   V_{\rm term}(DNs/U^{21}).
\]
Every fixed logarithmic derivative is bounded uniformly in
\(D,N,U\).  On a cutoff transition its argument is bounded, and a
logarithmic derivative of the exponential is a polynomial in its
argument times that exponential.  Logarithmic Fourier inversion
therefore gives
\[
 \begin{split}
 H_{D,N}(xy)&=\frac1{2\pi}\int_{\mathbb R}
      \widehat H_{D,N}(\nu)x^{i\nu}y^{i\nu}\,d\nu,\\
 \int_{|\nu|>cT_1}|\widehat H_{D,N}(\nu)|
       (1+|\nu|)^J\,d\nu&\ll_{J,N_0,c}T_1^{-N_0}
 \end{split}
\]
for every fixed \(J,N_0\).  The coefficient \(L^1\)-norm on the
full line is also uniformly bounded.  Trivial bounds for the two
finite polynomials have a fixed \(U\)-power because their lengths
are bounded by \(U^{21+o(1)}\).  After \(\tau>0\) is fixed, choose
\(N_0\) so that the discarded Fourier tail is \(o(1)\).

The absolute value of the remaining sum of integrals is bounded
below by a positive constant.  The number of dyadic pairs and the
uniform \(L^1\)-norm show that, for one pair and one
\(|\nu|\le cT_1\), the product of the two unnormalized blocks has
absolute value \(\gg(\log U)^{-2}\).  Both profiles have the
same height \(\gamma-\nu\):
\[
 \begin{split}
 W_M(x)&=W_1(x)V_{\le}(Dx/D_*)x^{-\sigma-i(\gamma-\nu)},\\
 W_S(y)&=W_2(y)y^{-\sigma-i(\gamma-\nu)}.
 \end{split}
\]
The first extra cutoff remains part of the inverse annular profile
and creates no second frequency.  Removing the factors
\(D^{1/2-\sigma}\) and \(N^{1/2-\sigma}\) by central normalization
gives
\[
 |M_rS_m|^2\gg(\log U)^{-4}(DN)^{2\sigma-1}
 \ge U^{\delta(r+m)-\epsilon}
\]
for sufficiently large \(U\).  The support inequalities give the
two length bounds in Equation~\eqref{eq:detector-note}.

Apply Lemma~\ref{lem:detector-dyads} to these profiles, choosing its
loss smaller than the present \(\epsilon\).  Comparison of the
product lower bound with the two upper bounds gives
\[
 \delta\{m-\min(m,1-m)\}
   =2\delta(m-1/2)_+\ll\epsilon.
\]
Since \(\delta\ge1/50\), this implies
\(m\le1/2+O(\epsilon)\) with an absolute constant.
The support inequality then gives
\(r\ge t-1/2-O(\epsilon)\).
Dividing the product lower bound in turn by each individual upper
bound gives the two individual lower bounds in
Equation~\eqref{eq:saturated-witness}.  Only the dyadic support
errors are \(O(1/\log U)\); all other losses are arbitrarily small
fixed powers.  Lemma~\ref{lem:smooth-calculus} permits the stated
rowwise choices with a fixed polynomial height factor.
\end{proof}

\section{A sextic-sieve row count}\label{sec:sextic-row-count}

The zero detector supplies a large inverse polynomial for each row above
the floor bin.  We now turn that lower bound into a count of physical rows.
We first prove the required squarefree sextic large sieve in the primary
convention, using the norm-recursion method of Blomer, Goldmakher, and
Louvel~\cite[Section 3]{BGL}. We then fix the part of a physical row whose
prime valuations are at least two and apply the sieve to its squarefree
factor. The size of the fixed part also gives an independent upper bound
for the number of rows; the better of the two bounds produces the required
exponent.

\subsection{The sextic large sieve in the primary convention}

We continue to use the fixed finite set $S$ of prime ideals, containing
the primes above $6$ and all fixed conductor primes.  All ideal indices in
the next lemma are prime to $S$ and are represented by their primary
generators.  In particular, ``squarefree'' refers to ideals of
$\mathcal O=\mathbb Z[\omega]$, not to their rational norms.

\begin{lemma}[Sextic large sieve]\label{lem:sextic-large-sieve}
Let $K,D\ge1$, let $\varsigma\in\{-1,1\}$, and let $(c_n)$ be any
sequence of complex numbers indexed by the squarefree ideals outside $S$.
The sequence is fixed independently of the row $k$.  For every
$\epsilon>0$,
\begin{equation}\label{eq:sextic-large-sieve}
 \sum_{\substack{k\ {\rm sf}\\q_k\le K}}
 \left|\sum_{\substack{n\ {\rm sf}\\q_n\le D}}
       c_n\chi_n(k)^{\varsigma}\right|^2
 \ll_{S,\epsilon}(KD)^\epsilon
       \bigl\{K+D+(KD)^{2/3}\bigr\}
       \sum_{\substack{n\ {\rm sf}\\q_n\le D}}|c_n|^2.
\end{equation}
Every symbol in this formula has its original zero value on a nonunit.
A fixed restriction of the row set is allowed.  A fixed restriction of
the coefficient support is allowed by extending that coefficient sequence
by zero.  Such a restriction may depend on an object fixed before the row
sum, but the coefficients may not otherwise depend on $k$.
\end{lemma}

\begin{proof}
We follow the higher-order large-sieve recursion of Blomer, Goldmakher,
and Louvel \cite[Section~3]{BGL}, giving the local verification for the
present primary normalization.  In the finite Fourier calculation the
individual residue characters act on elements; their unit-trivial quotient
is the character on ideals to which we apply Poisson summation.

For good ideals $a,b$, represented by their primary generators, put
\[
 F_b(a)=\chi_a(b),\qquad J=\{1,2,4\}.
\]
This uses the original zero-extended symbol, also when $a$ is not
squarefree.  It is multiplicative in both $a$ and $b$.  In every power
below, including a multiple of six, a nonunit still has value zero.
For $M>0$ let
\[
 \mathcal I_M=\{a\text{ good}:M<q_a\le2M\},\qquad
 \mathcal A_M=\{a\in\mathcal I_M:a\text{ squarefree}\}.
\]
For a sequence supported on $\mathcal A_N$ write
$\|\lambda\|_2^2=\sum_{b\in\mathcal A_N}|\lambda_b|^2$, and define
\begin{align*}
 D_j(M,N)&=\sup_{\|\lambda\|_2=1}
  \sum_{a\in\mathcal A_M}
  \left|\sum_{b\in\mathcal A_N}\lambda_bF_b(a)^j\right|^2,\\
 E_j(M,N)&=\sup_{\|\lambda\|_2=1}
  \sum_{a\in\mathcal I_M}
  \left|\sum_{b\in\mathcal A_N}\lambda_bF_b(a)^j\right|^2.
\end{align*}
An empty support gives norm zero.  These norms include all finite ray
classes.  We will prove the symmetric bound for $D_1$; the matrix in the
lemma is its transpose with the two lengths exchanged.

The squarefree norm $D_j$ is our target.  We also need $E_j$, since
Poisson summation produces arbitrary evaluation ideals.  Opening the square in its defining sum
and pairing the column characters will replace a row length $M$ by a dual
length of order $N^2/M$.  A power decomposition then returns the unrestricted
row sum to squarefree norms.  It selects either the first-power or the
second-power factor, so the exponents $j\in\{1,2,4\}$ are kept together:
this set is closed under $j\mapsto2j\pmod6$.  We will use this cycle to
improve a provisional exponent in the bound for $D_j$.

\medskip\noindent\emph{The paired summation formula.}
Use the fixed finite group
\[
 \mathcal T=\ker\bigl((\mathcal O/36\mathcal O)^\times
              \longrightarrow(\mathcal O/3\mathcal O)^\times\bigr),
 \qquad t(b)=b\bmod36\mathcal O.
\]
The primary normalization identifies these with the ray classes modulo
$36\mathcal O$: each unit orbit has a unique representative equal to one
modulo $3$.  In particular $t$ is multiplicative.  This subdivision is
fixed, since the primes above $2$ and $3$ belong to $S$.

For a global unit $u$ and a good prime $p$, reduction modulo $p$ gives
$\chi_p(u)=u^{(q_p-1)/6}$.  Since $q_p\equiv1\pmod6$, multiplication over
the prime factors of $b$ gives
\begin{equation}\label{eq:sextic-sieve-unit-class}
 \chi_b(u)=u^{(q_b-1)/6}.
\end{equation}
The exponent is read modulo six: expanding a product of integers
$1+6h$ proves the equality of exponents.  Thus equal $t$-classes have
the same restriction to the six global units.  They also have the same
residue modulo $4$, which controls the reciprocity factor
$\mathcal R$ of Lemma~\ref{lem:fixed-gauss-phase}.

Let $b,c$ be coprime squarefree good ideals with $t(b)=t(c)$ and let
$j\in J$.  The raw character on elements
\[
 \Theta_{b,c}^{(j)}(z)=\chi_b(z)^j\overline{\chi_c(z)^j}
       \quad (z\bmod bc)
\]
is trivial on global units by Equation~\eqref{eq:sextic-sieve-unit-class}.
It therefore defines a finite-order ray character on ideals by
\[
 \Psi_{b,c}^{(j)}((z))=\Theta_{b,c}^{(j)}(z).
\]
For fractional ideals prime to $bc$, the residue symbols are evaluated
on local unit fractions; the same unit cancellation makes this definition
independent of the generator.  At each good prime the sextic residue
character has exact order six, since the residue multiplicative group is
cyclic.  Hence its powers $j$ and $-j$ are nontrivial.  CRT shows that
$\Theta_{b,c}^{(j)}$ has raw conductor $bc$, and the ray character has the
same conductor: omitting a prime would contradict this nontriviality by
varying only its residue.  It is extended by zero on ideals meeting $bc$.
For the pair $b=c=1$ define $\Psi_{1,1}^{(j)}$ to be principal.

Sextic reciprocity, with its factor fixed by $t(b)=t(c)$, gives for every
good ideal $a$
\begin{equation}\label{eq:sextic-sieve-pair-values}
 \Psi_{b,c}^{(j)}(a)=F_b(a)^j\overline{F_c(a)^j}.
\end{equation}
For coprime inputs the two reciprocity factors cancel; on the other inputs
both sides are zero.  With the Gauss sums already defined in the arithmetic
preliminaries, CRT using $z=cv+bw$ gives
\begin{equation}\label{eq:sextic-sieve-pair-gauss}
 \begin{split}
 \Gamma_{b,c}^{(j)}
 &:=q_{bc}^{-1/2}\sum_{z\bmod bc}\Theta_{b,c}^{(j)}(z)e(z/(bc))\\
 &=\chi_b(c)^j\overline{\chi_c(b)^j}\gamma_j(b)\gamma_{-j}(c)
  =\eta_{t(b),j}\gamma_j(b)\gamma_{-j}(c),
 \end{split}
\end{equation}
where $\eta_{t(b),j}=\mathcal R(b,c)^j$ depends only on the common class.
Every $\gamma_{\pm j}$ here has modulus one by prime Gauss orthogonality
and CRT.  Formula~\eqref{eq:sextic-sieve-pair-gauss} retains
$\gamma_{-j}$ itself, including its factor at $-1$ under conjugation.

Choose once a nonnegative $W\in C_c^\infty((0,\infty))$ with $W\ge1$ on
$[1,2]$.  For the Fourier transform and self-dual measure in the arithmetic
preliminaries, put $w(z)=W(|z|^2)$ and define $\mathscr W$ by
\[
 \widehat w(y)=\mathscr W(|y|^2).
\]
The transform is radial, smooth at zero, and rapidly decreasing.  Thus
\begin{equation}\label{eq:sextic-sieve-transform-bounds}
 \mathscr W(x)\ll_A(1+x)^{-A},\qquad
 \int_{\mathbb R}|\mathcal M\mathscr W(\sigma+it)|\,dt<\infty
       \quad(\sigma>0).
\end{equation}
Here $\mathcal M$ denotes the Mellin transform.  The second assertion
follows by integration by parts in the Mellin integral; all logarithmic
derivatives are bounded at zero and rapidly
decreasing at infinity.  Scaling the Fourier transform gives
$M\mathscr W(M|y|^2)$ for $z\mapsto W(|z|^2/M)$.

For $bc\ne1$, finite Fourier inversion for the primitive raw character is
\[
 \sum_{v\bmod bc}\Theta_{b,c}^{(j)}(v)e(vy/(bc))
 =\overline{\Theta_{b,c}^{(j)}(y)}\sqrt{q_{bc}}\,
       \Gamma_{b,c}^{(j)}.
\]
For a nonunit $y$ the left side vanishes because one nontrivial local
character is summed against the trivial additive character.  For the stated
$e$ and $d\mu$, the trace pairing on the basis $(1,\omega)$ has matrix
$\left(\begin{smallmatrix}0&1\\1&-1\end{smallmatrix}\right)$, of determinant
$-1$, and the $d\mu$-covolume of $\mathcal O$ is one.  Thus the dual of
$bc\mathcal O$ is $(bc)^{-1}\mathcal O$, while the covolume of
$bc\mathcal O$ is $q_{bc}$.
Coset Poisson summation, followed by division by the six generators of
each ideal, therefore gives
\begin{equation}\label{eq:sextic-sieve-pair-poisson}
 \sum_{\mathfrak a\ne0}W(q_{\mathfrak a}/M)\Psi_{b,c}^{(j)}(\mathfrak a)
 =\frac{M\Gamma_{b,c}^{(j)}}{\sqrt{q_{bc}}}
   \sum_{\mathfrak a\ne0}\mathscr W(Mq_{\mathfrak a}/q_{bc})
                 \overline{\Psi_{b,c}^{(j)}(\mathfrak a)}.
\end{equation}
Both sums here are over all nonzero integral ideals.  The division by six
uses unit-triviality of $\Theta_{b,c}^{(j)}$ and radiality of the weight.
The zero frequency vanishes since this character is nontrivial.

\medskip\noindent\emph{Poisson transformation of the squared norm.}
For $\lambda$ supported on one $t$-class in $\mathcal A_N$, set
\[
 Q_j(M,N;\lambda)=
 \sum_{\substack{b,c\in\mathcal A_N\\(b,c)=1}}
   \lambda_b\overline{\lambda_c}
   \sum_{\mathfrak a\ne0}W(q_{\mathfrak a}/M)
                         \Psi_{b,c}^{(j)}(\mathfrak a).
\]
We claim, for every $\varepsilon>0$,
\begin{equation}\label{eq:sextic-sieve-pair-norm}
 \begin{split}
 |Q_j(M,N;\lambda)|
 &\ll_{S,\varepsilon}(2+MN)^\varepsilon\|\lambda\|_2^2\\
 &\quad\times
 \left\{M+\frac MN
   \max_{\substack{1\le H\\
       H\ll_{S,\varepsilon}(2+MN)^\varepsilon N^2/M}}
       E_j(H,N)\right\}.
 \end{split}
\end{equation}
The maximum runs over dyadic shells and is zero if empty.  If $N\le1$,
only the unit good ideal can occur and the annular ideal count proves the
$M$ term.  Also $Q_j=0$ when $M$ is below a fixed positive constant,
because $W$ has a fixed upper support endpoint and nonzero ideals have norm
at least one.

Suppose $N>1$ and normalize $\|\lambda\|_2=1$.  Then every pair in $Q_j$
is nontrivial and $q_{bc}\asymp N^2$.  Apply
Equation~\eqref{eq:sextic-sieve-pair-poisson} and take the triangle
inequality over the dual ideals.  For a small $\delta>0$ put
$Z=(2+MN)^\delta$.  Cauchy gives $(\sum_b|\lambda_b|)^2\ll N$.
By Equation~\eqref{eq:sextic-sieve-transform-bounds}, the part with
$q_{\mathfrak a}>ZN^2/M$ is, for any $A>1$, at most
\[
 M\sum_{q_{\mathfrak a}>ZN^2/M}(Mq_{\mathfrak a}/N^2)^{-A}
 \ll N^2Z^{1-A}.
\]
The same bound holds if the cutoff is below one.  Taking $A$ large makes
this an arbitrary negative power error.

In the remaining sum write $\mathfrak a=a_0\mathfrak s$, with $a_0$ good
and $\mathfrak s$ supported on $S$, and choose one generator $s_0$ for
each fixed $\mathfrak s$.  Equations~\eqref{eq:sextic-sieve-pair-values}
and~\eqref{eq:sextic-sieve-pair-gauss} give the separated factors
\[
 \overline{\Psi_{b,c}^{(j)}(a_0\mathfrak s)}
 =\overline{F_b(a_0)^j}F_c(a_0)^j
       \overline{\chi_b(s_0)^j}\chi_c(s_0)^j.
\]
All $\chi_b(s_0)$ have modulus one.  Split $a_0$ into shells
$a_0\in\mathcal I_H$, where $H\ll ZN^2/(Mq_{\mathfrak s})$, and insert
Mellin inversion for $\mathscr W$ on $\Re w=\delta$.  Apart from a
bounded factor depending on $M,q_{a_0},q_{\mathfrak s}$, the two
coefficient sequences are
\[
 A_b=\lambda_b\gamma_j(b)\overline{\chi_b(s_0)^j}
                    q_b^{-1/2+\delta+it},\qquad
 B_b=\overline{\lambda_b}\gamma_{-j}(b)\chi_b(s_0)^j
                    q_b^{-1/2+\delta+it}.
\]
They are fixed across the $a_0$ sum and have absolute values
$\ll N^{-1/2+\delta}|\lambda_b|$.  Detecting $(b,c)=1$ by M\"obius inversion
and applying Cauchy in $a_0$ gives
\begin{align*}
 &\sum_{a_0\in\mathcal I_H}
  \left|\sum_{\substack{b,c\in\mathcal A_N\\(b,c)=1}}
       A_bB_c\overline{F_b(a_0)^j}F_c(a_0)^j\right|\\
 &\quad\le E_j(H,N)\sum_{\mathfrak d}
   \left(\sum_{\mathfrak d\mid b}|A_b|^2\right)^{1/2}
   \left(\sum_{\mathfrak d\mid b}|B_b|^2\right)^{1/2}
 \ll N^{-1+3\delta}E_j(H,N).
\end{align*}
Here the divisor bound absorbs the sum over $\mathfrak d$.  Conjugating
the entire first polynomial uses the same $E_j$ norm.  The factors with
$H<1$ have just the unit evaluation ideal and satisfy $E_j(H,N)\ll N$.
There are only a fixed power of $\log(2+ZN^2)$ choices of
$\mathfrak s$ and of the dyadic shell, because $S$ is fixed and a nonzero
$Q_j$ has $M$ bounded below.  Equation~\eqref{eq:sextic-sieve-transform-bounds}
integrates the $t$ variable.  Choosing $\delta$ sufficiently small in
terms of $\varepsilon$ proves Equation~\eqref{eq:sextic-sieve-pair-norm},
including the allowed global loss on its $M$ term.

We now apply this paired estimate to $E_j$.  Partition its inner sum by
$t$, insert $W$, and open the square.  Write the two squarefree indices as
$db,dc$, where $d$ is their gcd, $(b,c)=1$, and $(d,bc)=1$.  Their residual
classes agree because $t$ is multiplicative.  Index multiplicativity and
Equation~\eqref{eq:sextic-sieve-pair-values} give on every good $a$
\[
 F_{db}(a)^j\overline{F_{dc}(a)^j}
   =1_{(a,d)=1}\Psi_{b,c}^{(j)}(a).
\]
Let $\mathfrak s_S=\prod_{\mathfrak p\in S}\mathfrak p$.  M\"obius inversion
for $(a,\mathfrak s_Sd)=1$ writes $a=\mathfrak r\mathfrak a$ with
$\mathfrak r\mid\mathfrak s_Sd$.  Choose one generator $r_0$ of each
fixed $\mathfrak r$.  The value of $\Psi$ at $\mathfrak r$ separates as
$\chi_b(r_0)^j\overline{\chi_c(r_0)^j}$, so the common residual sequence is
\[
 \lambda'_b=\lambda_{db}1_{(b,d)=1}\chi_b(r_0)^j,
 \qquad |\lambda'_b|\le|\lambda_{db}|.
\]
It is supported on $\mathcal A_{N/q_d}$ in one class.  Apply
Equation~\eqref{eq:sextic-sieve-pair-norm} with lengths $M/q_{\mathfrak r}$
and $N/q_d$.  The summed coefficient mass is at most
\[
 \sum_d\sum_{\mathfrak r\mid\mathfrak s_Sd}\sum_b|\lambda_{db}|^2
 \le d_{\mathcal O}(\mathfrak s_S)
       \sum_m d_{\mathcal O}(m)^2|\lambda_m|^2
 \ll_{S,\delta}(2+N)^\delta\|\lambda\|_2^2.
\]
Writing $\Delta_{\mathrm r}=q_{\mathfrak r}$ and $\Delta_{\mathrm d}=q_d$, we have
$1\le\Delta_{\mathrm d}\le2N$ and $1\le\Delta_{\mathrm r}\le q_{\mathfrak s_S}\Delta_{\mathrm d}$.
We obtain the recursive interface
\begin{equation}\label{eq:sextic-sieve-norm-reflection}
 \begin{split}
 E_j(M,N)
 &\ll_{S,\varepsilon}(2+MN)^\varepsilon
 \max_{\substack{1\le\Delta_{\mathrm d}\le2N\\
                 1\le\Delta_{\mathrm r}\le q_{\mathfrak s_S}\Delta_{\mathrm d}}}
 \left\{\frac M{\Delta_{\mathrm r}}\right.\\
 &\qquad\left.+
 \frac{M\Delta_{\mathrm d}}{N\Delta_{\mathrm r}}
 \max_{\substack{1\le H\\
       H\ll_{S,\varepsilon}(2+MN)^\varepsilon
              N^2\Delta_{\mathrm r}/(M\Delta_{\mathrm d}^2)}}
       E_j(H,N/\Delta_{\mathrm d})\right\}.
 \end{split}
\end{equation}
The maxima may be restricted to nonempty ranges.  This is the combination
of the gcd and Poisson steps in \cite[Lemmas~3.2--3.3]{BGL}; every sequence
used here is common within its norm sum, and $S$ remains fixed.

\medskip\noindent\emph{The initial and power bounds.}
Row inclusion gives $D_j\le E_j$.  Hilbert space duality and sextic
reciprocity, splitting one matrix variable by its finite $t$-class, give
\begin{equation}\label{eq:sextic-sieve-norm-symmetry}
 D_j(M,N)\ll_S D_j(N,M).
\end{equation}
The phases are unit-valued and the noncoprime entries are zero in both
orientations.  We also have the initial estimate
\begin{equation}\label{eq:sextic-sieve-norm-initial}
 D_j(M,N)\ll_S M^2+N\qquad(M,N\ge1).
\end{equation}
Indeed, for the squarefree row $a$, finite Fourier inversion gives on
every $b$, including nonunits,
\[
 \chi_a(b)^j=\frac1{\sqrt{q_a}\gamma_{-j}(a)}
       \sum_{x\bmod a}\overline{\chi_a(x)^j}e(xb/a).
\]
Cauchy costs at most $\#(\mathcal O/(a))^\times/q_a\le1$.  The reduced
fractions $x/a$ from $a\in\mathcal A_M$ are distinct modulo $\mathcal O$
and separated by
at least $1/(2M)$: a nonzero numerator of $x/a-y/c-h$ has absolute value
at least one, while $|ac|\le2M$; equality modulo $\mathcal O$ forces the
same primary denominator and residue.  The dual of
Lemma~\ref{lem:planar-additive-sieve}, with the coefficient generators in
the ball $q_b\le2N$, proves Equation~\eqref{eq:sextic-sieve-norm-initial}.

We need the following near-monotonicity, also used in the large-sieve
recursions of \cite[Lemma~4.4 and Remark~5]{GL} and \cite[Lemma~3.1]{BGL}:
\begin{equation}\label{eq:sextic-sieve-norm-monotone}
 D_j(M_1,N)\ll_S D_j(M_2,N)
 \quad\text{if }M_2\ge C M_1\log(2M_1N),\quad M_1,M_2,N\ge1.
\end{equation}
Here is a direct verification.  The assertion is immediate for a zero norm.
Otherwise choose a maximizing unit coefficient vector for $D_j(M_1,N)$,
and split the row shell at $3M_1/2$; one part
carries at least half its mass.  Put $L=M_2/M_1$.  For the first part use
good prime ideals of norms in $(L,4L/3]$, and for the second use norms in
$(2L/3,L]$.  Then $ap\in\mathcal A_{M_2}$ when $p\nmid a$.  The fixed-field
prime ideal theorem supplies $P\gg_S L/\log L$ such primes.  An ideal of
norm at most $2x$ contains at most
$d_x=\log(2x)/\log(2L/3)$ of them.  For
$S_a(v)=\sum_bv_bF_b(a)^j$ and $p\nmid a$, multiplicativity gives
\[
 S_a(\lambda)=S_{ap}\bigl((\lambda_b\overline{F_b(p)^j})_b\bigr)
                  +S_a\bigl((\lambda_b1_{p\mid b})_b\bigr).
\]
The first coefficient is zero at $p\mid b$ and cancels the factor at $p$
otherwise.  Squaring and summing over the chosen rows and primes gives
\[
 \frac{P-d_{M_1}}2D_j(M_1,N)
 \le2P D_j(M_2,N)+2d_ND_j(M_1,N).
\]
For the stated threshold with $C$ sufficiently large,
$d_{M_1}+4d_N\le P/2$, proving
Equation~\eqref{eq:sextic-sieve-norm-monotone}.

We finish with the power decomposition and recursion of
\cite[Lemmas~3.4--3.5]{BGL}.  Suppose, simultaneously for $j\in J$, that
for every positive loss
\begin{equation}\label{eq:sextic-sieve-inductive-bound}
 D_j(M,N)\ll_{S,\varepsilon}(MN)^\varepsilon
       \{M^\alpha+N+(MN)^{2/3}\},\qquad \alpha>4/3,
\end{equation}
for $M,N\ge1$; the unit shells satisfy the same bound directly.  The
initial estimate gives this with $\alpha=2$.  Write a good evaluation ideal
uniquely as
\[
 a=a_1a_2^2a_3^3a_4^4a_5^5a_6^6,
\]
where $a_1,\ldots,a_5$ are pairwise coprime and squarefree, and $a_6$ is
arbitrary.  Insert
$a_i\in\mathcal I_{X_i}$ and put $X=\prod_iX_i$; here $X_i\ge1/2$ and
$\prod_iX_i^i\asymp M$.  Choose $\ell\in\{1,2\}$ with
$X_\ell=\max(X_1,X_2)$.  After the other factors are fixed, the coefficient
\[
 \lambda'_b=\lambda_b\prod_{i\ne\ell}F_b(a_i)^{ij}
\]
has absolute value at most $|\lambda_b|$, and the remaining kernel is
exactly that of $D_{j\ell\bmod6}(X_\ell,N)$.  Enlarging its fixed positive
row restriction gives this norm bound.  This includes the zero of the
sixth power factor.  The set $J$ is closed under $j\mapsto j\ell\bmod6$.
Counting the fixed ideals and forgetting their coprimalities in positive
sums gives
\begin{equation}\label{eq:sextic-sieve-norm-power}
 E_j(M,N)\ll_{S,\varepsilon}(MN)^\varepsilon
 \max_{\boldsymbol X}
 \min\left\{M^{1/3},(M/X_\ell)^{2/3}\right\}
             D_{j\ell\bmod6}(X_\ell,N).
\end{equation}
To verify the factor, the number of fixed choices is $O(X/X_\ell)$.
For $\ell=1$ it is at most a constant times both
$(M/X_1)^{1/2}\ll(M/X_1)^{2/3}$ and
$(MX_2/X_1)^{1/3}\le M^{1/3}$; for $\ell=2$ it is at most a constant
times $(MX_1^2/X_2^2)^{1/3}\le M^{1/3}\ll(M/X_2)^{2/3}$.
The number of boxes is a fixed power of $\log(2+M)$.

Substituting Equation~\eqref{eq:sextic-sieve-inductive-bound} into
Equation~\eqref{eq:sextic-sieve-norm-power} yields
\[
 E_j(M,N)\ll_{S,\varepsilon}(MN)^\varepsilon
                    (M^\alpha+M^{1/3}N).
\]
The three terms before simplification are bounded by $M^\alpha$,
$M^{1/3}N$, and $(MN)^{2/3}$; the last is bounded by one of the first two
according as $N\le M$ or $N\ge M$.  Use this estimate in
Equation~\eqref{eq:sextic-sieve-norm-reflection}.  All auxiliary nonempty
scales are bounded by fixed powers of $2+MN$: here $N/\Delta_{\mathrm d}\ge1/2$ and
$H(N/\Delta_{\mathrm d})\ll_{S,\varepsilon}(2+MN)^{3+\varepsilon}$.
The unit column shell satisfies the same estimate directly.  Thus
preliminary losses can be chosen smaller to give any specified final loss.
The two scale factors are
\[
 \Delta_{\mathrm r}^{\alpha-1}\Delta_{\mathrm d}^{1-2\alpha}
       \ll_S\Delta_{\mathrm d}^{-\alpha}\le1,
 \qquad (\Delta_{\mathrm r}\Delta_{\mathrm d})^{-2/3}\le1.
\]
Consequently, for $M,N\ge1$,
\begin{equation}\label{eq:sextic-sieve-norm-recurrence}
 D_j(M,N)\le E_j(M,N)\ll_{S,\varepsilon}(MN)^\varepsilon
       \{M+N^{2\alpha-1}M^{1-\alpha}+(MN)^{2/3}\}.
\end{equation}

Put $h=2-3/(3\alpha-1)=(2\alpha-5/3)/(\alpha-1/3)$ and
$\beta=2-2/(3\alpha-1)$.  If $M\ge N^h$,
the middle term in Equation~\eqref{eq:sextic-sieve-norm-recurrence} is at
most $(MN)^{2/3}$.  If $M<N^h$, apply
Equation~\eqref{eq:sextic-sieve-norm-monotone} with a fixed multiple of
$N^h\log(2MN)$ in place of $M$.  Equation~\eqref{eq:sextic-sieve-norm-recurrence}
there gives $(MN)^\varepsilon N^\beta$, since
$2(h+1)/3=\beta$ and $h<\beta$.  Thus in both cases
\[
 D_j(M,N)\ll_{S,\varepsilon}(MN)^\varepsilon
                   \{M+N^\beta+(MN)^{2/3}\}.
\]
Symmetry gives Equation~\eqref{eq:sextic-sieve-inductive-bound} with
$\beta$ in place of $\alpha$, simultaneously for every $j\in J$.
Starting from $2$, the map $\alpha\mapsto2-2/(3\alpha-1)$ decreases to
$4/3$.  For a requested loss take finitely many iterations until the
remaining exponent gap is smaller than a fixed fraction of that loss, and
absorb it into $(MN)^\varepsilon$.  Combining the bound and its symmetric
form, using
$M^{4/3}\le(MN)^{2/3}$ when $M\le N$ and the analogous inequality when
$N\le M$, proves
\begin{equation}\label{eq:sextic-sieve-norm-shell}
 D_j(M,N)\ll_{S,\varepsilon}(MN)^\varepsilon
               \{M+N+(MN)^{2/3}\}\qquad(j\in J).
\end{equation}

Dyadic subdivision of both norm balls, with Cauchy across the column
shells, converts Equation~\eqref{eq:sextic-sieve-norm-shell} to the same
bound for the ball matrix with entries $F_b(a)=\chi_a(b)$.  The logarithmic
factors are absorbed by a smaller preliminary loss, and the unit shells
are bounded directly.  The matrix in Equation~\eqref{eq:sextic-large-sieve}
for $\varsigma=1$ is the transpose of this matrix at lengths $(D,K)$.
A complex matrix and its transpose have the same operator norm, by the
adjoint identity followed by whole-vector conjugation.  The bound is
symmetric in the two lengths, so it gives precisely
$K+D+(KD)^{2/3}$.  Conjugating the entire inner sum gives
$\varsigma=-1$, with unchanged zeros.  Finally, a fixed row restriction
removes nonnegative terms, and a fixed coefficient restriction is imposed
by zero extension.  This proves all assertions of the lemma.
\end{proof}

\subsection{Physical rows and their inverse witnesses}

With the sieve proved, it remains to apply it to the detector witnesses.
We apply the lemma only to a squarefree factor of the physical row.  The
separation of squarefree and powerful row factors is also used in the
proof of Corollary 1.4 of \cite{BGL}; here we keep the actual norm of the
powerful factor because it controls both the moment and the number of rows.

\begin{proposition}[Sextic-sieve row envelope]\label{prop:sextic-row-count}
Fix the arithmetic data $\mathcal A$, the bounded physical range
$d_{\min}\le d\le d_{\max}$, and the height and loss hypotheses of
Lemmas~\ref{lem:buffered-bins} and~\ref{lem:detector-dyads}.  Put
$U=Z^d$ and retain the notation $T_1$ of those lemmas.  Let $\mathcal B$
be any set of retained sixth-power-free element rows $u$ with
$q_u\asymp U$, all in one bin $(i,a)$ with $a>51/100$.  Put
$\delta=2a-1$, so $1/50<\delta\le1$.

For every $\epsilon>0$, after reducing the preliminary losses in
Proposition~\ref{prop:detector-witness}, for all sufficiently large $Z$,
\begin{equation}\label{eq:sextic-row-count}
 \#\mathcal B\ll_{\mathcal A,\epsilon}
 U^{R(\delta)+\epsilon}(1+T_1)^{A_{\mathcal A}},
 \qquad
 R(\delta)=\min\left\{1,
       \max\left(1-\frac\delta2,\frac43-\delta\right)\right\}.
\end{equation}
The finite order $A_{\mathcal A}$ is uniform in the physical dyad, the bin,
and all moving rows.  It may be fixed before the positive exponent in
$T_1=Z^\tau$ is chosen.  The estimate is valid for each fixed retained
tuple of external parameters and for the original row restrictions.  The
implied constant may depend on the fixed comparison constants and finitely
many of the shared profile seminorms.  The rowwise profile choices are
precisely those permitted by
Proposition~\ref{prop:detector-witness} and
Lemma~\ref{lem:smooth-calculus}; no arbitrary row-dependent coefficients
are allowed.

The floor bin $a=51/100$ is excluded from the witness assertion.  For that
bin the elementary bound $\#\mathcal B\ll_{\mathcal A}U$ applies; it agrees
with the numerical value $R(1/50)=1$.
\end{proposition}

\begin{proof}
Fix a small preliminary loss $\epsilon_0>0$.  Apply
Proposition~\ref{prop:detector-witness} with $t=1$ to every row in
$\mathcal B$.  Subdivide by its presentation labels in
$\psi_{u,\nu,\varsigma}(n)=\nu(n)\chi_n(u)^{\varsigma}$ and its dyadic
pair of polynomial lengths.  There are a fixed finite number of
presentation labels and $O_{\mathcal A}((\log U)^2)$ pairs.  Within one such
subdivision, the inverse length $D=U^r$ and the pair $(\nu,\varsigma)$
are fixed.  In this subdivision write
\[
 M_u(D;W)=D^{-1/2}\sum_n\mu(n)\nu(n)\chi_n(u)^{\varsigma}W(q_n/D)
\]
for the inverse polynomial of that presentation.  The witness gives
\begin{equation}\label{eq:sextic-witness-spike}
 \frac12-O(\epsilon_0)\le r\le1+O(1/\log U),
 \qquad |M_u(D;W_u)|^2\gg_{\mathcal A,\epsilon_0}
       U^{\delta r-\epsilon_0}.
\end{equation}
The notation $W_u$ retains the permitted rowwise norm-profile parameters.
The polynomial here is the original one for the displayed presentation,
not the polynomial of its primitive inducing character.  We first bound
it when those parameters are fixed, and then handle their rowwise choice.

For each sixth-power-free row, factor its ideal uniquely as
\[
 (u)=\mathfrak s\mathfrak w\mathfrak k,
 \quad
 \mathfrak s=\prod_{p\in S}p^{v_p(u)},\quad
 \mathfrak w=\prod_{\substack{p\notin S\\2\le v_p(u)\le5}}p^{v_p(u)},
 \quad
 \mathfrak k=\prod_{\substack{p\notin S\\v_p(u)=1}}p.
\]
Thus $\mathfrak k$ is squarefree, $(\mathfrak k,\mathfrak w)=1$, and
$\mathfrak w$ is powerful, meaning that each of its positive prime
valuations is at least two.  There are only finitely many possibilities
for $\mathfrak s$, because its valuations lie in $\{0,\ldots,5\}$ on
the fixed set $S$.  Choose one generator $s_{\mathfrak s}$ for each of
these ideals.  Writing $w,k$ for the primary generators of
$\mathfrak w,\mathfrak k$, there is a unique unit $\upsilon$ such that
\[
 u=\upsilon s_{\mathfrak s}wk.
\]
We fix $\mathfrak s$ and $\upsilon$ whenever a row sum is taken.

Insert dyadic ranges $q_{\mathfrak w}\asymp V$, including the unit ideal
in the bounded range $V=1$.  Only $1\le V\ll_{\mathcal A}U$ can occur,
and there are $O_{\mathcal A}(\log U)$ such ranges.  The original annulus
$q_u\asymp U$ gives the actual residual annulus
\begin{equation}\label{eq:sextic-actual-row-annulus}
 q_{\mathfrak k}=\frac{q_u}{q_{\mathfrak s}q_{\mathfrak w}}
       \asymp_{\mathcal A}\frac UV.
\end{equation}
We keep this restriction when forming the sum.  It will be enlarged to
the ball $q_k\ll_{\mathcal A}U/V$ only when applying a positive row bound.
Let $\mathcal U_V$ denote the rows of the current witness subdivision in
this powerful-part dyad, with all original physical restrictions retained.

The number of powerful ideals of norm at most $V$ is $O(V^{1/2})$.
Indeed, every powerful ideal has a unique expression
$\mathfrak v^2\mathfrak y^3$ with $\mathfrak y$ squarefree; the two
factors need not be coprime.  The ideal count from the arithmetic
preliminaries gives
\[
 \#\{\mathfrak w:q_{\mathfrak w}\le V,\ \mathfrak w\ {\rm powerful}\}
 \ll V^{1/2}\sum_{\mathfrak y}q_{\mathfrak y}^{-3/2}
 \ll V^{1/2}.
\]
The last series converges by the same ideal count.  For each fixed
$\mathfrak w$ in its dyad, the ball containing
Equation~\eqref{eq:sextic-actual-row-annulus} has $O_{\mathcal A}(U/V)$
ideals.  A nonempty annulus has $U/V$ bounded below by a fixed positive
constant, which also covers the unit residual ideal.  Since the
factorization of $(u)$ and its unit are unique, the number of rows in
this powerful-part dyad satisfies
\begin{equation}\label{eq:sextic-stratum-cardinality}
 \#\mathcal U_V\ll_{\mathcal A}U V^{-1/2}.
\end{equation}
Discarding any of the restrictions defining $\mathcal U_V$ in this upper
count can only add rows.

For fixed $\mathfrak w,\mathfrak s,\upsilon$, put
$v=\upsilon s_{\mathfrak s}w$.  Multiplicativity in the numerator gives
the exact identity
\begin{equation}\label{eq:sextic-frozen-mask}
 \chi_n(u)^{\varsigma}
   =\chi_n(v)^{\varsigma}\chi_n(k)^{\varsigma}.
\end{equation}
It includes the zero extensions: if $n$ meets either factor, both sides
are zero, and otherwise it is ordinary multiplicativity.  In particular,
the inverse polynomial for a fixed profile $W$ is the row sum in
Lemma~\ref{lem:sextic-large-sieve} with coefficients
\[
 c_n=D^{-1/2}\mu(n)\nu(n)\chi_n(v)^{\varsigma}W(q_n/D).
\]
The factor $\mu(n)$ keeps the columns squarefree.  The factor
$\chi_n(v)^{\varsigma}$ retains every original zero at $(n,v)>1$ and is
fixed across the residual row $k$.  The only remaining zero depending on
both $n$ and $k$ is the zero of the displayed sieve kernel itself.  The
restriction $(k,w)=1$, and any restriction inherited from the physical
row set for fixed external parameters, is a fixed row restriction in this
application.  We do not enlarge $S$ to contain the primes of $w$.
Consequently the constant in the sieve is independent of $w$, even though
its fixed coefficient mask may have large norm.

The common annular support and ideal counting give, uniformly for the
fixed profile parameters,
\[
 \sum_n|c_n|^2
 \le D^{-1}\sum_{q_n\asymp D}|W(q_n/D)|^2\ll_{\mathcal A}1.
\]
The same statement holds for each of the finitely many profile derivatives
used below, with the corresponding fixed seminorm.  Apply
Lemma~\ref{lem:sextic-large-sieve} after enlarging the positive residual
row sum from its actual annulus to $q_k\ll_{\mathcal A}U/V$.  A fixed
enlargement makes the row bound at least one on every nonempty quotient
range and includes the annular column support.  Summing the
result over the $O(V^{1/2})$ choices of $\mathfrak w$ and the finitely
many choices of $\mathfrak s,\upsilon$ yields
\begin{equation}\label{eq:sextic-stratum-fixed-moment}
 \sum_{u\in\mathcal U_V}|M_u(D;W)|^2
 \ll_{\mathcal A,\epsilon_1}(UD)^{\epsilon_1}V^{1/2}
 \left\{\frac UV+D+\left(\frac{UD}{V}\right)^{2/3}\right\}.
\end{equation}
All coefficient sequences in this derivation are fixed within the row
sum to which the sieve is applied.  They may differ for different frozen
values of $\mathfrak w$, which are summed only after those positive bounds.

We now restore the rowwise choice of $W_u$ in
Equation~\eqref{eq:sextic-witness-spike}.  Here the detector was used with
$t=1$, so $D_*=U$.  After the presentation and dyadic pair are fixed, its
inverse profiles have the form
\[
 W_{\rm dyad}(x)V_{\le}(Dx/U)x^{-\sigma-i\omega},
 \qquad \omega=\gamma-\nu_F,
\]
where $W_{\rm dyad}$ is the fixed smooth dyadic cutoff and $\nu_F$ is
the Fourier frequency from the detector.
The factor $V_{\le}(Dx/U)$ is common to the row sum; the varying parameters
are $\sigma\in[51/100,1]$ and $|\omega|\le(3i+1)T_1$.
Differentiating in either parameter inserts a power of $\log x$, which is
bounded on the fixed annulus.  Cover these two parameter ranges by unit boxes.  There are at most a fixed power of
$1+T_1$ such boxes.  The parameter Sobolev inequality in
Lemma~\ref{lem:smooth-calculus} bounds the supremum on a box by a finite
sum of integrals of squared profile derivatives.  Sum over the rows before
these integrals.  At each fixed parameter value, differentiation changes
only the fixed profile coefficient, inserting the allowed logarithmic
weights or profile derivatives.  Equation~\eqref{eq:sextic-stratum-fixed-moment}
therefore applies to every such integral.  This proves
\begin{equation}\label{eq:sextic-stratum-moment}
 \sum_{u\in\mathcal U_V}|M_u(D;W_u)|^2
 \ll_{\mathcal A,\epsilon_1}(UD)^{\epsilon_1}(1+T_1)^{A_{\mathcal A}}
 V^{1/2}\left\{\frac UV+D+
       \left(\frac{UD}{V}\right)^{2/3}\right\}.
\end{equation}
The derivative order and hence $A_{\mathcal A}$ are fixed before choosing
$\tau$.  The height intervals and the total allowance for additional
frequencies are exactly those in the shared detector estimates; the
Sobolev step does not assign a new frequency allowance or change a contour.
It does not permit a coefficient to be selected separately for each $k$.

It remains to combine this moment with the independent count
Equation~\eqref{eq:sextic-stratum-cardinality}.  Write
\[
 V=U^o,\qquad D=U^r,\qquad
 e_o(r)=\frac o2+\max\left\{1-o,r,\frac{2(1-o+r)}3\right\}.
\]
These are the parameters of the actual dyads.  A nonempty dyad has
$0\le o\le1+O_{\mathcal A}(1/\log U)$; the unit dyad has $o=0$.
Since the witness lengths stay bounded, the arbitrarily small power of
$UD$ in Equation~\eqref{eq:sextic-stratum-moment} can be written as an
arbitrarily small power of $U$.  Dividing that equation by the spike in
Equation~\eqref{eq:sextic-witness-spike}, and also using
Equation~\eqref{eq:sextic-stratum-cardinality}, gives for this subdivision
\begin{equation}\label{eq:sextic-stratum-envelope}
 \#\mathcal U_V
 \ll U^{\min\{1-o/2,e_o(r)-\delta r\}+O(\epsilon_0)+\epsilon_1}
       (1+T_1)^{A_{\mathcal A}},
\end{equation}
after decreasing the sieve loss denoted by $\epsilon_1$ if necessary.

We compute the largest exponent in this expression first on
$0\le o\le1$ and $1/2\le r\le1$.  Put $x=1-o$ and $A=(1+x)/2$.  Then
\[
 e_o(r)-\delta r
 =\max\left\{
 A-\delta r,\quad
 \frac{1-x}{2}+(1-\delta)r,\quad
 \frac12+\frac x6+\left(\frac23-\delta\right)r
 \right\}.
\]
The minimum of $A$ with a maximum of three real numbers is the maximum
of its minima with those numbers.  The first such minimum is
$A-\delta r\le1-\delta/2$.  For fixed $r$, the second is the minimum of
an increasing and a decreasing affine function of $x$.  They cross at
$x=(1-\delta)r\in[0,1]$, so its maximum over $x$ is
\[
 \frac{1+(1-\delta)r}{2}\le1-\frac\delta2.
\]
Both functions in the third minimum increase with $x$.  Its maximum over
$x$ is therefore
\[
 \min\left\{1,\frac23+\left(\frac23-\delta\right)r\right\}.
\]
The affine expression in $r$ takes the values $1-\delta/2$ and
$4/3-\delta$ at the two endpoints.  It follows, and equality is attained
at one of those endpoints, that
\begin{equation}\label{eq:sextic-envelope-optimization}
 \max_{\substack{0\le o\le1\\1/2\le r\le1}}
 \min\{1-o/2,e_o(r)-\delta r\}
 =\min\left\{1,\max\left(1-\frac\delta2,\frac43-\delta\right)\right\}.
\end{equation}
This computation is uniform for $0\le\delta\le1$.

The functions just used are maxima and minima of affine functions whose
slopes are uniformly bounded on a fixed neighborhood of these compact
ranges.  Replacing the actual $o$ by its nearest point in $[0,1]$, and
the actual witness $r$ by its nearest point in $[1/2,1]$, changes the
exponent by at most $O_{\mathcal A}(\epsilon_0+1/\log U)$, by
Equation~\eqref{eq:sextic-witness-spike} and the actual dyad bounds.
Thus this replacement is made only in the final optimization, not in the
row sum or its sieve application.  Choose $\epsilon_0$ and the preliminary
sieve loss small enough in terms of the prescribed $\epsilon$.  The
$O((\log U)^2)$ witness pairs and $O(\log U)$ powerful-part dyads cost
only another arbitrarily small power.  Summing
Equation~\eqref{eq:sextic-stratum-envelope} proves
Equation~\eqref{eq:sextic-row-count}.

Finally, the bin $a=51/100$ need not contain an actual zero, so no use of
Proposition~\ref{prop:detector-witness} is made there.  Counting all
elements in its physical annulus gives $O_{\mathcal A}(U)$ rows.  Since
$1/50<1/3$, the displayed formula for $R$ gives $R(1/50)=1$, as claimed.
\end{proof}

For use in the high estimate, the row envelope has the explicit form
\[
 R(\delta)=
 \begin{cases}
  1,&0\le\delta\le1/3,\\
  4/3-\delta,&1/3\le\delta\le2/3,\\
  1-\delta/2,&2/3\le\delta\le1.
 \end{cases}
\]
The argument uses only the inverse witness.  No estimate for a product
with the plain witness is needed in this row count.

\section{Analytic estimates for the high expansion}
\label{sec:shared-analytic-estimates}

The row envelope now enters the Poisson representation of the base probe.
For the first-stage application, retain the supposition
$\Delta_1=\beta_*-11/12>0$ and set
$X=Y=Z^{1/2}$.  Write $s$ for the first Mellin variable called $x$ in
Section~\ref{sec:local-euler}.  Equations~\eqref{eq:probe-poisson-identity}
and~\eqref{eq:probe-high} give the actual integral to be estimated:
\[
\begin{split}
 I_\eta(Z^{1/2},Z^{1/2},Z)
 =\frac1{(2\pi i)^3}\int_{(3)}\int_{(3)}\int_{(2)}
 &\mathcal W(Z^{1/2},Z^{1/2},Z;s,w,z)\\
 &\cdot\sum_u^{(6)}q_u^{-z}\overline{\xi(u)}
 \frac{\zeta_F^S(6z)L^S(w,\chi_\bullet(u))}
      {L^S(s,\eta\overline{\chi_\bullet(u)})}
 \mathcal H_{\eta,u}(s,w,z)\,dz\,dw\,ds.
\end{split}
\]
Here
\[
 \mathcal W(X,Y,Z;s,w,z)=X^{1/2-z}Z^{s+z-1}Y^{w-1}
 e^{(s+z-1)^2}M(z)\widehat W_1(w).
\]
The rows are the nonzero sixth-power-free elements, with their unit factors
and the physical restriction $(u,S)=1$.  Both character presentations retain
their original zeros at primes dividing $u$.  The coefficientwise calibration
of the base probe is already included in this identity.

The row $u=1$ contains the reciprocal of the target function.  For the intermediate row norms we will keep one buffered bin fixed, move its integral to
\[
 \Re s=a+16e,\qquad \Re w=1-a-6e,\qquad \Re z=17/50,
\]
and apply the envelope $R(2a-1)$ there.  Apart from small real losses and a fixed height factor, the numerator on
this contour costs $U^{a-1/2}$ for a row norm $q_u\asymp U$; the row envelope controls how many
such terms occur.  We first identify the principal signal, then carry out
this contour move and its concrete first-stage estimate.  Principal residues
and direct bounds for the outer row norms complete the comparison.

\subsection{The principal row and the normalization}

We identify the row whose numerator is principal.  A physical row
with a prime factor outside $S$ has nonprincipal numerator by
\hyperref[par:physical-row-ramification]{the ramification argument in
Section~\ref*{sec:detector}}.  Because the physical mask imposes $(u,S)=1$,
the remaining rows are units.  If a unit $u\ne1$ had a sixth root in $F$,
that root would have valuation zero at every prime, hence would be a unit
of $F$; every unit of $F$ has sixth power one.  Thus
$F(u^{1/6})/F$ is nontrivial.  It is finite Galois because $F$ contains
the sixth roots of unity.  Chebotarev applied to a nonidentity Frobenius
class supplies a prime outside the fixed set $S$ where the associated
sextic character is nontrivial \cite[Theorem 1.1]{TZ}.  By the Kummer
description in Lemma~\ref{lem:fixed-numerator-ray}, this is
$\chi_\bullet(u)$.  Therefore $u=1$ is the sole principal numerator row.
A denominator attached to a bounded unit row may be principal; its
reciprocal has a zero at one and will be kept on a global line below.

Define
\begin{equation}
 H_\eta(s)=\mathcal H_{\eta,1}(s,1,1/6),\qquad
 c_S=\widehat W_1(1)M(1/6)
       \left(\operatorname*{Res}_{v=1}\zeta_F^S(v)\right)^2/6.
 \label{eq:shared-principal-data}
\end{equation}
Lemma~\ref{lem:local-euler} makes $H_\eta$ holomorphic on $\Re s>7/8$.
Its uniform estimate $H_\eta=1+O(P_0^{-4/5})$ permits a choice of $P_0$
before the target, because the local bound is uniform in the target unit
phases.  Fix $P_0$ large enough that
\begin{equation}
 \sup_{\Re s>7/8}|H_\eta(s)-1|\le\frac12.
 \label{eq:shared-principal-nonvanishing}
\end{equation}
Enlarging $S$ by these primes leaves the fixed ray group $T$ unchanged,
as the calibration in the local identity holds coefficientwise.  The
excluded set and hence the particular function $H_\eta$ may differ between
applications, but within one application they are exactly those of
the exact high representation being used.

The constant $c_S$ is positive.  Indeed $\widehat W_1(1)>0$ because
$W_1$ is nonnegative and nonzero, and $M(1/6)>0$ by
Equation~\eqref{eq:radial-mellin-positive}.  The simple pole of
$\zeta_F^S$ has positive residue: deletion multiplies the positive residue
of $\zeta_F$ by $\prod_{p\in S}(1-q_p^{-1})>0$.

For the first stage put
\[
 C_{\mathrm I}(s)=s-\frac23,\qquad
 J_{\mathrm I,\eta}(Z)=\frac{I_\eta(Z^{1/2},Z^{1/2},Z)}{c_S}.
\]
Let $f_{\mathrm I,\eta}$ be the integral in
Equation~\eqref{eq:common-residue} with $C=C_{\mathrm I}$,
$\mathcal S=S$, and the function $H_\eta$ just defined.  The two scalar poles
at $w=1$ and $z=1/6$ will give $c_S f_{\mathrm I,\eta}$; our remaining
objective is a common power saving for every other term, measured relative
to $Z^{C_{\mathrm I}(\beta_*)}$.

\subsection{The exact identity used in contour moves}

The contour proof uses the full holomorphic correction in the displayed
integral.  To reuse that proof with the compensated probe, we record the
identity and bounds that it requires.  In the present application the
correction is $\mathcal H_{\eta,u}$, independent of $Z$, and the parameters
below are $l_x=l_y=h=1/2$ and $\ell=0$.

The scalar quotient in Equation~\eqref{eq:probe-high} is the part of the
high expansion that determines both the contour move and the principal
residues.  We keep that quotient fixed and state explicitly what may be
changed around it.  Put
\[
 \begin{split}
 \mathcal D_1(\epsilon_0)=\{(s,w,z):{}&\Re s\ge51/100,\quad
       \Re z\ge17/50,\quad \Re w\ge-1/100,\\
       &\Re(s+w)\ge1+\epsilon_0\},\qquad \epsilon_0>0,\\
 \mathcal D_2=\{(s,w,z):{}&\Re s\ge7/8,\quad
       \Re z\ge33/200,\quad \Re w\ge19/20\}.
 \end{split}
\]
These are the two regions in Lemma~\ref{lem:local-euler}.  A real box below
means a compact rectangular subset of $\mathbb R^3$ for
$(\Re s,\Re w,\Re z)$.

\begin{definition}[Data for an exact high representation]
\label{def:stage-high-data}
Fix real numbers $l_x,l_y>0$ and $\ell\ge0$ such that
\begin{equation}
 X=Z^{l_x},\qquad Y=Z^{l_y},\qquad h=1-l_x+\ell>0,
 \qquad
 C(s)=s+\frac{l_x}{2}-1+\frac h6
     =s-\frac56+\frac{l_x}{3}+\frac\ell6.
 \label{eq:stage-mellin-exponent}
\end{equation}
These real numbers are fixed before the target character.  For each
primitive finite-order target $\eta$, fix the data $S,\xi,W_0,W_1$ of
Section~\ref{sec:probe}, independently of $Z$.  Data for an exact high
representation consist of the following objects and assertions.

First, $\mathscr I_\eta(Z)$ is an independently specified complex-valued
quantity for every sufficiently large real $Z$.  In each application it
is given by a finite linear combination, for that $Z$, of the completed
sums in Equation~\eqref{eq:general-probe}, allowing specified restrictions
on the completed index $A$ and specified rescalings of its three positive
scales.  The number of terms and coefficients in this finite combination
may depend on $Z$.  Any direct estimate for $\mathscr I_\eta$ refers to
this expression, not to a separately defined high integral.

Second, for every nonzero sixth-power-free element $u$ with $(u,S)=1$ and
every such $Z$, there is a function
$\mathfrak H_{\eta,u,Z}(s,w,z)$.  It is holomorphic on a neighborhood of
each point of $\mathcal D_2$ and of every $\mathcal D_1(\epsilon_0)$.
For every $D>0$ and every real box $\mathcal K$ contained in one of these
regions, there are finite constants $B_{\eta,\mathcal K,D}$ and
$J_{\eta,\mathcal K,D}$ such that, uniformly for
$1\le q_u\le Z^D$, all imaginary parts, and real parts in $\mathcal K$,
\begin{equation}
 |\mathfrak H_{\eta,u,Z}(s,w,z)|
 \ll_{\eta,\mathcal K,D}
 Z^{B_{\eta,\mathcal K,D}}
 (1+|\Im s|+|\Im w|+|\Im z|)^{J_{\eta,\mathcal K,D}}.
 \label{eq:stage-all-height-majorant}
\end{equation}
The constants and exponents are independent of $Z,u$ and of any later
order of integration by parts.  No nonvanishing of $\mathfrak H$ or of
any of its local factors is assumed.

Finally, the following identity is asserted for the independently given
$\mathscr I_\eta(Z)$, with absolute convergence of the sum and integrals
on the displayed lines:
\begin{equation}
\begin{split}
 \mathscr I_\eta(Z)=\frac1{(2\pi i)^3}
 \int_{(3)}\int_{(3)}\int_{(2)}
 &\mathcal W(X,Y,Z;s,w,z)
 \sum_u^{(6)}q_u^{-z}\overline{\xi(u)}\\
 &\cdot
 \frac{\zeta_F^S(6z)L^S(w,\chi_\bullet(u))}
      {L^S(s,\eta\overline{\chi_\bullet(u)})}
 \mathfrak H_{\eta,u,Z}(s,w,z)\,dz\,dw\,ds,
\end{split}
\label{eq:stage-high-identity}
\end{equation}
where the sum has the physical restrictions of
Equation~\eqref{eq:probe-poisson-identity}, and
\[
 \mathcal W(X,Y,Z;s,w,z)
 =X^{1/2-z}Z^{s+z-1}Y^{w-1}
       e^{(s+z-1)^2}M(z)\widehat W_1(w).
\]
Thus Equation~\eqref{eq:stage-high-identity} is a hypothesis to be proved
for the physical expression, not its definition.
\end{definition}

For the base probe, the physical expression is
$I_\eta(Z^{l_x},Z^{l_y},Z)$, $\ell=0$, and
$\mathfrak H_{\eta,u,Z}=\mathcal H_{\eta,u}$.
Equations~\eqref{eq:probe-poisson-identity} and~\eqref{eq:probe-high}
prove the required identity, and Lemma~\ref{lem:local-euler} gives the
holomorphy and Equation~\eqref{eq:stage-all-height-majorant}.  More
generally, the function $\mathfrak H$ in the definition always means the
\emph{full correction after the displayed scalar quotient is removed}.
If that correction is a sum of products of local factors, the products
themselves must be holomorphic in the stated regions.  Writing a quotient
by an individual local factor at its zeros does not establish this
condition.  The decompositions used only to estimate a retained contour
will be stated separately below.

\subsection{External tails on vertical lines and horizontal joins}

The product of our three Mellin tests decays in three independent height
directions.  The following estimate turns that decay into bounds both off a
large box and on the horizontal sides of a contour rectangle.

\begin{lemma}[External integrated and trace tails]
\label{lem:external-contour-tails}
Let $m\ge2$, let $\mathcal K$ be a compact set of real contour parameters,
and, for $\boldsymbol r\in\mathcal K$ and $Z\ge2$, let
$K_{\boldsymbol r,Z}$ be a nonnegative Borel measurable kernel on
$\mathbb R^m$.  Suppose it is jointly Borel measurable in
$(\boldsymbol r,\boldsymbol t)$ and, for every integer $L\ge0$,
\[
 K_{\boldsymbol r,Z}(\boldsymbol t)
 \ll_L(1+|\boldsymbol t|)^{-L}
\]
uniformly in $\boldsymbol r,Z,\boldsymbol t$.  Let
$\mathcal D\subseteq\mathbb R^m$ be Borel measurable, and suppose a Borel
measurable factor $F_Z$ satisfies on $\mathcal D$
\begin{equation}
 |F_Z(\boldsymbol t)|\le C Z^B(1+|\boldsymbol t|)^J,
 \label{eq:external-arithmetic-majorant}
\end{equation}
where $B,J,C$ are fixed independently of $Z$ and of the integer $N$ below.
Write $\widehat{\boldsymbol t}_j$ for the vector obtained by omitting the
$j$th coordinate of $\boldsymbol t$.  Uniformly for $T\ge1$ and
$\boldsymbol r\in\mathcal K$,
\begin{align}
 \int_{\mathcal D\cap\{\max_j|t_j|>T\}}
 K_{\boldsymbol r,Z}(\boldsymbol t)|F_Z(\boldsymbol t)|\,d\boldsymbol t
 &\ll_N Z^B T^{-N},
 \label{eq:external-integrated-tail}\\
 \int_{\mathcal D\cap\{t_j=T\}}
 K_{\boldsymbol r,Z}(\boldsymbol t)|F_Z(\boldsymbol t)|\,
 d\widehat{\boldsymbol t}_j
 &\ll_N Z^B(1+|T|)^{-N}.
 \label{eq:external-trace-tail}
\end{align}
The second assertion holds for either sign of $T$ and every $j$, using
coordinate Lebesgue measure on $t_j=T$.  Both assertions remain valid after
integration over a real contour interval of bounded length when the
hypotheses are uniform there and the kernels, domain indicators and
arithmetic factors are jointly Borel measurable in that real parameter and
the remaining coordinates.
\end{lemma}

\begin{proof}
Multiply the kernel bound by
Equation~\eqref{eq:external-arithmetic-majorant}.  Integrating
$(1+|\boldsymbol t|)^{J-L}$ outside the box proves
Equation~\eqref{eq:external-integrated-tail} when $L>N+J+m$.
On $t_j=T$, integration in the other $m-1$ coordinates gives
\[
 \int_{\mathbb R^{m-1}}
 (1+|T|+|\widehat{\boldsymbol t}_j|)^{J-L}
 d\widehat{\boldsymbol t}_j
 \ll(1+|T|)^{J-L+m-1}.
\]
Taking $L>N+J+m-1$ proves Equation~\eqref{eq:external-trace-tail}.
Restricting to $\mathcal D$ decreases these positive integrals, and a bounded
real interval contributes only its length.
\end{proof}

For our triple integral, take
\[
 (y_1,y_2,y_3)=(\Im(s+z),\Im z,\Im w),\qquad
 K_{\boldsymbol r,Z}(\boldsymbol t)
 =e^{-y_1^2}|M(r_z+iy_2)|\,|\widehat W_1(r_w+iy_3)|.
\]
The change of height variables has determinant one.  On a fixed real box
with $r_z$ in a compact subinterval of $(0,\infty)$, the Mellin estimates of
Section~\ref{sec:poisson-representation} give arbitrary polynomial decay of $M$ and
$\widehat W_1$.  The Gaussian has the same property.  Their product is
therefore $O_L((1+|\boldsymbol y|)^{-L})$ for every $L$, hence also
$O_L((1+|\boldsymbol t|)^{-L})$.  The omitted factor
$e^{(\Re(s+z)-1)^2}$ is bounded on that real box.

After the $w$ residue only $s,z$ remain.  The same proof applies to the
kernel
\begin{equation}
 K^{(2)}_{\boldsymbol r,Z}(\boldsymbol t)
 =e^{-y_1^2}|M(r_z+iy_2)|,
 \qquad(y_1,y_2)=(\Im(s+z),\Im z).
 \label{eq:external-two-dimensional-kernel}
\end{equation}
This supplies the trace bound needed for the subsequent $z$-join.

The same kernel test handles additional external separating variables.
For $m\ge3$, let $\boldsymbol y=A\boldsymbol t$ with a fixed
$A\in\mathrm{GL}_m(\mathbb R)$, and multiply the three-factor kernel by a
Borel density $|v_{\boldsymbol r,Z}(y_4,\ldots,y_m)|$, jointly Borel in
$(\boldsymbol r,\boldsymbol v)$, satisfying
\begin{equation}
 \sup_{\boldsymbol r,Z,\boldsymbol v}
 (1+|\boldsymbol v|)^L|v_{\boldsymbol r,Z}(\boldsymbol v)|<\infty
 \quad\text{for every integer }L\ge0.
 \label{eq:external-density-contract}
\end{equation}
For $m=3$ the density is one on the zero-dimensional space.  The product is
again rapidly decreasing in all $m$ coordinates, since $A$ and its inverse
are fixed.  For the two-factor kernel, the identical argument uses a density
on $\mathbb R^{m-2}$ and $m\ge2$, with density one when $m=2$.
Thus both product-kernel forms have the integrated and trace bounds above.

Uniform annular profiles separated by logarithmic Fourier inversion
satisfy Equation~\eqref{eq:external-density-contract} by repeated
integration by parts, as in Lemma~\ref{lem:smooth-calculus}.  Translate every
pure norm twist into its Mellin argument first.  The additional height
coordinates then append a block triangular matrix with fixed inverse to
the three-dimensional change of variables.

An internal coefficient measure known only through a fixed weighted $L^1$
norm is not thereby covered by the pointwise density hypothesis or by its
trace conclusion.  Such a measure remains integrated at its already fixed
moment order inside the factor $F_Z$.  Any additional external coordinate
on which a trace estimate is used must separately satisfy
Equation~\eqref{eq:external-density-contract}.

The domain $\mathcal D$ in the lemma is important.  Suppose a reciprocal
is bounded only while the imaginary part of its argument
$\gamma+\sum_j a_jt_j$ lies in a buffered interval.  A coordinate with
$a_j\ne0$ may be integrated only over the range that preserves this
condition; the lemma does not extend the reciprocal bound beyond it.
Coordinates with $a_j=0$ may be extended when the other factors satisfy
Equation~\eqref{eq:external-arithmetic-majorant} there.  All factors on an
extended axis must use their global or absolute estimates, not an estimate
valid only on a retained interval.  In particular, an integrated tail alone
does not justify a horizontal join: that join uses
Equation~\eqref{eq:external-trace-tail}.

Increasing $N$ in this lemma increases only the decay orders of the external
smooth tests.  It does not differentiate $F_Z$.  Thus a fixed moment-profile
order or a fixed arithmetic height exponent in $F_Z$ is unchanged when $N$
is chosen later.  This is the same order distinction made in
Lemma~\ref{lem:smooth-calculus}.

\subsection{Moving a fixed bin}

We first move the full row integral.  The row envelope will be applied only
after this step, on the retained contour.

\begin{lemma}[Contour transformation for a fixed bin]
\label{lem:fixed-bin-contour}
Assume Definition~\ref{def:stage-high-data}.  Fix
$\sigma_0\in[7/8,1)$ with $\beta_*>\sigma_0$, put $z_0=17/50$, and let
$U=Z^d$ with $0<d_{\min}\le d\le d_{\max}<\infty$.
Let $\mathcal B$ be a finite set of retained physical rows $q_u\asymp U$
in one bin $(i,a)$ of Lemma~\ref{lem:buffered-bins}, fixed before any
Mellin variable is moved.  The comparison constants in this annulus are
fixed.  Take $0<e<10^{-3}$ and $T_1=Z^\tau>2$, where
$0<\tau\le d_{\min}/100$.  The real ranges and $e$ are independent of the
target.

Fix positive constants $c_s,c_w,c_z\le1/2$, independently of $Z$ and the
rows.  Retain the original heights
$|\Im s|\le c_sT_1$, $|\Im w|\le c_wT_1$ and
$|\Im z|\le c_zT_1$.  The contribution of $\mathcal B$ on the starting
lines in Equation~\eqref{eq:stage-high-identity} equals its integral on
these retained segments of
\begin{equation}
 \Re s=a+16e,\qquad \Re w=1-a-6e,\qquad \Re z=z_0,
 \label{eq:stage-central-contours}
\end{equation}
up to $O_{\eta,N}(Z^{B_\eta}T_1^{-N})$ for every fixed $N$.  The finite
exponent $B_\eta$ is fixed before $N$.  An empty row set contributes zero.
The transformation uses the full holomorphic correction; no decomposition
of that correction or subdivision depending on a contour point is needed.
\end{lemma}

\begin{proof}

Section~\ref{sec:detector} gives $a\le\beta_*$, including the floor bin.
Isolate the finite sum over $\mathcal B$
on the absolute lines in Equation~\eqref{eq:stage-high-identity}.  Move
$z$ from $2$ to $z_0$, keep $w=3$, and move $s$ from $3$ to
$\beta_*+20e$.  These moves stay in $\mathcal D_1(\epsilon_0)$ for a fixed
positive $\epsilon_0$, the two numerator arguments remain to the right
of their poles, and the reciprocal stays in $\Re s>\beta_*$.  Its global
bound follows from Lemmas~\ref{lem:logarithmic-control}
and~\ref{lem:deleted-euler-factors}, using the conductor and deletion
radical bounds in Section~\ref{sec:detector}.  The correction obeys
Equation~\eqref{eq:stage-all-height-majorant}.  Lemma~\ref{lem:external-contour-tails}
with $T\to\infty$ therefore justifies these moves.

Now move $w$ from $3$ to $1-a-6e$ while $s$ stays on that global line.
Throughout the move,
\[
 \Re(s+w)\ge1+(\beta_*-a)+14e\ge1+14e.
\]
The numerator character of every retained row is nonprincipal, so its
$L$-function is entire.  Also $\Re w\ge-6e>-1/100$ and
$\Re z=z_0$.  Thus the correction remains holomorphic in
$\mathcal D_1(\epsilon_0)$, for example with $\epsilon_0=10e$, and no
pole is crossed.  On a $w$-join the reciprocal is still global.  The
global upper strip bound for the numerator, the deleted-factor bound and
Equation~\eqref{eq:stage-all-height-majorant} give a majorant of the form
Equation~\eqref{eq:external-arithmetic-majorant} on the axes integrated
there.  The trace estimate makes the joins tend to zero.

Before moving $s$ farther, restrict the three original Mellin heights to the stated box.  On the discarded
part keep $s$ on $\Re s=\beta_*+20e$.  There the global reciprocal bound,
the global numerator bound, the deleted factors and the all-height
correction bound give $Z^{B_\eta}$ times a fixed polynomial in the heights:
the row range is bounded by $q_u\ll Z^{d_{\max}}$, and the number of rows
is $O(U)$.  The integrated tail in
Lemma~\ref{lem:external-contour-tails} gives
$O_{\eta,N}(Z^{B_\eta}T_1^{-N})$.

Move only the retained $s$ segment to $a+16e$.  On this rectangle,
\[
 \Re(s+w)\ge1+10e,
\]
and the other inequalities defining $\mathcal D_1(10e)$ still hold.
The real part of the reciprocal argument is at least $a+16e>a+6e$.
The stated height box places both scalar arguments
strictly inside $|\Im v|\le(3i+2)T_1$ for sufficiently large $Z$.
Lemma~\ref{lem:buffered-bins} therefore excludes its zeros throughout
the retained rectangle.  Since $6z_0>1$, the scalar zeta factor has no
pole on this move.  Thus no pole is crossed.

For an $s$-join, its imaginary coordinate is fixed at a constant multiple
of $T_1$.  The scalar denominator in Equation~\eqref{eq:stage-high-identity}
depends only on $s$, so its buffered estimate remains valid when the
$\Im z$ and $\Im w$ integrations are extended to their whole axes.
On those extended axes use the global numerator estimate and
Equation~\eqref{eq:stage-all-height-majorant}.  The trace estimate then gives
$O_{\eta,N}(Z^{B_\eta}T_1^{-N})$.  No $w$- or $z$-join in this argument
extends an $s$-axis that has been moved into a merely buffered region.

\end{proof}

The exact high representation is a triple integral with a majorant for its
full correction.  We can therefore perform this transformation before
introducing any auxiliary separating variable.  If a central estimate later
uses such variables, their retained domains and discarded parts must be
justified in that estimate; they are not new coordinates of the contour
identity just proved.

\subsection{Applying the row envelope}

For the base correction, Lemma~\ref{lem:local-euler} verifies the hypotheses
of Lemma~\ref{lem:fixed-bin-contour}.  Indeed
$|\mathcal H_{\eta,u}|\ll_\epsilon q_u^\epsilon$ uniformly in all three
heights in both Euler regions; for $q_u\le Z^D$, taking $\epsilon=1$ gives
Equation~\eqref{eq:stage-all-height-majorant} with $B=D$ and height order
zero.  We now estimate the retained integral, using
$l_x=l_y=h=1/2$ and $\ell=0$ throughout this subsection.

Fix
\[
 d_{\min}=\frac1{63},\qquad \zeta=\frac1{1000},\qquad
 d_{\max}=h+\zeta=\frac{501}{1000}.
\]
For a dyad $U=Z^d$ in this range, fix one dynamic bin $(i,a)$ before moving
any Mellin variable, and put $\delta=2a-1$.  Proposition~\ref{prop:sextic-row-count}, with requested loss
$\epsilon_r>0$, gives its cardinality at most
$U^{R(\delta)+\epsilon_r}(1+T_1)^{A_\eta}$ up to a fixed constant.
For the floor this is the direct count $O_\eta(U)$, since
$R(1/50)=1$; it uses no witness.

The numerator $L^S(w,\chi_\bullet(u))$ is precisely the original
zero-extended presentation with $\nu=1$ in Section~\ref{sec:detector}.
On the retained line $\Re w=1-a-6e$, Lemma~\ref{lem:buffered-bins} bounds it
by
$U^{\delta/2+12e+\epsilon_n}(3+T_1)^C$ for any $\epsilon_n>0$.
This also holds in the floor bin.  The central point lies in
$\mathcal D_1(10e)$, so the stronger local estimate in
Lemma~\ref{lem:local-euler} gives
$|\mathcal H_{\eta,u}|\ll_{e,\epsilon_H}U^{\epsilon_H}$ there, for any
$\epsilon_H>0$.  Consequently
\[
 \sum_{u\in\mathcal B}
 |L^S(w,\chi_\bullet(u))\mathcal H_{\eta,u}(s,w,z)|
 \ll_\eta
 U^{R(\delta)+\delta/2+12e+\epsilon_r+\epsilon_n+\epsilon_H}
 (1+T_1)^{A'_\eta}.
\]
Here $A'_\eta$ is finite and is fixed before the final external tail order.

Put $\varepsilon_c=12e+\epsilon_r+\epsilon_n+\epsilon_H$.
The buffered reciprocal costs $U^{\varepsilon_d}$ for any
$\varepsilon_d>0$, and $\zeta_F^S(6z_0)$ is absolutely bounded.
The tests have bounded joint $L^1$ norm in the three independent height
coordinates above.  The outside powers, including $q_u^{-z_0}$, are
\[
 Z^{a/2-3/4+z_0/2+13e}U^{-z_0}.
\]
Multiplying these bounds gives, for every $\varepsilon_p>0$, a retained
bin contribution at most
\begin{equation}
 \ll_\eta Z^{1/4+E_{\mathrm I}(d)+13e
       +(1+d)\varepsilon_c+d\varepsilon_d+\varepsilon_p}
 (1+T_1)^{A'_\eta}.
 \label{eq:balanced-central-contribution}
\end{equation}
Here we have harmlessly enlarged the bound by
$Z^{\varepsilon_c+\varepsilon_p}$ so that the losses use the same form as
the later general estimate, and
\[
 E_{\mathrm I}(d)=-\frac34+\frac{\delta+R(\delta)}2
 +\left(d-\frac12\right)
       \left(R(\delta)+\frac\delta2-\frac{17}{50}\right).
\]
There is no additional Mellin integral in this application.  Its scalar
buffered arguments have heights $\Im s$ and $\Im w$.  The witness
frequencies and profile choices in the row count stay within the detector's
existing allowance and do not enter either scalar argument.

At $d=h=1/2$, the three pieces of the row envelope give, respectively,
\[
 E_{\mathrm I}(h)=
 \begin{cases}
  -1/4+\delta/2,&0\le\delta\le1/3,\\
  -1/12,&1/3\le\delta\le2/3,\\
  -(1-\delta)/4,&2/3\le\delta\le1.
 \end{cases}
\]
The first two pieces are at most $-1/12$.  In the third, the exact bin
ceiling from Section~\ref{sec:detector} is
\[
 \delta\le2\beta_*-1=\frac56+2\Delta_1.
\]
It follows in every piece that
\[
 E_{\mathrm I}(h)-\Delta_1
 \le-\frac1{24}-\frac{\Delta_1}{2}.
\]
This is the comparison with $C_{\mathrm I}(\beta_*)$ required here;
$E_{\mathrm I}(h)$ itself need not be negative.
The frequency slope has the three forms
\[
 R(\delta)+\frac\delta2-\frac{17}{50}
 =\begin{cases}
  33/50+\delta/2,&0\le\delta\le1/3,\\
  149/150-\delta/2,&1/3\le\delta\le2/3,\\
  33/50,&2/3\le\delta\le1.
 \end{cases}
\]
It therefore lies in $[33/50,62/75]$.  Positivity controls all $d\le h$,
and the extension to $d\le h+\zeta$ costs at most $(62/75)\zeta$.
Thus, before adjustable losses, every central dyad has saving at least
\[
 m_c=\frac1{24}-\frac{62}{75}\zeta=\frac{1021}{25000}
\]
relative to $C_{\mathrm I}(\beta_*)$.

\subsection{Central estimates for a correction split into pieces}

The preceding calculation multiplied one aggregate absolute row bound by
the outside Mellin powers.  We record its form when the full correction is
estimated by several pieces and the available bound varies between sets of
rows.  The contour has already been justified for the fixed bin; these sets
will be used only to estimate its retained integrand.

\begin{lemma}[A retained integral and its exponent]
\label{lem:bin-contour-accounting}
Assume the data of Definition~\ref{def:stage-high-data}.  Fix
$\sigma_0\in[7/8,1)$ with $\beta_*>\sigma_0$, and put $z_0=17/50$.
Let $0<d_{\min}<d_{\max}<\infty$, $U=Z^d$ with
$d_{\min}\le d\le d_{\max}$, and let $\mathcal B$ be a finite set of
physical rows $u$ with $q_u\asymp U$ in one retained bin $(i,a)$ of
Lemma~\ref{lem:buffered-bins}.  The comparison constants in $q_u\asymp U$
are fixed.  Put $\delta=2a-1$, take $0<e<10^{-3}$ as in that lemma, and
let $T_1=Z^\tau>2$ with $0<\tau\le d_{\min}/100$.
The real ranges and $e$ are chosen independently of the target.

The bin $\mathcal B$ is fixed before any Mellin variable is moved.  On the retained central contours of
Equation~\eqref{eq:stage-central-contours}, suppose that there is a decomposition
\[
 \mathfrak H_{\eta,u,Z}(s,w,z)
       =\sum_{\lambda\in\Lambda}
             \mathfrak H^{(\lambda)}_{\eta,u,Z}(s,w,z),
\]
where $\Lambda$ is finite, nonempty and fixed before $Z$.  This equality is required
only on the retained contours; the summands need not be holomorphic away
from them.  For each $\lambda$ and each retained $(s,w,z)$, suppose
$\mathcal B$ is partitioned into at most $C(1+\log Z)^D$ sets
$\mathcal B_{\lambda,\nu}(s,w,z)$, with fixed $C,D$.  For each set at each
retained point let $R,g$ be real numbers in a fixed bounded range.  Assume
that, for fixed
$\varepsilon_c\ge0$ and a finite $A_\eta\ge0$,
\begin{equation}
 \sum_{u\in\mathcal B_{\lambda,\nu}(s,w,z)}
  \left|L^S(w,\chi_\bullet(u))
       \mathfrak H^{(\lambda)}_{\eta,u,Z}(s,w,z)\right|
 \ll_\eta U^{R+\delta/2+\varepsilon_c}
 Z^{\ell(z_0-1/2)+g+\varepsilon_c}(1+T_1)^{A_\eta}.
 \label{eq:stage-central-bound}
\end{equation}
The bound is uniform in the retained point, row set and moving labels.
The number $\varepsilon_c$ and the bounded range for $R,g$ are fixed before
the target; $R,g$ may depend on $d,a$, the pointwise set and the global number
$\beta_*$, but not otherwise on $\eta$.  Let $\mathcal P_Z$ be the set of
all pairs $(R,g)$ that occur at any retained point in this decomposition.

Here ``retained'' has the following precise height meaning.  List once all
external coordinates $v_1,\ldots,v_m$ that enter a buffered $L$-value,
reciprocal, or logarithmic derivative in the verification of
Equation~\eqref{eq:stage-central-bound}, and include all three original
Mellin heights, with zero coefficients when they do not occur in an
argument.  After pure-twist translations, every such argument
has imaginary part
$\gamma_j+\sum_{k=1}^m a_{jk}v_k$, where the finite matrix $(a_{jk})$ is
fixed and $|\gamma_j|\le(3i+1)T_1$.  Restrict
\begin{equation}
 |v_k|\le c_kT_1,\qquad
 c_k=\frac1{2m(1+\max_j|a_{jk}|)}.
 \label{eq:stage-height-allocation}
\end{equation}
Coordinates with zero coefficients may be restricted by the same rule.
Then every added height is at most $T_1/2$ in total.  Fixed bounded
enlargements are included by increasing the lower threshold for $Z$.
If extra Mellin integrals are used to establish
Equation~\eqref{eq:stage-central-bound}, that inequality is required for
its full left side after those integrations.  Any discarded parts must have
been left on their global or absolute lines, bounded by
Lemma~\ref{lem:external-contour-tails} or
Lemma~\ref{lem:smooth-calculus}, and included in the displayed bound before
the hypothesis is asserted.  The allowance $T_1/2$ is not renewed at
successive estimates.

The common ideal exponent associated with a pair $(R,g)$ is
\begin{equation}
 E_{\sigma_0}(d;R,g)=a-\sigma_0+h(z_0-1/6)-a l_y-\ell/2+g
                  +d(R+\delta/2-z_0).
 \label{eq:stage-high-exponent}
\end{equation}
If $\mathcal B$ is empty its contribution is zero.  Otherwise put
\[
 E^{\max}_{\sigma_0,Z}(d)
   =\sup_{(R,g)\in\mathcal P_Z}E_{\sigma_0}(d;R,g).
\]
For every $\varepsilon_d,\varepsilon_p>0$, the total contribution of
$\mathcal B$ on the central contours is, for sufficiently large $Z$,
\begin{equation}
 \begin{split}
 \ll_\eta{}& Z^{C(\sigma_0)+E^{\max}_{\sigma_0,Z}(d)+(16-6l_y)e}\\
 &\cdot Z^{(1+d)\varepsilon_c+d\varepsilon_d+\varepsilon_p}
       (1+T_1)^{A_\eta}.
 \end{split}
 \label{eq:stage-central-contribution}
\end{equation}
The supremum is finite because the pairs lie in a fixed bounded range.
The power $\varepsilon_p$ absorbs the pointwise logarithmic multiplicity.
No separate integral for a pointwise piece or row set is asserted.
All discarded portions and horizontal joins in this contour move are
$O_{\eta,N}(Z^{B_\eta}T_1^{-N})$ for every fixed $N$, where $B_\eta$ is
finite and is fixed before $N$.
\end{lemma}

\begin{proof}
Apply Lemma~\ref{lem:fixed-bin-contour} to the original triple integral,
using for its three box constants the corresponding $c_k$ in
Equation~\eqref{eq:stage-height-allocation}.  These constants are positive
and at most $1/2$.  The full correction and the fixed bin therefore give
the retained integral and the asserted errors before any pointwise pieces
are introduced.  Any auxiliary integrations used for the central bound
are subject to the additional hypotheses in the statement.

Now use the central decomposition and form the
pointwise sets $\mathcal B_{\lambda,\nu}$.  At each retained point apply
the triangle inequality to the original row sum, and then apply
Equation~\eqref{eq:stage-central-bound} to the sets present at that point.
There are at most a fixed multiple of $(1+\log Z)^D$ such sets pointwise.
Bound each of their exponents by the supremum over $\mathcal P_Z$ and
integrate only the original holomorphic row sum.  Thus neither measurability
of an individual pointwise subdivision nor one common label set across
contour points is needed.  On the retained contours,
Lemma~\ref{lem:buffered-bins} bounds the reciprocal by
$O_{\eta,e,\varepsilon_d}(U^{\varepsilon_d})$.  The tests have a uniformly
bounded joint $L^1$ norm, since the transformation
$(\Im s,\Im z,\Im w)\mapsto(\Im(s+z),\Im z,\Im w)$ is invertible.
The scalar factor $\zeta_F^S(6z_0)$ is absolutely bounded.  It remains to
compute the real power of $Z$.

The outside factor, including $q_u^{-z_0}$ and the real displacements in
Equation~\eqref{eq:stage-central-contours}, contributes
\[
 l_x(1/2-z_0)+a+z_0-1-a l_y-dz_0+(16-6l_y)e.
\]
Equation~\eqref{eq:stage-central-bound} adds
$d(R+\delta/2+\varepsilon_c)+\ell(z_0-1/2)+g+\varepsilon_c$.
The reciprocal adds $d\varepsilon_d$.  Subtracting
$C(\sigma_0)=\sigma_0+l_x/2-1+h/6$ and using
$h=1-l_x+\ell$ gives Equation~\eqref{eq:stage-high-exponent} and the
remaining terms in Equation~\eqref{eq:stage-central-contribution}.
The pointwise logarithmic multiplicity is bounded by
$Z^{\varepsilon_p}$ for sufficiently large $Z$.  This proves the claim.
\end{proof}

The lemma separates two uses of the correction.  Its holomorphy and
all-height majorant justify the contour move before the rows are subdivided
by their pointwise sizes.  The pieces in
Equation~\eqref{eq:stage-central-bound} are used only after the move; they
may be defined using local divisions whose nonvanishing has been proved on
that retained region.  Such a division supplies no continuation or tail
bound outside that region.  For the base correction there is only one
piece, and Lemma~\ref{lem:buffered-bins} supplies the reflected numerator
factor $U^{\delta/2+12e+\epsilon}$ appearing in the central hypothesis.

\subsection{Extracting the principal signal}

The intermediate-row estimate leaves the principal row and the two outer
norm ranges.  We first cross the two scalar poles in the principal row.
The following formulation also records the precise condition under which
a different correction has the same target signal.

\begin{lemma}[Extraction of the principal signal]
\label{lem:principal-residue-interface}
Assume Definition~\ref{def:stage-high-data} and
Equation~\eqref{eq:shared-principal-nonvanishing}.  Fix
$\sigma_0\in[7/8,1)$ with $\beta_*>\sigma_0$ and $0<e<10^{-3}$.
Suppose that for every $\epsilon>0$, uniformly in all imaginary parts on
\[
 \Re s=\beta_*+e,\qquad 19/20\le\Re w\le1+e,
 \qquad 33/200\le\Re z\le1/6+e,
\]
one has
\begin{equation}
 |\mathfrak H_{\eta,1,Z}(s,w,z)|
 \ll_{\eta,e,\epsilon} Z^{\ell\Re z+\epsilon}
  (1+|\Im s|+|\Im w|+|\Im z|)^{J_{\eta,e,\epsilon}},
 \label{eq:principal-correction-bound}
\end{equation}
with finite $J_{\eta,e,\epsilon}$ fixed before any external tail order.
Suppose also that $A_\eta(Z)\ne0$ for every sufficiently large real $Z$,
that $|A_\eta(Z)|^{-1}\ll_{\eta,\epsilon}Z^\epsilon$ for every
$\epsilon>0$, and that on $\Re s=\beta_*+e$, uniformly in $\Im s$,
\begin{equation}
 \mathfrak H_{\eta,1,Z}(s,1,1/6)
 =H_\eta(s)Z^{\ell/6}A_\eta(Z)(1+\mathcal R_{\eta,Z}(s)).
 \label{eq:principal-correction-residue}
\end{equation}
Here either $\mathcal R_{\eta,Z}$ is identically zero, or there is a number
$\mu>0$, chosen independently of $\eta$, such that
$|\mathcal R_{\eta,Z}(s)|\ll_\eta Z^{-\mu}$ on that entire line.

Let $\mathscr P_\eta(Z)$ be the $u=1$ term of
Equation~\eqref{eq:stage-high-identity}, and define
\[
 f_\eta(Z)=\frac1{2\pi i}\int_{(2)}
    Z^{C(s)}e^{(s-5/6)^2}\frac{H_\eta(s)}{L_F^S(s,\eta)}\,ds.
\]
Then $c_S>0$, and, for every $\epsilon>0$,
\begin{align}
 \frac{\mathscr P_\eta(Z)}{c_SA_\eta(Z)}-f_\eta(Z)
 \ll_{\eta,e,\epsilon}{}&
 Z^{C(\beta_*)+(1+h)e-l_y/20+\epsilon}
 +Z^{C(\beta_*)+e-h/600+\epsilon}\notag\\
 &+Z^{C(\beta_*)+e-\mu+\epsilon}.
 \label{eq:principal-remainder-bound}
\end{align}
The last term is omitted when $\mathcal R_{\eta,Z}$ is identically zero.
The implied constants and lower thresholds may depend on the target, but
the displayed real exponents do not.
\end{lemma}

\begin{proof}
For $u=1$, the scalar factor in the high identity is
\[
 \frac{\zeta_F^S(6z)\zeta_F^S(w)}{L_F^S(s,\eta)}.
\]
Isolate this term on the absolute contours.  Move to
$\Re s=\beta_*+e$, $\Re w=1+e$ and $\Re z=1/6+e$.
The reciprocal stays in $\Re s>\beta_*$, and both zeta arguments stay
to the right of one.  The paths lie in $\mathcal D_2$, so the correction
is holomorphic.  The all-height majorant and
Lemma~\ref{lem:external-contour-tails} justify the horizontal limits.

Move $w$ to $19/20$, crossing its simple pole at $w=1$.  In that residue
move $z$ to $33/200=1/6-1/600$, crossing its simple pole at $z=1/6$.
The unresidued $w$ integral keeps $\Re z=1/6+e$.  All these paths lie in
$\mathcal D_2$, and $M(z)$ is holomorphic for $\Re z>0$.
Thus the two stated scalar poles are the only poles crossed.  The reciprocal
remains on its global line throughout, even when $\eta$ is principal.
Every extended vertical $w$- or $z$-axis used to bound a tail lies strictly
on one side of its scalar pole; $w=1$ and $z=1/6$ occur only as residues,
and the horizontal joins have large nonzero height.  On those axes the
zeta functions have fixed polynomial bounds.  Consequently the integrated
and trace estimates of Lemma~\ref{lem:external-contour-tails} apply without
an unremoved pole in their majorants.
After the $w$ residue, its Mellin factor has become the constant
$\widehat W_1(1)$.  The remaining heights use the two-dimensional kernel
in Equation~\eqref{eq:external-two-dimensional-kernel}, with
$(y_1,y_2)=(\Im(s+z),\Im z)$; its trace estimate justifies the subsequent
$z$-join.

On the principal contours, Equation~\eqref{eq:principal-correction-bound}
and the outside powers give the raw exponent
\[
 \frac{l_x}{2}+\Re s-1+h\Re z+l_y(\Re w-1).
\]
For the unresidued $w$ integral this is
$C(\beta_*)+(1+h)e-l_y/20$; for the leftover $z$ integral in the
$w$ residue it is $C(\beta_*)+e-h/600$.  The reciprocal on
$\Re s=\beta_*+e$ has an arbitrarily small power of its fixed target
conductor and a fixed polynomial in height by
Lemmas~\ref{lem:logarithmic-control} and~\ref{lem:deleted-euler-factors}.
The tests integrate those height powers.  Requesting the small powers in
the correction and in $A_\eta^{-1}$ to be sufficiently small gives the
first two terms of Equation~\eqref{eq:principal-remainder-bound}.

The product of the scalar residues is $c_S$: the residue of
$\zeta_F^S(6z)$ at $1/6$ is one sixth of the residue of $\zeta_F^S(v)$
at one.  Positivity of $c_S$ was proved when the normalization was defined.

At the double residue the outside power is
\[
 X^{1/3}Z^{s-5/6}Z^{\ell/6}=Z^{C(s)},
 \qquad \Phi(s+1/6-1)=e^{(s-5/6)^2}.
\]
Equation~\eqref{eq:principal-correction-residue} therefore makes the
normalized double residue
\[
 \frac1{2\pi i}\int_{(\beta_*+e)}
 Z^{C(s)}e^{(s-5/6)^2}\frac{H_\eta(s)}{L_F^S(s,\eta)}
          (1+\mathcal R_{\eta,Z}(s))\,ds.
\]
When present, the error factor is estimated on this line; the global
reciprocal bound and Gaussian give
$O_{\eta,e,\epsilon}(Z^{C(\beta_*)+e-\mu+\epsilon})$.
No continuation of that error factor is required.

Move only the main integral right to $\Re s=2$.  The reciprocal is
holomorphic for $\Re s>\beta_*$ by the definition of $\beta_*$ and
absolute Euler convergence beyond one.  At a principal pole its reciprocal
has a zero, so that case adds no residue.  The function $H_\eta$ is
holomorphic there by Lemma~\ref{lem:local-euler}, and is bounded there by
Equation~\eqref{eq:shared-principal-nonvanishing}.  The global reciprocal
estimate is polynomial in
height uniformly on the fixed real strip, while the Gaussian is
$O(e^{-(\Im s)^2})$.  Rectangular contours therefore have vanishing
horizontal sides, and the shifted integral is exactly $f_\eta(Z)$.
This also covers $\beta_*=1$.  The three remainders give the asserted bound.
\end{proof}

The function $H_\eta$ in this lemma is the correction of the unmodified
scalar Euler product at the double residue.  A different full correction
is permitted only when it verifies
Equation~\eqref{eq:principal-correction-residue} with a nonzero normalizer.
For the base correction that equation holds with $A_\eta=1$ and
$\mathcal R_{\eta,Z}=0$.  The normalized physical quantity in any
application is $\mathscr I_\eta/(c_SA_\eta)$; the same normalizer must be
used for its direct estimate and for this principal comparison.  The raw
central and outer-row bounds acquire only an arbitrarily small additional
power after this division, by the hypothesis on $A_\eta^{-1}$.

\paragraph{The base correction.}

Take $l_x=l_y=h=1/2$ and $\ell=0$.  On the principal rectangle of Lemma~\ref{lem:principal-residue-interface},
the local correction is bounded uniformly in every height because that
rectangle lies in $\mathcal D_2$.  Thus
Equation~\eqref{eq:principal-correction-bound} holds with $\ell=0$ and
height order zero.  At the double residue it satisfies exactly
\[
 \mathfrak H_{\eta,1,Z}(s,1,1/6)=H_\eta(s)
       =H_\eta(s)Z^{\ell/6}\cdot1\cdot(1+0).
\]
Hence $A_\eta=1$ and the residue error is identically zero.  In the row
$u=1$, the outside factor $q_u^{-z}\overline{\xi(u)}$ is one; the sole
factor $1/6$ is already in $c_S$ from the residue of $\zeta_F^S(6z)$.
Lemma~\ref{lem:principal-residue-interface} therefore has just its two
remainder terms.  Their savings before losses are
\[
 m_w=\frac{l_y}{20}=\frac1{40},\qquad
 m_z=\frac h{600}=\frac1{1200}.
\]

\subsection{Small and large row norms}

The buffered bin estimate is needed only on a bounded interval of positive
row exponents.  The following direct bounds cover its two complements.
They ask for explicit absolute estimates on the full correction and do not
use its central decomposition.

\begin{lemma}[Outer row norms]
\label{lem:outer-row-tails}
Assume Definition~\ref{def:stage-high-data}.  Fix
$\sigma_0\in[7/8,1)$ with $\beta_*>\sigma_0$, $0<e<10^{-3}$,
$z_0=17/50$, and $d_{\min}>0$.  For a dyadic number $U\ge1$, let
$\mathscr R_\eta(U;Z)$ denote the contribution in
Equation~\eqref{eq:stage-high-identity} of physical rows
$U\le q_u<2U$, omitting $u=1$.

Suppose that for every $\epsilon>0$, for all such rows with
$q_u\le2Z^{d_{\min}}$, and uniformly in all imaginary parts on
$(\Re s,\Re w,\Re z)=(\beta_*+e,1/2,z_0)$,
\begin{equation}
 |\mathfrak H_{\eta,u,Z}(s,w,z)|
 \ll_{\eta,e,\epsilon}
 Z^{\ell z_0+\epsilon}q_u^\epsilon
 (1+|\Im s|+|\Im w|+|\Im z|)^{J_{\eta,e,\epsilon}}.
 \label{eq:small-correction-bound}
\end{equation}
Then, for every $\epsilon>0$ and $1\le U\le Z^{d_{\min}}$,
\begin{equation}
 |\mathscr R_\eta(U;Z)|
 \ll_{\eta,e,\epsilon}
 Z^{C(\beta_*)+h(z_0-1/6)-l_y/2+e+\epsilon}
       U^{63/50+\epsilon}.
 \label{eq:small-row-bound}
\end{equation}
In particular the sum of these dyads is
\begin{equation}
 \ll_{\eta,e,\epsilon}
 Z^{C(\beta_*)+h(z_0-1/6)-l_y/2+e+(63/50)d_{\min}+\epsilon}.
 \label{eq:small-row-sum}
\end{equation}

For the other end, suppose that for every fixed $v>2$ and $\epsilon>0$,
for every physical row, uniformly in all imaginary parts on
$(\Re s,\Re w,\Re z)=(2,2,v)$,
\begin{equation}
 |\mathfrak H_{\eta,u,Z}(s,w,z)|
 \ll_{\eta,v,\epsilon}
 Z^{\ell v+\epsilon}q_u^\epsilon
 (1+|\Im s|+|\Im w|+|\Im z|)^{J_{\eta,v,\epsilon}}.
 \label{eq:large-correction-bound}
\end{equation}
Write $B_0=l_x/2+1+l_y$.  Then, for every dyad $U$,
\begin{equation}
 |\mathscr R_\eta(U;Z)|
 \ll_{\eta,v,\epsilon}Z^{B_0+hv+\epsilon}U^{1+\epsilon-v}.
 \label{eq:large-row-bound}
\end{equation}
For every fixed $\zeta>0$, if $\epsilon<v-1$, its sum over
$U>Z^{h+\zeta}$ is
\begin{equation}
 \ll_{\eta,v,\epsilon}
 Z^{B_0+(h+\zeta)(1+\epsilon)-\zeta v+\epsilon}.
 \label{eq:large-row-sum}
\end{equation}
Thus a fixed sufficiently large $v$ makes this last contribution smaller
than any prescribed power of $Z$.
\end{lemma}

\begin{proof}
Every row under consideration has nonprincipal numerator by the
classification preceding Lemma~\ref{lem:principal-residue-interface}.
The denominator may be principal for a bounded unit row; its reciprocal
is nevertheless holomorphic and bounded on $\Re s=\beta_*+e$ by the
principal specialization of Lemma~\ref{lem:logarithmic-control}.
For all rows, the conductor and deletion radical bounds in
Section~\ref{sec:detector}, together with
Lemma~\ref{lem:deleted-euler-factors}, give an arbitrarily small power of
$q_u$ and a fixed height polynomial for that reciprocal.

For a small dyad move its finite sum to
$(\Re s,\Re w,\Re z)=(\beta_*+e,1/2,z_0)$.
The reciprocal stays global.  The numerator is entire, and the scalar
zeta argument stays to the right of one.  These paths may be taken in
$\mathcal D_1(3/8)$: at the final point
$\Re(s+w)=\beta_*+e+1/2>1+3/8$, and the other inequalities are immediate.
The all-height correction bound and
Lemma~\ref{lem:external-contour-tails} justify the moves.  On the final
lines, Equation~\eqref{eq:hecke-growth} and the deletion bound give
$U^{3/5+\epsilon}$ times a fixed height polynomial for the nonprincipal
numerator.  There are $O(U)$ element rows in the dyad, including unit
factors, and $q_u^{-z_0}\ll U^{-z_0}$.  The correction is bounded by
Equation~\eqref{eq:small-correction-bound}.  The tests integrate all fixed
height powers.  The resulting exponent outside the row power, relative
to $C(\beta_*)$, is
\[
 h(z_0-1/6)-l_y/2+e,
\]
and the row power is $1+3/5-z_0=63/50$.  Requesting the component small
powers to sum to the displayed $\epsilon$ proves
Equation~\eqref{eq:small-row-bound}.  The row exponent is positive, so
dyadic summation up to $Z^{d_{\min}}$ gives
Equation~\eqref{eq:small-row-sum}, after decreasing the preliminary losses.

For each fixed $Z$ and large dyad, move its finite row sum to the absolute
lines $(2,2,v)$.  The path from $(3,3,2)$ remains in $\Re s,\Re w\ge2$
and $\Re z\ge2$, so the scalar factors are holomorphic there.  For this
fixed dyad choose $D$ large enough to contain its rows in $q_u\le Z^D$;
Equation~\eqref{eq:stage-all-height-majorant} and the external trace bound
make its horizontal limits vanish.  Constants used only to justify this
equality may depend on the fixed dyad.  The estimate on the final lines
is uniform in the dyad by Equation~\eqref{eq:large-correction-bound}.
The scalar $L$-factors
are absolutely bounded there, as is $\zeta_F^S(6z)$, and
Equation~\eqref{eq:large-correction-bound} applies.  The outside power,
including the correction but not the row count, is
\[
 l_x(1/2-v)+1+v+l_y+\ell v=B_0+hv.
\]
The $O(U)$ rows and $q_u^{-v}$ give $U^{1-v}$; the tests integrate the
fixed height polynomial.  This proves Equation~\eqref{eq:large-row-bound}.
Because $1+\epsilon-v<0$, summing its geometric dyadic tail gives
Equation~\eqref{eq:large-row-sum}.  For fixed $\zeta>0$, the coefficient
of $v$ in that exponent is $-\zeta$, proving the last assertion.
The original row series is absolutely convergent on its starting lines,
and the displayed bounds give an absolutely summable final tail, so the
individual dyadic contour identities may be summed.
The number $v$ and the finite test orders it requires are fixed before
$Z$ tends to infinity.
\end{proof}

\paragraph{The first-stage outer ranges.}

Use $l_x=l_y=h=1/2$, $\ell=0$, $d_{\min}=1/63$ and
$\zeta=1/1000$, as in the intermediate-row estimate.  For the small rows, the line $(\beta_*+e,1/2,z_0)$ lies in
$\mathcal D_1(3/8)$, since $\beta_*>11/12$.  The local bound
$\mathcal H_{\eta,u}\ll_\epsilon q_u^\epsilon$, uniform in all heights,
verifies Equation~\eqref{eq:small-correction-bound} for all the prescribed
rows, with height order zero.  Equation~\eqref{eq:small-row-sum} then has
the following relative exponent before its losses:
\[
 h(z_0-1/6)-\frac{l_y}{2}+\frac{63}{50}d_{\min}
 =\frac{13}{150}-\frac14+\frac1{50}
 =-\frac{43}{300}.
\]
For the large rows, $(2,2,v)$ with $v>2$ lies in $\mathcal D_2$.  The same
all-height local bound verifies Equation~\eqref{eq:large-correction-bound}
for every physical row.  Here $B_0=l_x/2+1+l_y=7/4$, so
Equation~\eqref{eq:large-row-sum} has exponent
\[
 \frac74+\frac{501}{1000}(1+\epsilon)-\frac v{1000}+\epsilon.
\]
It tends to $-\infty$ as the fixed number $v$ increases.  For example,
$v=4000$ makes it less than $-1$ when $\epsilon\le10^{-6}$, whereas
$C_{\mathrm I}(\beta_*)\ge1/4$.  This more than supplies the saving needed
below.  The three ranges $U\le Z^{d_{\min}}$,
$Z^{d_{\min}}<U\le Z^{h+\zeta}$, and $U>Z^{h+\zeta}$ cover every physical
dyad.  The row $u=1$ was already assigned to the principal term.

\section{The \texorpdfstring{$11/12$}{11/12} conclusion}
\label{sec:eleven-twelfths-conclusion}

For the normalized base probe the central estimate, principal extraction,
and outer-row estimates now give the following margins before adjustable
losses:
\[
\begin{array}{c|c}
 \text{contribution}&\text{saving relative to }C_{\mathrm I}(\beta_*)\\\hline
 \text{intermediate rows}&1021/25000\\
 \text{unresidued principal }w\text{ integral}&1/40\\
 \text{remaining principal }z\text{ integral}&1/1200\\
 \text{small rows}&43/300\\
 \text{large rows }(v=4000)&>1
\end{array}
\]
The smallest margin comes from the principal $z$ remainder.  We retain a
common positive margin after all real losses, then choose the analysis
height for each target.  The physical sum and the signal do not depend on
that height.

\subsection{Choosing the final height}

The preceding estimates separate real powers from finite powers of the
analysis height $T_1$.  The following elementary step records the order
of choices needed to obtain one power saving for every target.

\begin{lemma}[Late choice of height and external order]
\label{lem:late-height-closure}
Fix $\sigma_0\in(1/2,1)$ with $\Delta_0=\beta_*-\sigma_0>0$ and an
affine function $C(s)=s+c$.  Suppose that all real parameters and finite
structural choices in an application have been fixed independently of
$\eta$.  Suppose there are common numbers $m>0$ and
$0<\omega<\Delta_0$ such that, for every primitive finite-order target,
functions $J_\eta,f_\eta$ independent of $T_1$ satisfy
\[
 |J_\eta(Z)|\ll_\eta Z^{C(\sigma_0)+\omega}.
\]
Assume that after a finite sum of estimates, for every integer $N\ge0$,
\begin{equation}
 |J_\eta(Z)-f_\eta(Z)|
 \ll_{\eta,N} Z^{C(\beta_*)-m}(1+T_1)^{A_\eta}
       +Z^{B_\eta}T_1^{-N},
 \label{eq:late-height-hypothesis}
\end{equation}
where $A_\eta\ge0$ and $B_\eta$ are finite and independent of $N$.
Suppose there is a number $\tau_{0,\eta}>0$, fixed after the real choices
and the finite profile orders for the target but before $N$ and $Z$, such
that the estimate is valid for sufficiently large $Z$ whenever
$T_1=Z^\tau$ and
$0<\tau\le\min\{d_{\min}/100,\tau_{0,\eta}\}$, with a fixed
$d_{\min}>0$.
This additional ceiling may encode a height condition in a preceding
estimate, such as the one in Lemma~\ref{lem:detector-dyads}.
The lower threshold may depend on $\eta,N,\tau$.  Increasing $N$ is assumed
to change only external test seminorms, not $A_\eta$ or the already fixed
real powers in Equation~\eqref{eq:late-height-hypothesis}.

Then $\tau$ and $N$ can be chosen after $\eta$ so that
\[
 |J_\eta(Z)-f_\eta(Z)|\ll_\eta Z^{C(\beta_*)-m/2}.
\]
In particular the common saving $\sigma=m/2$, together with the displayed
low estimate, has the target quantifier required by
Proposition~\ref{lem:continuation-criterion}.
\end{lemma}

\begin{proof}
For the fixed target choose
\[
0<\tau_\eta\le
 \min\left\{\frac{d_{\min}}{100},\tau_{0,\eta},
                 \frac{m}{4(A_\eta+1)}\right\},
 \qquad T_1=Z^{\tau_\eta}.
\]
Then $(1+T_1)^{A_\eta}\ll_\eta Z^{m/4}$, so the first term in
Equation~\eqref{eq:late-height-hypothesis} is
$O_\eta(Z^{C(\beta_*)-3m/4})$.  Next choose a fixed integer $N$ such that
\[
 B_\eta-N\tau_\eta<C(\beta_*)-m/2.
\]
This is possible because $\tau_\eta>0$ and $B_\eta$ is finite.  The second
term is then bounded by the required power.  Increase the lower threshold
for $Z$ after these choices.  Both $m$ and $\omega$ were fixed before the
target, whereas $\tau_\eta,N$ and the threshold may depend on it.  Since
$J_\eta$ and $f_\eta$ do not contain $T_1$, this proves the asserted
family-wide exponent without changing either function.
\end{proof}

\begin{proposition}[Balanced high estimate]
\label{prop:balanced-high}
Under the supposition $\Delta_1>0$, for every primitive finite-order target
$\eta$ and all sufficiently large $Z$,
\[
 |J_{\mathrm I,\eta}(Z)-f_{\mathrm I,\eta}(Z)|
 \ll_\eta Z^{C_{\mathrm I}(\beta_*)-1/4800}.
\]
The displayed saving is independent of the target; the implied constant and
lower threshold may depend on it.
\end{proposition}

\begin{proof}
Use $l_x=l_y=h=1/2$, $\ell=0$ and $C=C_{\mathrm I}$.
The exact identity and local correction were verified at the start of
Section~\ref{sec:shared-analytic-estimates};
Equation~\eqref{eq:balanced-central-contribution} estimates its intermediate
rows, and the principal and outer specializations there give the margins
listed above.  It remains to choose their losses.

\paragraph{Order of choices.}
We give a common loss budget.  Put
\[
 m_0=\frac1{1200},\qquad \epsilon_*=10^{-6},\qquad
 m=\frac1{2400},\qquad \omega=\frac{\Delta_1}{2}.
\]
The error-free central, small, principal, and large savings just proved are
all at least $m_0$.  Fix the displayed geometry, row ranges, and $v=4000$
before the target.  Request loss $\epsilon_*$ in the row count, numerator,
local correction, buffered reciprocal, central multiplicity, and each
principal or outer estimate.  Reserve a further loss $\epsilon_*$ for the
$O(\log Z)$ physical dyads.  The witness-pair and powerful-part logarithms
are already included in the requested row-count loss; the $O_e(1)$ bins
and fixed presentation choices cost constants.

To obtain the requested row-count loss, first choose its preliminary witness,
dyadic, and sieve losses sufficiently small in terms of $\epsilon_*$.
These choices are independent of the target.  Indeed the witness length
error has an absolute coefficient, and the affine slopes in the optimization
of Proposition~\ref{prop:sextic-row-count} are uniformly bounded.  The
target-dependent constants in the actual dyadic endpoints multiply only
$1/\log U$, which changes a fixed constant or lower threshold.  Apply
Lemma~\ref{lem:detector-dyads} on the enclosing pre-saturation ranges
$0\le r,m\le22$: the detector's terminal product cutoff is $O(U^{21})$
before it proves the shorter witness lengths.  Its number $e_0$ depends
only on the requested dyadic loss and these bounded ranges.  We may thus
fix, still before the target,
\[
 0<e<\min\{10^{-3},\epsilon_*,e_0\}.
\]

With $\epsilon_r=\epsilon_n=\epsilon_H=\epsilon_*$, this gives
$\varepsilon_c\le15\epsilon_*$.  Since $d\le501/1000<1$, the explicit
central losses in Equation~\eqref{eq:balanced-central-contribution}, together
with the reserved physical-dyad loss, are at most
\[
 13\epsilon_*+2(15\epsilon_*)+\epsilon_*+\epsilon_*+\epsilon_*
 =46\epsilon_*<\frac1{4800}=\frac{m_0}{4}.
\]
The two principal losses are at most $(3/2+1)\epsilon_*$ and
$2\epsilon_*$; the small-row loss is at most $2\epsilon_*$.  Each is less
than $m_0/4$.  The displayed large-row estimate already uses its requested
loss.  Therefore all these terms, after division by the fixed $c_S$, retain
at least $3m_0/4=1/1600$ before the height factor.  We use only
$m=1/2400$, leaving a further power $1/4800$ available below.

It remains to check when the height hypotheses hold.  Let
$\epsilon_{\rm det}>0$ be the fixed dyadic loss chosen above, and, after
fixing a target, let $A_{{\rm det},\eta}$ dominate the finitely many orders
in its uses of Lemma~\ref{lem:detector-dyads}.  Set
\[
 \tau_{0,\eta}
 =\frac{d_{\min}\epsilon_{\rm det}}
        {20(A_{{\rm det},\eta}+1)}>0.
\]
For $0<\tau\le\tau_{0,\eta}$ and sufficiently large $Z$,
\[
 (1+Z^\tau)^{A_{{\rm det},\eta}}
 \le Z^{d_{\min}\epsilon_{\rm det}/10}
 \le U^{\epsilon_{\rm det}/10}
 \qquad(d\ge d_{\min}).
\]
This verifies the detector height condition uniformly over the physical
range.  The fixed factor from $1+Z^\tau\le2Z^\tau$ is absorbed by the lower
threshold.  We also impose $\tau\le d_{\min}/100$ as required by the bins.
The internal tail orders used to establish the witnesses are chosen after
this $\tau$ and are separate from the final external order $N$ below.
Their increase does not change the retained profile or height orders.
Any fixed constants introduced by those internal choices are absorbed into
the unused power $1/4800$ by increasing the lower threshold, which may depend
on $\eta$ and $\tau$.

For every final integer $N\ge0$, Lemma~\ref{lem:fixed-bin-contour}
bounds the discarded high portions and joins by
$O_{\eta,N}(Z^{B_\eta}T_1^{-N})$, with $B_\eta$ fixed before $N$.
The bounded real and row ranges and the finite set of bins permit one such
$B_\eta$ for all the central dyads.
Summing the $O(\log Z)$ dyads can be absorbed by increasing $B_\eta$ by one,
again before $N$.  Combining the preceding estimates gives
\[
 |J_{\mathrm I,\eta}(Z)-f_{\mathrm I,\eta}(Z)|
 \ll_{\eta,N}Z^{C_{\mathrm I}(\beta_*)-m}(1+T_1)^{A_\eta}
       +Z^{B_\eta}T_1^{-N}
\]
for some finite $A_\eta,B_\eta$, whenever
$0<\tau\le\min\{d_{\min}/100,\tau_{0,\eta}\}$ and $Z$ is sufficiently
large.  Both functions are independent of $T_1$.
Proposition~\ref{prop:balanced-low}, applied with loss $\omega$ and divided
by the same fixed $c_S$, supplies
\[
 |J_{\mathrm I,\eta}(Z)|\ll_\eta
 Z^{1/4+\omega}=Z^{C_{\mathrm I}(11/12)+\omega},
 \qquad 0<\omega<\Delta_1.
\]
All hypotheses of Lemma~\ref{lem:late-height-closure} are now verified.
It gives the saving $m/2=1/4800$ asserted in the proposition.
\end{proof}

\subsection{Transfer to Dirichlet \texorpdfstring{$L$-functions}{L-functions}}

The continuation criterion concerns the Hecke family over $F$.  We prove once that a strict zero-free
half-plane for that family has the corresponding Dirichlet consequence.

\begin{proposition}[Quadratic transfer]
\label{prop:hecke-dirichlet-transfer}
Let $\sigma_0\in[1/2,1)$.  Suppose every primitive finite-order Hecke
$L$-function over $F=\mathbb Q(\sqrt{-3})$ is zero-free on
$\Re s>\sigma_0$, with its principal pole at one allowed.  Then every
finite-order Hecke $L$-function over $F$ and every Dirichlet $L$-function
is zero-free on that same strict half-plane, again allowing the principal
pole at one.
\end{proposition}

\begin{proof}
Passing from a primitive Hecke character to one that it induces changes
only finitely many factors $1-\eta(\mathfrak p)N\mathfrak p^{-s}$.
They are nonzero for $\Re s>0$, so the Hecke assertion extends to all
finite-order characters.  The same observation for factors
$1-\chi(p)p^{-s}$ reduces the Dirichlet assertion to a primitive
Dirichlet character $\chi$ of conductor $q$.

Let $\chi_{-3}$ be the quadratic character of conductor three, and let
$\eta$ be the finite-order Hecke character given by
$\mathfrak a\mapsto\chi(N\mathfrak a)$ on ideals coprime to $3q$.
It is a ray character: if $\alpha\equiv1\pmod{q\mathcal O}$, then
$N\alpha\equiv1\pmod q$, so the norm character is trivial on the
corresponding principal ray subgroup.  Let $S_{\mathbb Q}$ be the rational
primes dividing $3q$, and let $\mathcal S_{\mathbb Q}$ be the primes of
$F$ above them.  Superscripts by these sets denote deletion of those Euler
factors.

For $p\notin S_{\mathbb Q}$, the local factors agree as follows.  If $p$
splits in $F$, then $\chi_{-3}(p)=1$, the two prime ideals have norm $p$,
and the Hecke factor is $(1-\chi(p)p^{-s})^{-2}$.  This is the product of
the Dirichlet factors for $\chi$ and $\chi\chi_{-3}$.  If $p$ is inert,
then $\chi_{-3}(p)=-1$, the unique prime ideal has norm $p^2$, and its
factor is
\[
 (1-\chi(p)^2p^{-2s})^{-1}
  =(1-\chi(p)p^{-s})^{-1}(1+\chi(p)p^{-s})^{-1}.
\]
These exhaust the primes outside $S_{\mathbb Q}$.  Absolute Euler
convergence for $\Re s>1$, followed by uniqueness of meromorphic
continuation, gives
\begin{equation}
 L_F^{\mathcal S_{\mathbb Q}}(s,\eta)
 =L^{S_{\mathbb Q}}(s,\chi)
      L^{S_{\mathbb Q}}(s,\chi\chi_{-3}).
 \label{eq:hecke-dirichlet-factorization}
\end{equation}
The product character on the right may be imprimitive; deletion of
$S_{\mathbb Q}$ makes the identity independent of that choice.  Every
deleted factor is nonzero for $\Re s>0$.

Both Dirichlet factors are holomorphic in $0<\Re s<1$.  A zero of
$L(s,\chi)$ there would therefore give a zero of the Hecke factor, with
no cancellation by a pole, and the assumed Hecke half-plane excludes it
when $\Re s>\sigma_0$.  Absolute Euler convergence handles $\Re s>1$.
On $s=1+it$ with $t\ne0$, neither Dirichlet factor has a pole, so the same
product argument applies.

At $s=1$, a pole-zero cancellation could occur only if one primitive
inducing character among $\chi$ and $\chi\chi_{-3}$ were principal.  The
other would then be $\chi_{-3}$.  Its Dirichlet series converges at one
by bounded partial sums of the nonprincipal periodic character, and
grouping consecutive terms gives
\[
 \begin{split}
 L(1,\chi_{-3})
 &=\sum_{n\ge0}\left(\frac1{3n+1}-\frac1{3n+2}\right)\\
 &=\int_0^1\frac{1-x}{1-x^3}\,dx
   =\int_0^1\frac{dx}{1+x+x^2}
   =\frac{\pi}{3\sqrt3}>0.
 \end{split}
\]
The first integral follows by monotone convergence of the nonnegative
paired integrands.  Thus there is no zero in this last case either; a
principal pole is allowed.  This proves the strict half-plane assertion
with no claim on its boundary.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:eleven-twelfths}]
Suppose that $\beta_*>11/12$.  Lemma~\ref{lem:local-euler} and
Equation~\eqref{eq:shared-principal-nonvanishing} give the required
holomorphic, nonzero $H_\eta$ on $\Re s>11/12$ for every
primitive target.  The low and high bounds established in
Proposition~\ref{prop:balanced-high} and its proof verify
Proposition~\ref{lem:continuation-criterion} with
\[
 \sigma_0=\frac{11}{12},\qquad C=C_{\mathrm I},\qquad
 \omega=\frac{\Delta_1}{2},\qquad \sigma=\frac1{4800}.
\]
These two positive losses are independent of the target, while the allowed
constants, excluded sets, and thresholds may depend on it.  The continuation
criterion contradicts the supposition.  Therefore $\beta_*\le11/12$.
By its definition and absolute Euler convergence in $\Re s>1$, every
primitive finite-order Hecke $L$-function over $F$ is zero-free in the strict
half-plane $\Re s>11/12$, with the principal pole allowed.

Proposition~\ref{prop:hecke-dirichlet-transfer}, applied at the same boundary,
extends this assertion to all finite-order Hecke characters and all Dirichlet
$L$-functions, including $\zeta(s)$.  It preserves the strict half-plane and
the allowed principal pole.  This proves the theorem.
\end{proof}

\part{The seven-eighths zero-free half-plane}
\label{part:seven-eighths}

\section{The compensated probe}\label{sec:compensation}

Theorem~\ref{thm:eleven-twelfths} applies to every primitive character
entering the supremum in Equation~\eqref{old-eq:1.1b}, and therefore gives
$\beta_*\le11/12$.  Suppose for contradiction throughout Part~II that
$\beta_*>7/8$, and put
\begin{equation}\label{eq:part-II-bootstrap}
 \Delta:=\beta_*-\frac78,\qquad 0<\Delta\le\frac1{24},
 \qquad
 \kappa:=2\beta_*-1=\frac34+2\Delta\le\frac56.
\end{equation}
The exact bin ceiling in Section~\ref{sec:detector} applies, since
$\beta_*>7/8>51/100$.  For every retained nonprincipal row and its bin
$(i,a)$ it gives
\begin{equation}\label{eq:part-II-bin-ceiling}
 a\le\beta_*,\qquad \delta=2a-1\le\kappa\le\frac56.
\end{equation}
Indeed, the finite set defining $M_i(u)$ consists of the floor $51/100$ and
real parts of actual zeros, all at most $\beta_*$.  This uses only the
definition of the supremum, not its attainment.  The boundary case
$\delta=5/6$ remains part of the argument.

For the second application of Proposition~\ref{lem:continuation-criterion},
write throughout Part~II
\begin{equation}\label{eq:part-II-mellin-target}
 C(s)=C_{\mathrm{II}}(s):=s-\frac{11}{16},\qquad
 C(7/8)=\frac3{16}.
\end{equation}
Our task is to construct a normalized probe $J_{\mathrm{II},\eta}$ from
the finite expression below.  It is distinct from the balanced probe
$J_{\mathrm I,\eta}$.  Its low-side target is
$|J_{\mathrm{II},\eta}(Z)|\ll_\eta Z^{3/16+\omega}$ for a common
$0<\omega<\Delta$; its high side must satisfy the signal estimate in
Equation~\eqref{eq:high-probe-contract} with this $C$.  The nonzero scalar
normalization will be specified after the principal term is evaluated.

For each primitive target $\eta$, use an admissible fixed instance of the
data $T,S,b_*,\xi,\tau,\Xi,W_0,W_1$ from Section~\ref{sec:probe}.  The
permitted choice of $P_0$ will be made in the parameter order below.  These
data may differ from those used in Part~I, but within this application they
are independent of $Z$ and are the same in the physical probe and its high
representation.

We first define the two-term modification on the original probe, before
moving any contour.  The subsequent low and high estimates will concern
this same finite expression.

Use the fixed geometry
\begin{equation}
 \begin{gathered}
 b=\frac18,\qquad h=\frac{13}{16},\qquad \ell=\frac16,\\
 l_x=1+\ell-h=\frac{17}{48},\qquad
 l_y=l_x+b=\frac{23}{48},\\
 X=Z^{l_x},\qquad Y=Z^{l_y},\qquad
 M=l_x+l_y=\frac56,\qquad M+\ell=1.
 \end{gathered}
 \label{old-eq:5.1}
\end{equation}
Fix positive slot lengths $\ell_1,\ldots,\ell_K$ of total length
$\ell$, and put $P_i=Z^{\ell_i}$.  Each physical slot has a
nonnegative, nonzero smooth annular weight $W_i(q_p/P_i)$.  Its allowed
prime set is
\[
 \mathcal P_i(Z)=\{p\text{ prime}:p\notin S,\ p\in1_T,
                         \ q_p/P_i\in\operatorname{supp}W_i\}.
\]
The underlying window sets
$\{p\text{ prime}:q_p/P_i\in\operatorname{supp}W_i\}$ for distinct slots
are required to be disjoint before imposing the ray and $S$ restrictions.
In particular, the sets $\mathcal P_i(Z)$ are disjoint, and every tuple in
$\prod_i\mathcal P_i(Z)$ consists of distinct primes.
The number and lengths of the slots will be chosen later, but they are
fixed independently of $Z$.  A sum over $p\in1_T$ for slot $i$ below
always retains this same exclusion and annular support.

For a squarefree product $D$ of slot primes, let
$I_{\eta;D}(X,Y,Z)$ denote Equation~\eqref{eq:general-probe} with
the indicator $1_{D\mid cn^3}$ inserted in its completed row.
Thus $I_{\eta;1}=I_\eta$.  For a tuple
$(p_1,\ldots,p_K)\in\prod_i\mathcal P_i(Z)$ and
$J\subseteq\{1,\ldots,K\}$ write $p_J=\prod_{i\in J}p_i$, with
$p_\varnothing=1$.  The subset $J$ indexes the rescaled slots and $J^c$
the marked slots; this subset notation is distinct from the normalized
probe $J_{\mathrm{II},\eta}$.  Define the modified probe by the finite identity
\begin{equation}\label{eq:compensated-probe-definition}
\begin{split}
 I_{\eta,\mathrm{modified}}(Z)
 =\sum_{(p_i)\in\prod_i\mathcal P_i(Z)}\prod_{i=1}^K W_i(q_{p_i}/P_i)
 \sum_{J\subseteq\{1,\ldots,K\}}&(-1)^{|J|}q_{p_J}^{-3/2}
          \overline{\eta(p_{J^c})}\\
 &\cdot I_{\eta;p_{J^c}}
       (X/q_{p_J},Y/q_{p_J},Zq_{p_{J^c}}).
\end{split}
\end{equation}
Equivalently, at each prime $p$ the operation is the marked term
$\overline{\eta(p)}I_{\eta;p}(X,Y,Zq_p)$ minus the rescaled term
$q_p^{-3/2}I_\eta(X/q_p,Y/q_p,Z)$.  Formula
\eqref{eq:compensated-probe-definition} specifies their composition:
every slot window stays at its original scale $P_i$, including when
another slot changes the completed scale.  On $1_T$ one has
$\theta(p)=1$ for every $\theta\in\widehat T$.  Thus marking commutes
with the Fourier decomposition of $\overline G$ and preserves the
same excluded set and zero masks in every summand.
The subtraction is designed to cancel the scalar prime contribution on
the high side, leaving the sextic-character prime factor used by the moment
estimates. Section~\ref{sec:local-compensation} proves the exact identity
and bounds the remaining local errors.

\section{Coefficient conventions and finite correlations}
\label{sec:additional-arithmetic}

We record the additional coefficient conventions for the prime factors,
then prove a fixed-ray prime normalizer and the full finite Fourier
correlations needed below. The latter retain shared prime powers and will
be used in the additive Gram bound as well as the fourth moment. The
residue-symbol and fixed-data conventions of Part I remain in force.

\subsection{Additional coefficient conditions}

We use the plain and inverse polynomials, annular profiles, and fixed arithmetic
datum $\mathcal A$ from Section~\ref{sec:arithmetic}.  The following conditions
specify the extra prime factors and moving zero supports used in this part.

A prime slot of log-length $z_i$ is
\begin{equation}
 Q_{\psi,i}=Z^{-z_i/2}\sum_{p\ {\rm prime}}
       \psi(p)\nu_i(p)W_i(q_p/Z^{z_i}).
 \label{old-eq:1.2}
\end{equation}
Here $\nu_i$ is a fixed finite-ray character or a fixed finite linear
combination of such characters.  It is independent of the row and of the
other selected primes.  Distinct slots have disjoint underlying prime
supports before any common mask or row zero extension is imposed.  For the
fourth moment of Section~\ref{sec:plain}, a fixed finite group $\Theta$ of
ray characters is part of the data, and every character component with
nonzero coefficient in every positive-length slot must belong to $\Theta$.
The inverse moment of Section~\ref{sec:inverse} also allows a bounded
coefficient $a_i(p)$ chosen independently for each slot and independently
of the row and all other columns; the coefficient of a prime tuple is then
the product of its individual slot coefficients.  The $\Theta$ restriction
does not apply to these separate inverse-moment weights.

For an auxiliary fourth moment the rows are elements $0<q_k\ll Z^m$, and
\begin{equation}
 \psi_k(n)=\tau(n)\chi_n(k),\qquad M=m+q.
 \label{old-eq:1.3}
\end{equation}
The twist $\tau$ is common to the row sum after its outer labels are fixed,
and its displayed factorization into fixed finite-ray and moving
residue-symbol factors is part of the data.  Let $D_{\mathrm{mov}}$ be the
squarefree product of all good primes at which at least one displayed moving
factor has its natural zero on nonunits.  This includes a prime even when
that factor has exponent divisible by six, including exponent zero, or when
local characters cancel after multiplication.  We require
$q_{D_{\mathrm{mov}}}\le Z^q$, counting an overlapping prime once.

An additional puncture is an indicator $1_{(n,R)=1}$ with $R$ squarefree,
$q_R\le Z^B$, and $B$ in a prescribed bounded range.  It must be common to
the current row sum and to every plain and prime factor in that sum; it may
depend on previously frozen outer labels.  A prime may be removed from
$D_{\mathrm{mov}}$ and put in $R$ only through an exact factorization of its
displayed local factor into that coprimality indicator and the local or
fixed-ray character phases that remain.  Those phases must be retained, the
refactored zero must be removed from the displayed moving factor, and the
common puncture is deleted before the natural reflection in
Section~\ref{sec:plain}.  No zero prime may be omitted from both supports.
If a moving character still ramified at that prime remains, the prime stays
in $D_{\mathrm{mov}}$; only a redundant zero may be transferred to $R$.
A fixed finite-ray character here has its complete zero-extended presentation
and ray group fixed in the preceding sense; a character with moving conductor
cannot be relabeled as fixed.  A locally frozen moving twist factor or
redundant zero mask retains the stated moving-support and puncture
requirements.  In particular a $Z$-varying redundant mask cannot be included
in the fixed arithmetic datum.  The zero extension of $\chi_n(k)$ is allowed
to vary naturally with $k$, but cancellation between that varying row factor
and a fixed twist may not be recast as a separately chosen puncture for each row.  Externally
chosen row-dependent punctures and row-dependent column coefficients are
not part of this class.  This full-support convention governs every
fourth-moment invocation in Section~\ref{sec:plain}.

The common uniformity convention has the following additional clauses for
these coefficient classes.  Any required slot mesh depends only on the fixed
real log-length ranges, the strict margins, and the specified positive power
losses.  Finite seminorm orders, polynomial height orders, implied constants,
and lower thresholds may also depend on a specified fixed number of slots.
They are uniform over the moving moduli, the radicals included in $M$, the
admissible punctures, and the outer labels in their stated ranges, even when
an outer label is fixed during one row sum.  The constant $C$ in the definition
of a divisor-bounded multiplicity may also depend on this fixed slot count.
The independence of $C$ from $Z$, the current rows, and the averaged labels,
and any separate requirement that the multiplicity depend only on $f$, remain
as in Section~\ref{sec:arithmetic}.  The slot count is a separate fixed
parameter, and moving labels remain outside $\mathcal A$.

\subsection{Prime counting in a fixed ray class}

\begin{lemma}[Fixed-ray prime normalizer]\label{lem:ray-prime-normalizer}
Let $T$ be a fixed quotient of a ray class group of $F$, and write
$p\in1_T$ when the image of an unramified prime ideal is the identity.
For a fixed nonnegative, nonzero smooth annular weight $W$,
\begin{equation}\label{eq:ray-prime-normalizer}
 \sum_{p\in1_T}W(q_p/P)q_p^{-5/6}
 \sim\frac{P^{1/6}}{|T|\log P}
      \int_0^\infty W(y)y^{-5/6}\,dy.
\end{equation}
Deleting any further fixed finite set of primes does not change this
asymptotic.  Its lower threshold may depend on all the fixed data.
\end{lemma}

\begin{proof}
For the fixed abelian extension of $F$ corresponding to $T$, the prime ideal
theorem in a fixed Frobenius class gives
\[
 \pi_{1_T}(x):=\#\{p\in1_T:q_p\le x\}
       \sim\frac{\operatorname{Li}(x)}{|T|}.
\]
This is the fixed-extension consequence of Chebotarev in
\cite[Theorem 1.1]{TZ}; the extension and its conductor are fixed as
$x\to\infty$.  If $\operatorname{supp}W\subset[a,b]\subset(0,\infty)$,
the error $o(x/\log x)$ is uniform for $aP\le x\le bP$ as $P\to\infty$.
Stieltjes integration by parts therefore changes the left side of
Equation~\eqref{eq:ray-prime-normalizer} by $o(P^{1/6}/\log P)$ when
$d\pi_{1_T}(x)$ is replaced by $dx/(|T|\log x)$.  The resulting integral is
\[
 \frac{P^{1/6}}{|T|\log P}
 \int_a^b W(y)y^{-5/6}\frac{\log P}{\log(Py)}\,dy.
\]
The last ratio tends uniformly to one.  This proves the formula and its
positivity.  A fixed finite set is eventually outside the annular window.
No error exponent uniform in the ray conductor is used.
\end{proof}

\subsection{Full finite Fourier correlations}\label{sec:finite-correlations}

We now allow arbitrary prime powers in a modulus.  The two-argument notation
$G(a,k)$ below is a finite Fourier sum; it is distinct from the one-argument
finite-ray function $G(a)$ of Lemma~\ref{lem:fixed-gauss-phase}.  Define
\begin{equation}
 G(a,k)=q_a^{-1/2}\sum_{x\bmod a}\chi_a(x)e(kx/a),\qquad G(1,k)=1.
 \label{old-eq:2.4}
\end{equation}
All characters in this subsection have the zero extensions specified in
Section~\ref{sec:arithmetic},
and we put $v_p(0)=+\infty$.

\begin{lemma}[Prime-power Fourier sums]\label{lem:prime-power-fourier}
For $P=q_p$ and every integer $a\ge1$,
\begin{equation}
 \begin{aligned}
 |G(p^a,k)|&=P^{(a-1)/2}1_{v_p(k)=a-1},&&6\nmid a,\\
 G(p^a,k)&=P^{a/2}1_{p^a\mid k}
             -P^{a/2-1}1_{p^{a-1}\mid k},&&6\mid a.
 \end{aligned}
 \label{eq:gauss-local}
\end{equation}
\end{lemma}

\begin{proof}
Write a residue modulo $p^a$ as $x_0+py$, with $x_0\bmod p$ and
$y\bmod p^{a-1}$.  The sum over $y$ is zero unless $p^{a-1}\mid k$, and
equals $P^{a-1}$ otherwise.  In the latter case write $k=p^{a-1}k_0$.
The remaining normalized sum is
\[
 P^{a/2-1}\sum_{x_0\bmod p}\chi_p(x_0)^a e(k_0x_0/p).
\]
If $6\nmid a$, its inner sum vanishes when $p\mid k_0$ and otherwise has
absolute value $\sqrt P$.  If $6\mid a$, the inner sum is the sum of the
additive character over the units, namely $P1_{p\mid k_0}-1$.  These are
exactly the two cases in the statement.
\end{proof}

\begin{lemma}[Full correlation and common factors]\label{lem:full-correlation}
For primary moduli $u,v$ outside $S$ and $j\in\mathcal O$, put
\begin{equation}
 F(u,v;j)
 =\sum_{\substack{x\bmod u,\ y\bmod v\\
                    vx-uy\equiv j\pmod{uv}}}
       \chi_u(x)\overline{\chi_v(y)}.
 \label{old-eq:2.11}
\end{equation}
Then
\[
 F(u,v;j)=\frac1{\sqrt{q_uq_v}}\sum_{h\bmod uv}
 G(u,h)\overline{G(v,h)}e(-jh/(uv)).
\]
At zero frequency, $F(u,v;0)=0$ unless $u=v$, and
$F(u,u;0)=\varphi(u)$, where $\varphi(u)$ is the number of units modulo $u$.

Let $C=(u,v)$ and write $u=Cn_1$, $v=Cn_2$, where $(n_1,n_2)=1$.
Then $F(u,v;j)=0$ unless $C\mid j$.  For $j=Ck$,
\begin{equation}
 \begin{aligned}
 F(Cn_1,Cn_2;Ck)
 &=\chi_{n_1}(k)\overline{\chi_{n_2}(-k)}
      \mathcal R(n_1,n_2)L_C(n_1,n_2;k),\\
 L_C(n_1,n_2;k)
 &=\sum_{\substack{x,y\bmod C\\n_2x-n_1y\equiv k\pmod C}}
       \chi_C(x)\overline{\chi_C(y)}
   =\prod_{p^c\parallel C}L_{p^c}.
 \end{aligned}
 \label{eq:correlation-lift}
\end{equation}
For $P=q_p$ and $p\nmid n_1n_2$, the local factor is
\begin{equation}
 L_{p^c}=P^{c-1}\chi_p(n_1/n_2)^c
 \begin{cases}
 P-1,&p\mid k,\\
 -1,&p\nmid k,\ 6\nmid c,\\
 P-2,&p\nmid k,\ 6\mid c.
 \end{cases}
 \label{eq:correlation-local}
\end{equation}
If $p$ divides exactly one of $n_1,n_2$, the local factor is
$P^{c-1}(P-1)1_{6\mid c}1_{p\nmid k}$.  On the genuine residual locus
$(n_1,n_2)=1$, $L_C$ means the congruence sum in
Equation~\eqref{eq:correlation-lift}.  When $L_C$ is used on all residual
pairs, it instead denotes the artificial product extension of these local
formulas, with the local value defined to be zero if $p\mid(n_1,n_2)$.
Outside the genuine residual locus this is not the original congruence sum.
The genuine function and this artificial extension both satisfy
$|L_C|\le q_C$, and the extension is periodic modulo
$\operatorname{rad}C$ in each residual column.
\end{lemma}

\begin{proof}
Expanding both Gauss sums in the displayed Fourier transform leaves
\[
 \frac1{q_uq_v}\sum_{x\bmod u,y\bmod v}\chi_u(x)\overline{\chi_v(y)}
 \sum_{h\bmod uv}e\bigl(h(vx-uy-j)/(uv)\bigr).
\]
Additive orthogonality makes the inner sum $q_uq_v$ when the congruence holds
and zero otherwise.  If $j=0$ and a summand is nonzero, $x$ and $y$ are units
modulo $u$ and $v$.  Reducing the congruence modulo $u$ gives $u\mid v$, and
reducing modulo $v$ gives $v\mid u$.  Since the generators are primary,
$u=v$.  The congruence then says $x=y\bmod u$ and gives $\varphi(u)$.  This
argument uses the zero masks also when a local character power is principal.

The divisibility by $C$ is immediate.  After division by $C$, the congruence
for $j=Ck$ is
\[
 n_2x-n_1y\equiv k\pmod{Cn_1n_2}.
\]
Reduction modulo $n_1$ and $n_2$ gives, with zero values retained,
\[
 \chi_{n_1}(x)=\chi_{n_1}(k)\overline{\chi_{n_1}(n_2)},\qquad
 \overline{\chi_{n_2}(y)}
 =\overline{\chi_{n_2}(-k)}\chi_{n_2}(n_1).
\]
The product of the two unit factors is $\mathcal R(n_1,n_2)$.
It remains to identify the multiplicity of lifts of the congruence modulo
$C$.  This can be checked at each prime.  If $c=v_p(C)$ and neither residual
modulus contains $p$, there is no additional lift.  If, say,
$d=v_p(n_1)>0$ and $p\nmid n_2$, then $y\bmod p^c$ is free and the congruence
uniquely determines $x\bmod p^{c+d}$ from it.  Reduction modulo $p^c$ is
exactly the common congruence in Equation~\eqref{eq:correlation-lift}.
The case $p\mid n_2$ is symmetric, and this argument includes $c=0$.
The Chinese remainder theorem thus gives a bijection with the common
solutions used in $L_C$, even when $C$ meets one residual modulus.

For the local calculation put $k_p=\mathcal O/(p)$ and $A=\chi_p^c$, a
character of $k_p^\times$ extended by zero.  Since at least one of
$n_1,n_2$ is a unit at $p$, each
solution modulo $p$ has $P^{c-1}$ lifts modulo $p^c$.  If both are units and
$p\mid k$, the unit variables are proportional and the field sum is
$A(n_1/n_2)(P-1)$.  If $p\nmid n_1n_2k$, put
$y=(k/n_1)t$ and $x=(k/n_2)(1+t)$.  The field sum becomes
\[
 A(n_1/n_2)\sum_{t\ne0,-1}A((1+t)/t)
 =A(n_1/n_2)\sum_{z\in k_p^\times\setminus\{1\}}A(z).
\]
It is $-A(n_1/n_2)$ for nonprincipal $A$ and
$(P-2)A(n_1/n_2)$ for principal $A$.  If $p\mid n_1$ and $p\nmid n_2$, the
field equation forces $x=k/n_2$.  For $p\mid k$ its character is zero.  For
$p\nmid k$ the remaining sum over $y$ is zero unless $A$ is principal, in
which case it is $P-1$.  The other case is symmetric.  This proves all local
formulas on the genuine residual locus.  Extending them by the stipulated
zero when both residuals meet $p$ gives functions of the residual columns
modulo $p$, each of absolute value at most $P^c$.  Their product proves the
final assertions for the artificial extension as well; no congruence-sum
identity is asserted at a newly added pair.
\end{proof}

For some applications it is preferable to remove all primes common to the
two full moduli, with their entire multiplicities.  The next form leaves one
common row character on each remaining product.

\begin{lemma}[Complete-common-support correlation]
\label{lem:complete-support-correlation}
Suppose $u=Da$, $v=Eb$ are primary and outside $S$, with
$(a,b)=1$ and $(ab,DE)=1$.  Then, for every $j\in\mathcal O$,
\begin{equation}
 \begin{split}
 F(Da,Eb;j)={}&F(D,E;j)\mathcal R(a,E)
       \overline{\mathcal R(b,D)}\mathcal R(a,b)\\
 &\hspace{8mm}\cdot\chi_a(j)\overline{\chi_b(-j)}.
 \end{split}
 \label{eq:correlation-child-character}
\end{equation}
The moduli $D,E$ may contain any prime powers, including shared auxiliary
prime factors.
\end{lemma}

\begin{proof}
At primes of $a$ and $b$, solving the congruence in
Equation~\eqref{old-eq:2.11} gives respectively
$\chi_a(j)\overline{\chi_a(Eb)}$ and
$\overline{\chi_b(-j)}\chi_b(Da)$.  These statements remain valid when a
character of $j$ is zero; only the displayed unit factors are inverted.
At primes of $DE$ change variables $x'=bx\bmod D$, $y'=ay\bmod E$.
The congruence becomes $Ex'-Dy'\equiv j\pmod{DE}$ and its character factor
changes by $\overline{\chi_D(b)}\chi_E(a)$.  The remaining unit factor is
\[
 \frac{\chi_E(a)}{\chi_a(E)}
 \frac{\chi_b(D)}{\chi_D(b)}
 \frac{\chi_b(a)}{\chi_a(b)}
 =\mathcal R(a,E)\overline{\mathcal R(b,D)}\mathcal R(a,b).
\]
All denominators here are symbols of units by the hypotheses.  The Chinese
remainder theorem and the definition of $F(D,E;j)$ complete the proof.
\end{proof}

The hypotheses of Equation~\eqref{eq:correlation-child-character} are not
removed by extending $\mathcal R$ as a bicharacter.  To state exactly the
extension used with M\"obius inversion, fix $D,E,j$ and define
$\widetilde F_{D,E}(a,b;j)$ to be the right side of that equation for all
primary $a,b$ outside $S$ with $(ab,DE)=1$, retaining its zero symbols even
when $(a,b)>1$.  For any finite coefficient array $c(a,b)$ on this locus,
\[
 \sum_{(a,b)=1}c(a,b)F(Da,Eb;j)
 =\sum_{a,b}c(a,b)\widetilde F_{D,E}(a,b;j)
                \sum_{t\mid a,\ t\mid b}\mu(t).
\]
Indeed the full inner divisor sum is $1_{(a,b)=1}$, and the two correlation
expressions agree on that locus.  The value of $\widetilde F_{D,E}$ elsewhere
is artificial, not a formula for $F(Da,Eb;j)$.  In particular the full
divisor sum must be inserted before a factorwise estimate separates the two
residual columns; extra common primes introduced by an individual divisor
term do not become primes of the genuine $D,E$ correlation.

\section{Marked completion and reflected row energy}
\label{sec:marked}\label{sec:marked-energy}

The compensated low estimate and the inverse moment use the same marked
completion. We establish its reflected mean square here. The low estimate
will use the case with no additional squarefree label or puncture; the
more general form allows the labels and masks introduced by the inverse
moment's Poisson transformations.

We retain the notation of Section~\ref{sec:arithmetic}.  In particular,
ideal variables have their multiplicative primary generators, the fixed
excluded set \(S\) contains the primes over \(6\), and every power of a
residue symbol is zero on nonunits, even when its exponent is divisible by
six.  Element rows need not be squarefree or prime to \(S\).

\subsection{Completed sums and product-form marks}

Fix a finite index set \(I\) and pairwise disjoint lists
\(\mathcal P_i\) of primes outside \(S\), with
\(q_p\asymp Z^{z_i}\) for \(p\in\mathcal P_i\). The lists and their
bounded individual coefficients \(a_i(p)\) are independent of the
current row and squarefree label. Their nominal lengths \(z_i\) lie in
fixed bounded ranges.

Here and below a \emph{fixed puncture} means a function
\[
 \rho(n)=1_{(n,\mathfrak r_\rho)=1},
\]
where \(\mathfrak r_\rho\) is squarefree and is required to be fixed only
within the indicated current row and label sums.  A bounded product of such
functions has this form with \(\mathfrak r_\rho\) equal to the radical of
the product of their moduli.  The norm of this radical will be bounded
explicitly.  It
may depend on previously fixed outer ideals, but not on either current
averaging variable.

Let \(f\) be a squarefree ideal outside \(S\), with
\(q_f\asymp Z^V\) for a nonnegative \(V\) in a fixed bounded range,
and let \(k\in\mathcal O\).
For a fixed multiplicative finite ray character \(\nu\) whose full
zero-extended defining modulus has prime support in \(S\), define \(a(n)\)
only for squarefree primary \(n\) outside \(S\), and define \(\Psi_k(n)\)
for every primary \(n\) outside \(S\) by
\[
 a(n)=\bar\alpha(n)\gamma_2(n)\nu(n)\rho(n),\qquad
 \Psi_k(n)=\nu(n)\rho(n)\chi_n(k)\chi_n(f)^4.
\]
The finite character, puncture, and all slot lists are independent of
\(k,f\).  A fixed function on a finite ray group is always expanded into
genuine group characters before this definition is used; thus \(\nu\) is
multiplicative even when an earlier step produced a finite linear
combination of characters.

For a subcollection \(I'\subset I\) of disjoint prime lists with bounded
coefficients, define its mark by
\begin{equation}\label{old-eq:4.2}
 \mathfrak d_{I'}(A)=
 \sum_{(p_i)\in\prod_{i\in I'}\mathcal P_i}
      \prod_{i\in I'}a_i(p_i)1_{p_i\mid A}.
\end{equation}
We omit \(I'\) from the notation when it is understood.  Because the lists
are disjoint, a tuple has a squarefree product.  For every fixed
\(\epsilon>0\), the number of tuples dividing \(A\) is
\(O_\epsilon(q_A^\epsilon)\).  The coefficient in
Equation~\eqref{old-eq:4.2} is a product of functions of the individual
primes; this property, not merely the pointwise divisor bound, will be
used when some slots are assigned to an extracted factor.

For a smooth annular \(W\), put \(V_*(y)=y^{1/2}W(y)\) and define the completed marked sum
\begin{equation}\label{eq:marked-completion}
 \mathcal T_{\mathfrak d}(X;k)=
 \sum_{\substack{n\ {\rm sf}\\b}}
 \frac{\bar\alpha(n)\gamma_2(n)\Psi_k(n)\,
       \bar\alpha(b)^3\Psi_k(b)^3}{\sqrt{q_n}\,q_b}
 \mathfrak d(nb^3)V_*(q_nq_b^3/X).
\end{equation}
Both primal ideals avoid \(S\); they may share primes.  The mark belongs
to the whole index \(nb^3\).  With \(\mathfrak d=1\), this is the Mellin
completion on the left of Equation~\eqref{eq:reflection}.  The Gaussian
Mellin test from that Equation is also permitted when only the completed
estimate is used.

For the reflection formula we use Lemma~\ref{lem:reflection-row-sectors},
only for a nonzero row.  Write \(k=u k_S k_{\rm good}\) as in that Lemma.
After fixing the unit, the \(S\)-valuations modulo six, and the good row ray
class, its literal sector character is
\[
 \Psi_0^{[k]}(A)=
 \begin{cases}
 \nu(A)\chi_A(u k_S)\mathcal R(A,k_{\rm good}),
       &A\equiv1\pmod3,\ (A,S)=1,\\
 0,&\text{otherwise}.
 \end{cases}
\]
The zero branch is evaluated before either character.  The Lemma supplies
one finite family and hence one common \(L\), with prime support \(S\), before
the good primes vary.  Write \(M_{\rm ref}=\lambda^{12}L^4\); the full
reflection sector modulus is \(M_{\rm ref}^2\).  All good primes of
\(k,f,\mathfrak r_\rho\) remain in the local prime set, with their exponents
combined modulo six and every zero retained.  The fourth power at \(f\) has
no extra reciprocity sign, since \(\mathcal R(A,f)^4=1\).  The puncture is
the product of the local zero powers at its good primes.  Thus the local
presentation gives exactly \(\Psi_k(n)\Psi_k(b)^3\) on the stated primal
support, including at shared good primes; no moving good prime enters \(L\).

\subsection{Whole-index marked reflection}\label{par:reflection-marks}

The next corollary adds the product-form marks to the exact reflection in
Part I.  Its branch data are important: an inactive marked prime is absent
from the conductor, so the marked transform is not obtained by multiplying
one fixed-conductor dual sum by independent local factors.

\begin{corollary}[Completed reflection with whole-index marks]
\label{cor:marked-reflection}
Fix an invocation of Proposition~\ref{prop:completed-reflection}, with base
local prime set \(\mathcal P_0\), powers \(j_p\), fixed multiplier \(\phi\),
and test \(V\).  Put \(M_{\rm ref}=\lambda^{12}L^4\) for this invocation.
Let \(\mathcal Q\) be a finite set of distinct good primes
disjoint from \(\mathcal P_0\).  In the direct completed coefficient sum,
insert the factor \(\prod_{q\in\mathcal Q}1_{q\mid nb^3}\); denote this sum
by \(\mathcal T_{\mathcal Q}(X;\Psi,V)\).  It has the same normalization as
Equation~\eqref{eq:marked-completion}, with \(\Psi_k,V_*\) replaced by
\(\Psi,V\).  The primal squarefree and cube ideals may share primes.

There is an exact finite expansion
\begin{equation}\label{eq:marked-reflection}
 \begin{split}
 \mathcal T_{\mathcal Q}(X;\Psi,V)
 =\sum_{\mathcal B}\zeta_{\mathcal Q}
 \sum_{0\ne\mu\in\lambda^{-4}\mathcal O}
 &\frac{d(\mu)\alpha(\mu)\vartheta(\lambda^4\mu)}{\sqrt{q_\mu}}
 \prod_{p\in\mathcal A_0}B_p(\lambda^4\mu)\\
 &\cdot\prod_{q\in\mathcal A_{\mathcal Q}}
       q_q^{-1/2}\chi_q(\lambda^4\mu)^{-2}
 V^\sharp\!\left(\frac{q_\mu X}{q_c^2}\right).
 \end{split}
\end{equation}
Here a branch \(\mathcal B\) specifies \(h_0\bmod L\), the base active and
inactive sets \(\mathcal A_0,\mathcal I_0\), and a partition
\(\mathcal Q=\mathcal A_{\mathcal Q}\sqcup\mathcal I_{\mathcal Q}\).
All base primes with \(j_p\ne0\) are active.  Put
\[
 r=\prod_{p\in\mathcal A_0\cup\mathcal A_{\mathcal Q}}p,
 \qquad c=c_F(h_0)r.
\]
The base factors \(B_p\) and the canonical functions \(d,\vartheta\) are
those of Proposition~\ref{prop:completed-reflection} for this active radical.
With every \(\sigma_p,\epsilon_p,\omega_{p,j}\) computed using this branch's
whole \(c\), the scalar is
\begin{equation}\label{eq:marked-reflection-phase}
 \begin{split}
 \zeta_{\mathcal Q}={}&-\frac{i}{81}\bar\alpha(c)^2
       \widehat\phi(h_0)\bar\kappa_F
       \prod_{p\in\mathcal I_0}(1-q_p^{-1})
       \prod_{q\in\mathcal I_{\mathcal Q}}q_q^{-1}\\
 &\cdot\prod_{p\in\mathcal A_0}\chi_p(\sigma_p)^{-2}\omega_{p,j_p}
       \prod_{q\in\mathcal A_{\mathcal Q}}
             \chi_q(\sigma_q)^{-2}(-\omega_{q,0}).
 \end{split}
\end{equation}
In particular \(|\zeta_{\mathcal Q}|\le1/81\).  The functions
\(d,\vartheta\) and the scalar \(\kappa_F\) have the canonical dependence
on \(h_0\) and \(r\bmod M_{\rm ref}^2\) from Part I.  Every dual sum is
absolutely convergent and has the same full source support as there.
If a marked prime instead belongs to the base local set, the corresponding
primal term is identically zero and is removed before this formula is used.
\end{corollary}

\begin{proof}
For every good prime and every \(A\in\mathcal O\), the zero convention gives
the pointwise identity \(1_{q\mid A}=1-\chi_q(A)^0\).  Expand its product over
\(\mathcal Q\), and apply Proposition~\ref{prop:completed-reflection} to each
of the resulting finitely many completed sums.  For a fixed \(h_0\) and
active radical, the Proposition uses the same \(c_F,c,d,\vartheta,\kappa_F\),
other local phases, and kernel whether an absent prime is omitted from the
local set or included with exponent zero in its inactive branch.  Thus the
two inactive contributions at \(q\) combine with coefficient
\(1-(1-q_q^{-1})=q_q^{-1}\).  The active contribution comes only from the
subtracted zero-mask transform and has local factor
\(q_q^{-1/2}\chi_q(\lambda^4\mu)^{-2}\) and scalar
\(-\omega_{q,0}=\tau_{q,2}^{+}\chi_q(\epsilon_q)^{-2}\).
Collecting these choices at all marked primes proves
Equations~\eqref{eq:marked-reflection} and
\eqref{eq:marked-reflection-phase}.  The inactive prime is absent from
\(r\), hence from that branch's conductor and kernel scale.  This is finite
inclusion-exclusion between compatible branches, not a new theta identity.

If a marked prime is a base local prime, the mark forces it to divide
\(nb^3\), while the zero-extended base factor at that prime then vanishes,
including for exponent zero.  This proves the last assertion without any
division of a zero symbol.  It also covers the case where the primal
squarefree and cube ideals share the marked prime.
\end{proof}

For the row norm, the phase consequence of this corollary must be stated
with its exact scope.  For distinct active good primes \(p,q\),
Equation~\eqref{eq:reflection-cross-prime-phase} and cubic reciprocity give
\begin{equation}\label{eq:reflection-pair-phase}
 \chi_p(q)^{2j_p+2}\chi_q(p)^{2j_q+2}
       =\left(\frac qp\right)_3^{j_p+j_q+2}.
\end{equation}
For an active mark the exponent is \(j_q=0\); its changed sign occurs only
in its one-prime scalar.  A residual row prime with \(j_p=1\) and a marked
prime therefore have pair factor \((q/p)_3^3=1\).  A \(j_p=1\) row prime
and a moving \(j_q=4\) prime would instead leave \((q/p)_3\), which need
not be one.  All nonresidual base primes, especially every \(j=4\) prime,
are therefore fixed before the inner row norm.

Write \(R\) for the product of the residual \(j=1\) row primes and \(P\)
for the product of active marked primes.  Primal row-mark collisions are
removed first by the last assertion of the corollary, and are subsequently
represented by \(1_{(P,R)=1}\).  If the frozen active base product is
\(F_{\rm act}\), then \(r=F_{\rm act}RP\).  Fix separate classes of \(R\)
and \(P\) modulo the full \(M_{\rm ref}^2\), not merely their product and
not merely their smaller reciprocity classes.  This fixes the cusp data
without adding a pair-dependent restriction.  Row-row phases are row
scalars, mark-mark phases are tuple scalars, and their interactions with
frozen primes depend on only one of those sets.  The angular conductor
scalar factors in the same way.

After fixing the dual unit, its \(\lambda\)-valuation, and extracted frozen
factors, write the remaining dual part as \(nb^3\).  The moving columns are
exactly
\[
 \chi_R(nb^3)^3=\chi_R(nb)^3,\qquad
 \chi_P(nb^3)^{-2}=\chi_P(n)^{-2}1_{(P,b)=1}.
\]
These identities include every zero.  A zero caused by a row or a tuple
meeting an extracted frozen factor is a fixed restriction on that row or
tuple.  The source coefficient, fixed additive factor, and all frozen local
column factors are functions of the full extracted dual index alone.  Hence,
after common smooth separation, the tuple coefficient may be a general
bounded function of the tuple independent of \(R,n,b\), while the dual
coefficient is independent of \(R,P\).  The earlier product form of the
marks remains required when slots are assigned to extracted factors.

\subsection{A quadratic--cubic norm estimate}

Part I established the required orientation of Goldmakher--Louvel's
quadratic large sieve and proved the zero-preserving completed reduction in
Lemma~\ref{lem:completed-quadratic-reduction}.  We add Heath--Brown's cubic
large sieve:
\begin{equation}\label{eq:cubic-large-sieve}
 \sum_{\substack{a\equiv1\ (3)\\q_a\le U}}^{*}
 \left|\sum_{\substack{b\equiv1\ (3)\\q_b\le V}}^{*}
           c_b\left(\frac ba\right)_3\right|^2
 \ll_\epsilon (UV)^\epsilon
       \{U+V+(UV)^{2/3}\}\sum_b^{*}|c_b|^2,
\end{equation}
where both stars mean squarefree in \(\mathcal O\), and the coefficients
are arbitrary complex numbers \cite[Theorem 2]{HB}.  The indices need
not have squarefree rational norm, and there is no exclusion of rational
prime factors.  Conjugation and cubic reciprocity allow the opposite
orientation.  In this sieve and Equation~\eqref{eq:quadratic-large-sieve},
a fixed restriction on the row set
decreases the positive outer sum.  A fixed restriction on the coefficient
support is instead implemented by setting the omitted coefficients to zero
and applying the same theorem; no column-support monotonicity is claimed.
This observation does not allow an arbitrary pair-dependent mask; the
next lemma resolves the two such masks that it uses.

\begin{lemma}[A quadratic--cubic norm bound]\label{lem:hybrid-energy}
Let \(K,N,B,L\ge1\).  Let \(k,n,P\) be squarefree primary ideals with
\(q_k\ll K\), \(q_n\asymp N\), and \(q_P\asymp L\); let \(b\) be any
primary ideal with \(q_b\asymp B\).  The ideals \(k,P\) avoid the current
excluded set \(S\); additional fixed exclusions are allowed.  The source
ideals \(n,b\) may contain primes of \(S\) other than \(\lambda\).
Let \(a(P)\) and \(\beta(n,b)\)
satisfy
\[
 |a(P)|\le1,\qquad |\beta(n,b)|\le1.
\]
Here \(a\) is independent of \(k,n,b\), and \(\beta\) is independent of
\(k,P\).  Then
\begin{equation}\label{old-eq:4.6b}
 \begin{split}
 &\sum_k\left|
 \sum_P\frac{a(P)}{\sqrt{q_P}}1_{(P,k)=1}
 \sum_{n,b}\beta(n,b)\chi_k(nb)^3
                 \chi_P(n)^{-2}1_{(P,b)=1}
 \right|^2\\
 &\qquad\ll_\epsilon
 (KNBL)^\epsilon(K+NB)B\{N+L+(NL)^{2/3}\}.
 \end{split}
\end{equation}
Fixed restrictions on the \(k\)-set and fixed restrictions on the
\((n,b)\)- or \(P\)-supports are allowed, as are fixed ray sectors.
All such restrictions must preserve the two coefficient-independence
conditions above.  Divisor-bounded multiplicities may be included if
they preserve those conditions; otherwise they must first be removed by
the divisor Cauchy inequalities in the proof.
\end{lemma}

\begin{proof}
Write \(\mathcal H\) for the left side of Equation~\eqref{old-eq:4.6b}.
First remove the complete moving row-mark mask:
\[
 1_{(P,k)=1}=\sum_{D\mid(P,k)}\mu(D).
\]
For each row the number of such divisors is at most \(d_{\mathcal O}(k)\).
Rowwise divisor Cauchy, followed by the positive sum over rows, therefore
costs a permitted factor \((KNBL)^\epsilon\).  In a fixed \(D\)-summand write
\(P=DP'\) and \(k=Dk'\).  Squarefreeness gives the fixed restrictions
\((D,P')=(D,k')=1\).  Define
\[
 \begin{split}
 a_D(P')&=a(DP')1_{(D,P')=1},\\
 \beta_D(n,b)&=\beta(n,b)\chi_D(nb)^3\chi_D(n)^{-2}1_{(D,b)=1}.
 \end{split}
\]
All these factors retain their zeros and \(|\beta_D|\le1\).  Factoring
\(q_P^{-1/2}=q_D^{-1/2}q_{P'}^{-1/2}\) gives
\begin{equation}\label{eq:hybrid-mark-mask-reduction}
 \begin{split}
 \mathcal H\ll (KNBL)^\epsilon\sum_D\frac1{q_D}
 \sum_{k'}\left|
 \sum_{n,b}\beta_D(n,b)\chi_{k'}(nb)^3
 \sum_{P'}\frac{a_D(P')}{\sqrt{q_{P'}}}
       \chi_{P'}(n)^{-2}1_{(P',b)=1}
 \right|^2.
 \end{split}
\end{equation}
Here \(q_{k'}\ll K/q_D\), \(q_{P'}\asymp L/q_D\), and all inherited fixed
restrictions are retained.  The outer \(D\)'s are the squarefree good divisors
allowed by the split, so \(q_D\ll\min(K,L)\).  In particular, no \((P',k')\) mask is reinstated
inside an individual \(D\)-summand.  Its cancellation belongs to the complete
M\"obius sum already bounded by rowwise Cauchy.

For fixed \(D\), the product of \(\beta_D(n,b)\) and the inner \(P'\)-sum
is an arbitrary complex coefficient independent of \(k'\).  Apply
Equation~\eqref{eq:completed-quadratic-reduction} with row length \(K/q_D\).
Use its notation
\[
 b=gt^2,\qquad c=(n,g),\qquad n=cm,\qquad g=ch,
 \qquad q_c\asymp C,\quad q_g\asymp G,\quad q_t\asymp T,
\]
so \(B\asymp GT^2\), and write \(c=rc'\) for its complete \(c\)-mask
divisor.  The ideals \(c,m,h\) are pairwise coprime and squarefree; no
coprimality between \(g\) and \(t\) is imposed.  The cited reduction already
includes the natural \(k'\)-\(t\) restriction and the complete M\"obius
expansion of the \(k'\)-\(c\) mask.  It does not reinstate a mask between
the remaining row and \(c'\).

For one \(C,G,T\) block its positive output, inserted in
Equation~\eqref{eq:hybrid-mark-mask-reduction}, is at most
\begin{equation}\label{eq:hybrid-after-quadratic-reduction}
 \begin{split}
 (KNBL)^\epsilon\sum_D\frac1{q_D}T\sum_{q_t\asymp T}
 \sum_{\substack{r\ {\rm sf},\ (r,S)=1\\q_r\ll\min(K/q_D,C)}}
 &\left\{\frac K{q_Dq_r}+\frac{NG}{C^2}\right\}\frac C{q_r}
 \sum_{c',m,h}|\beta_D(rc'm,rc'ht^2)|^2\\
 &\quad\cdot\left|
 \sum_{P'}\frac{a_{D,r}(P')}{\sqrt{q_{P'}}}
       \chi_{P'}(c'm)^{-2}1_{(P',ht)=1}
 \right|^2,
 \end{split}
\end{equation}
where the positive \(c',m,h\)-sum keeps the original support and
\[
 a_{D,r}(P')=a_D(P')\chi_{P'}(r)^{-2}.
\]
The factorization used here is, including all zeros,
\[
 \chi_{P'}(rc'm)^{-2}1_{(P',rc'ht^2)=1}
 =\chi_{P'}(r)^{-2}\chi_{P'}(c'm)^{-2}1_{(P',ht)=1}.
\]
The character already supplies the missing zeros at \(r,c'\), and \(m\).
Thus the \(r\)-factor is tuple-only, including its zero when \((P',r)\ne1\).
The fixed \(D\)-phases remain inside \(\beta_D\) until this positive bound.
The quadratic reduction has already separated the finitely many squarefree
\(S\)-parts of \(m,h\) before sieving their good product; \(m,h\) in
Equation~\eqref{eq:hybrid-after-quadratic-reduction} are the original ideals.

We may now discard \(|\beta_D|^2\le1\).  For fixed \(h,t,D,r\), the
\(P'\)-coefficients are independent of \(j=c'm\), and ideal counting gives
\[
 \sum_{P'}\frac{|a_{D,r}(P')|^2\,1_{(P',ht)=1}}{q_{P'}}
 \ll_\epsilon (KNBL)^\epsilon.
\]
The product \(j=c'm=n/r\) is squarefree primary and has norm \(O(N/q_r)\).
It may contain a permitted prime of \(S\) other than \(\lambda\), since only
the quadratic column was stripped of its fixed \(S\)-part.  Grouping \(c',m\)
by \(j\) costs at most a divisor factor.  Enlarge only this positive
squarefree \(j\)-range.  Cubic reciprocity, conjugation, and
Equation~\eqref{eq:cubic-large-sieve}, followed by the \(O(G/C)\) choices
of \(h\), give, on \(q_D\asymp D_0,\ q_r\asymp r_0\),
\begin{equation}\label{old-eq:4.6c}
 (KNBL)^\epsilon\frac GC
 \left\{\frac N{r_0}+\frac L{D_0}
       +\left(\frac{NL}{r_0D_0}\right)^{2/3}\right\}.
\end{equation}
The fixed mask at \(ht\) is a coefficient restriction in this cubic sieve,
not a row-dependent mask.

The reduction supplies \(T\sum_t\), with \(O(T)\) possible \(t\), and
the factor \(C/q_r\).  Together with the \(h\)-count, these contribute
\[
 T^2(C/r_0)(G/C)\asymp B/r_0.
\]
There are \(O(r_0)\) possible \(r\) in its dyad, while the sum of
\(q_D^{-1}\) over its dyad is \(O(1)\), up to a permitted small power.
Both sieve factors increase when \(D_0,r_0\) are replaced by \(1\), and
\(NG/C^2\ll NB\).  Summing the logarithmically many dyads proves
Equation~\eqref{old-eq:4.6b}.  Subunit quotient ranges are empty, and bounded
ranges are included by changing the fixed annular constants.  This argument
has introduced neither a \((g,t)=1\) condition nor a new pair-dependent mask.
\end{proof}

\subsection{The reflected energy bound}

We next quantify the reflection of Equation~\eqref{eq:marked-completion}.
The following description also specifies the dyadic contribution used in
the statement.

\paragraph{Reflected block data.}\label{par:reflected-energy-data}
Fix \(f,\rho\) and put \(Q=\log_Zq_{\mathfrak r_\rho}\).  For an element
row \(0<q_k\ll Z^M\), define the maximal powerful part of its ideal by
\[
 k_{\rm pow}=\prod_{v_p(k)\ge2}p^{v_p(k)}.
\]
Split the valuation-one primes into the squarefree product
\(k_{\rm sup}\) supported on
\(S\cup\operatorname{supp}(f\mathfrak r_\rho)\) and the squarefree
product \(k_{\rm res}\) outside that set.  These three prime supports
are pairwise disjoint, and their product is the ideal of \(k\).
A powerful ideal here means that every positive prime valuation is at
least two.  Fix the unit of \(k\).  Insert smooth dyadic partitions in
\[
 q_{k_{\rm pow}}\asymp Z^O,\qquad q_{k_{\rm res}}\asymp Z^H.
\]
Choose these ideal centers nonnegative, with center zero for the unit dyad.
In particular, an actual annular factor
\(\chi_{\rm row}(q_{k_{\rm res}}/Z^H)\) is kept in the row sum.  The
bounded dyad includes \(k_{\rm res}=1\).  Choose fixed constants
\(C_k,C_O,C_H\ge1\), from these supports, such that
\[
 q_k\le C_kZ^M,\qquad q_{k_{\rm pow}}\ge Z^O/C_O,\qquad
 q_{k_{\rm res}}\ge Z^H/C_H.
\]
Since \(q_{k_{\rm sup}}\ge1\), the general norm inequality
Equation~\eqref{eq:actual-residual-row-dyad} gives here
\[
 H\le M-O+\frac{\log C_{\rm res}}{\log Z},
 \qquad C_{\rm res}=C_HC_kC_O.
\]
The inequality need not be an equality, because \(k_{\rm sup}\) may
have positive norm length.

Freeze the actual ideals \(k_{\rm pow},k_{\rm sup}\).  At every prime
outside \(S\), combine the local exponents of
\(\rho(n)\chi_n(k)\chi_n(f)^4\) modulo six, always retaining the zero
extension.  The primes of \(k_{\rm res}\) have exponent \(j=1\).
All other such primes, including every non-slot prime of exponent
\(j=4\), are now fixed.  In the reflection formula call a
prime active when it occurs in \(c\).  For a fixed choice of the active
zero-exponent primes and a fixed splitting of each \(j=4\) Ramanujan
factor, let:
\begin{center}
\begin{tabularx}{\linewidth}{@{}>{$}l<{$} X@{}}
 A_0 & be the log-norm of all active non-slot primes outside
             \(k_{\rm res}\),\\
 S_0 & be the log-norm of the small Ramanujan terms and active
             zero-mask terms among them,\\
 N_0 & be the log-norm of the \(j=4\) divisibility terms assigned
             to the squarefree dual ideal,\\
 B_0 & be the log-norm of the \(j=4\) divisibility terms assigned
             to the cube ideal but not the squarefree ideal.
\end{tabularx}
\end{center}
Here a log-norm is \(\log_Z\) of the norm of the indicated product.
The assignment uses
\(1_{p\mid nb^3}=1_{p\mid n}+1_{p\nmid n}1_{p\mid b}\).
Let \(z_a=\sum_{i\ {\rm active}}z_i\) be the sum of their nominal slot
lengths.  The product of the active primes has norm divided by \(Z^{z_a}\)
in a fixed compact interval, including for zero nominal lengths or the
empty list.  After extracting
the forced squarefree and cube primes, restrict the remaining dual ideals
to
\[
 q_n\asymp Z^v,\qquad q_b\asymp Z^{\ell_b},\qquad
 q_{\lambda^{h_\lambda}}=Z^{e_\lambda}.
\]
Choose the ideal centers \(v,\ell_b\) nonnegative.  These are source dual
ideals: they retain every prime of \(S\) permitted by
Equation~\eqref{old-eq:4.3}; no extra \(S\)-mask is placed on them.  Only
the displayed \(\lambda\)-valuation support is imposed.
The unit and the integer \(h_\lambda\ge-4\) are fixed in this dyad.
For these choices, denote by
\(\mathcal U_{v,\ell_b,e_\lambda}(k_{\rm res})\) the right side of
Equation~\eqref{eq:marked-reflection}, summed with the product-form tuple
coefficients, with the
specified local choices and dual restrictions, and the actual factor
\(\chi_{\rm row}(q_{k_{\rm res}}/Z^H)\).  Active slot tuples are still
summed in this definition.  This definition is made separately for
each fixed powerful and supported row part and each sector obtained by
fixing separate classes of the residual row product and active slot
product modulo the full reflection modulus \(M_{\rm ref}^2\).

\begin{lemma}[Reflected energy]\label{lem:reflected-energy}
Retain the \hyperref[par:reflected-energy-data]{reflected block data} above.
Let \(Z\ge2\), and suppose their log-lengths range over fixed bounded sets.
In the completed sum of Equation~\eqref{eq:marked-completion}, the fixed character,
puncture, and slot coefficients are independent of \(k,f\), the slot
supports are disjoint, and the slot coefficients are products as in
Equation~\eqref{old-eq:4.2}.  The fixed row restrictions after freezing
\(k_{\rm pow},k_{\rm sup}\) must be independent of the active slots and
dual variables, apart from the explicit zero mask
\(1_{(P,k_{\rm res})=1}\).  Fix a small \(\tau_{\rm ref}>0\), put
\(X=Z^{N_*}\).  Define \(T_d\) and call a dyad retained by the following
formula:
\begin{equation}\label{old-eq:4.4}
 \begin{gathered}
 T_d=2H+2A_0+2z_a-N_*-N_0-3B_0,\\
 v+3\ell_b+e_\lambda\le T_d+\tau_{\rm ref}
 \quad\text{for a retained dyad}.
 \end{gathered}
\end{equation}
Terms with
\[
 v+3\ell_b+e_\lambda>T_d+\tau_{\rm ref}
\]
have arbitrarily small total size, with a fixed polynomial height cost,
after a sufficiently far kernel contour shift and a threshold for \(Z\)
depending only on the fixed support intervals and \(\tau_{\rm ref}\).
Every retained dyad has coefficient exponent, per active
tuple,
\begin{equation}\label{old-eq:4.5}
 -v/2-\ell_b-e_\lambda/3-(S_0+B_0+z_a)/2.
\end{equation}
Define
\begin{equation}\label{old-eq:4.7}
 u=\min\{v,z_a,(v+z_a)/3\},
\end{equation}
and
\begin{equation}\label{old-eq:4.6}
 \begin{split}
 E_{\rm ref}={}&O/2+\max(H,v+\ell_b)-S_0-B_0+z_a-u-\ell_b
                     -2e_\lambda/3\\
 &-\tfrac12(T_d-v-3\ell_b-e_\lambda)_+.
 \end{split}
\end{equation}
For every \(\epsilon>0\), the contribution with fixed
\(k_{\rm pow},k_{\rm sup}\) satisfies
\[
 \sum_{k_{\rm res}\ {\rm sf}}
       |\mathcal U_{v,\ell_b,e_\lambda}(k_{\rm res})|^2
 \ll Z^{E_{\rm ref}-O/2+\epsilon}.
\]
Summing the fixed parts in their \(O\)-dyad and the local choices changes
this to \(O(Z^{E_{\rm ref}+\epsilon})\).  The bounds are uniform in the
moving ideals and punctures in the stated ranges, with finitely many
smooth seminorms and a fixed polynomial cost for norm-twist heights.
The negative value of \(e_\lambda\), when present, is
\(O(1/\log Z)\).  In particular, the formula is used only on actual
residual-row dyads satisfying Equation~\eqref{eq:actual-residual-row-dyad}.
\end{lemma}

\begin{proof}
First remove the inactive slots by triangle inequality in the row Hilbert
space.  At such a slot the absolute coefficient mass is
\[
 \sum_{p\in\mathcal P_i}q_p^{-1}\ll_\epsilon Z^\epsilon,
\]
uniformly under any fixed restriction, including for a bounded or zero-length
list.  It therefore suffices to prove a uniform bound with those primes fixed.
Each such branch has its own conductor, cusp sector, and kernel scale, with
the inactive prime absent from the conductor.  Its earlier collision exclusion
is now a fixed row restriction.

Fix \(f,\rho,k_{\rm pow},k_{\rm sup}\), the non-slot local choices, \(h_0\),
and separate classes of the residual row product and active slot product
modulo the full \(M_{\rm ref}^2\).  Also fix the unit and ramified exponent of
the dual index.  No non-slot \(j=4\) label remains averaged.  We use only the
algebraic frozen-base extraction in
Equation~\eqref{eq:unmarked-structural-block}, inside the original sum over
active tuples.  Corollary~\ref{cor:marked-reflection} and the subsequent phase
separation make that extraction simultaneous across the tuples: the base
coefficient is common to the residual rows and tuples after their separate
sectors are fixed, while the remaining scalar separates into a row factor and
a tuple factor.  We do not apply the numerical unmarked estimate separately
to each tuple.

The structural extraction gives the common central coefficient
\[
 Z^{-v/2-\ell_b-e_\lambda/3-(S_0+B_0)/2}
\]
times normalized inverse-root factors and bounded coefficients.  Its
\(N_0\)-assignment extracts a forced \(j=4\) prime only from the squarefree
dual ideal, even if the cube ideal shares it; the entire shared cube part
remains in its residual norm and coefficient.  Its \(B_0\)-assignment extracts
one occurrence from the cube ideal and leaves the factor \(q_p^{-1/2}\).
These are statements about the full source support, including the permitted
primes of \(S\).  Every active marked column in
Equation~\eqref{eq:marked-reflection} contributes \(q_p^{-1/2}\).
Since \(q_P/Z^{z_a}\) lies in a fixed compact interval, this proves the
per-tuple coefficient exponent in Equation~\eqref{old-eq:4.5}.

We now record the precise marked instance of the common profile.  Let
\(F_{\rm act},N_F,B_F\) be the actual frozen products with log-norms
\(A_0,N_0,B_0\).  Put \(R=k_{\rm res}\), \(P=\prod_{i\ {\rm active}}p_i\).
The branch denominator and a supported dual index are
\[
 c=c_FF_{\rm act}RP,\qquad
 \mu=u\lambda^{h_\lambda}N_Fn(B_Fb)^3.
\]
With the fixed block centers, define
\[
 \begin{gathered}
 y_R=q_R/Z^H,\quad y_n=q_n/Z^v,\quad
 y_b=q_b/Z^{\ell_b},\quad y_i=q_{p_i}/Z^{z_i},\\
 Y=\frac{q_{\lambda^{h_\lambda}N_FB_F^3}X Z^vZ^{3\ell_b}}
          {q_{c_FF_{\rm act}}^2Z^{2H}\prod_{i\ {\rm active}}Z^{2z_i}}
   =q_{c_F}^{-2}Z^{v+3\ell_b+e_\lambda-T_d}.
 \end{gathered}
\]
Multiplicativity gives the exact identity
\begin{equation}\label{eq:reflection-actual-annuli}
 \frac{q_\mu X}{q_c^2}
       =Yy_ny_b^3y_R^{-2}\prod_{i\ {\rm active}}y_i^{-2}.
\end{equation}
It remains true if \(N_F\) shares primes with \(b\).  All frozen factors
in \(Y\) are their actual products, and every moving norm remains a coordinate.
The actual row cutoff and the individual dual and slot cutoffs therefore place
the kernel argument in
\[
 [C_{\rm ker}^{-1},C_{\rm ker}]
       Z^{v+3\ell_b+e_\lambda-T_d}
\]
for one \(C_{\rm ker}\ge1\) depending only on the fixed support intervals and
the common fixed modulus.  In particular this comparison has not used any
shorter row added by positivity.

Put \(D_{\rm ker}=T_d-v-3\ell_b-e_\lambda\).  On this full product of actual
annuli, Lemma~\ref{lem:reflection-common-profile} permits
\[
 m_{\rm ker}=Z^{-(D_{\rm ker})_+/4}
\]
to be factored from the entire homogeneous linear row vector, at a factor
at most \(C_{\rm ker}^{1/4}\) in its fixed seminorm constants.  The normalized
profile has bounded fixed Euler seminorms; no inhomogeneous tuple seminorm is
claimed to become small.  The profile includes the normalized inverse roots
from the structural extraction and every other common smooth norm window.

The tail is removed on the genuine, unseparated sums.  If
\(\log Z\ge2\log C_{\rm ker}/\tau_{\rm ref}\), every whole dyad with center
excess greater than \(\tau_{\rm ref}\) has actual kernel argument greater
than \(Z^{\tau_{\rm ref}/2}\).  Apply Lemma~\ref{lem:lattice-kernel-tail}
with lattice \(\lambda^{-4}\mathcal O\),
\(a=X/q_c^2\), and \(U=Z^{\tau_{\rm ref}/2}\).  On the full source support,
the coefficient bound used in Part I gives
\[
 |d(\mu)|/\sqrt{q_\mu}\le27q_\lambda^{4/3}.
\]
Each base or marked local factor is at most \(q_p^{1/2}\) after its
indicator is discarded.  The bounded row, tuple, and local log-length ranges,
the local branch counts, and \(1+a^{-1}\) have a fixed polynomial cost
\(Z^{B_{\rm tail}}\), with \(B_{\rm tail}\) chosen independently of the
kernel order.  For any desired saving \(D>0\), the choice
\[
 A>1+\frac{2(B_{\rm tail}+D)}{\tau_{\rm ref}}
\]
in that shell lemma makes the absolute total of these whole dyads
\(O(Z^{-D})\) times a finite test seminorm.  Norm twists have the fixed
polynomial height cost of that seminorm.  The shell sum counts all discarded
dual indices, so no count at a retained dual length is used for this tail.
It is removed before Fourier absolutization.

For a retained dyad, apply the common-profile lemma and
Equation~\eqref{eq:reflection-common-minkowski} to the one joint profile in
Equation~\eqref{eq:reflection-actual-annuli}.  Its density is common to the
entire current row Hilbert space, including the active-tuple sum: the actual
row, dual, and individual slot norms are its coordinates, not parameters of
separately chosen measures.  The full fixed support boxes govern every invoked
weighted Fourier norm and height cost.  At a separated mode the norm powers
have absolute value one and preserve the row, tuple, and dual coefficient
independences.  Only inside this separated nonnegative row norm may the row
set be enlarged from the original annulus to \(q_R\ll Z^H\), using the same
separated formula on the added rows.  The kernel is never evaluated there.

After extracting the common coefficient but before using \(m_{\rm ker}^2\),
the separated squared norm is bounded by
\[
 Z^{-v-2\ell_b-2e_\lambda/3-S_0-B_0+\epsilon}\mathcal N,
\]
where
\begin{equation}\label{old-eq:4.6a}
 \begin{split}
 \mathcal N={}&
 \sum_{\substack{k_{\rm res}\ {\rm sf}\\q_{k_{\rm res}}\ll Z^H}}
 \left|
 \sum_{\boldsymbol p}
 \frac{a(\boldsymbol p)}{\sqrt{q_P}}1_{(P,k_{\rm res})=1}
 \sum_{\substack{n\ {\rm sf}\\b}}
 \beta(n,b)\chi_{k_{\rm res}}(nb)^3
              \chi_P(n)^{-2}1_{(P,b)=1}
 \right|^2,\\
 &P=\prod_{i\ {\rm active}}p_i,\qquad q_P\asymp Z^{z_a}.
 \end{split}
\end{equation}
The ideals \(n,b\) retain their stated dyadic norms and fixed coefficient
restrictions.  After extracting common bounds,
\(|a(\boldsymbol p)|,|\beta(n,b)|\le1\).

We verify the two independence hypotheses before invoking the norm bound.
Every residual prime has exponent \(j=1\), every active slot is a whole-index
\(j=0\) mark, and all other active primes are frozen.  The separate full
\(M_{\rm ref}^2\) classes fix the cusp coefficient and additive factor.
Equation~\eqref{eq:reflection-pair-phase} cancels every row-slot phase, while
the preceding marked-reflection discussion assigns all row-row, slot-slot,
and frozen interactions to a row scalar or a tuple scalar.  The angular
conductor scalar separates in the same fashion.  The remaining source and
frozen local columns depend only on the full extracted dual index; their
zeros at a frozen factor give fixed row, tuple, or dual restrictions.
Finally the exact zero-preserving column identities leave only
\(1_{(P,k_{\rm res})=1}\) and \(1_{(P,b)=1}\) in addition to the character
zeros.  Hence \(a\) is independent of \(k_{\rm res},n,b\), and \(\beta\)
is independent of \(k_{\rm res},\boldsymbol p\), exactly as required in
Equation~\eqref{old-eq:4.6a}.  The function \(a\) need not retain product
form at this norm step.

Apply Lemma~\ref{lem:hybrid-energy} with
\[
 K=Z^H,\qquad N=Z^v,\qquad B=Z^{\ell_b},\qquad L=Z^{z_a}.
\]
Aggregating tuples with the same \(P\) costs only a divisor factor.  Since
\[
 v+z_a-u=\max\{v,z_a,2(v+z_a)/3\},
\]
the hybrid bound and the outside coefficient give exponent
\[
 \max(H,v+\ell_b)-S_0-B_0+z_a-u-\ell_b-2e_\lambda/3.
\]
Only now does the squared scalar \(m_{\rm ker}^2\) supply the last term in
Equation~\eqref{old-eq:4.6}.  This proves the fixed-part assertion.

Every powerful ideal is uniquely \(x^2y^3\) with \(y\) squarefree.  Ideal
counting and \(\sum_yq_y^{-3/2}<\infty\) give \(O(Z^{O/2})\) such ideals
in the \(O\)-dyad.  The supported squarefree part divides the radical of
\(S f\mathfrak r_\rho\), so it has \(O(Z^\epsilon)\) choices in the stated
ranges; the local choices have the same divisor bound.  Rowwise divisor
Cauchy on the local expansion followed by the positive sum over these fixed
parts therefore adds \(O/2+\epsilon\) once.  The active tuples were already
inside the hybrid norm and are not counted again.  This proves the aggregate
assertion.
\end{proof}

The dependence on the original row length can be bounded without
identifying \(H\) with \(M-O\).  At one active non-slot prime the
contribution to
\(2A_0-N_0-S_0-4B_0\) is as follows:
\[
 \begin{array}{c|c}
 \text{local choice}&\text{coefficient of its log-norm}\\ \hline
 j\text{ odd, or }j=2&2\\
 j=0\text{ active}&1\\
 j=4\text{ small or assigned to }n&1\\
 j=4\text{ assigned to }b&-2\\
 j=0\text{ inactive}&0.
 \end{array}
\]
A residual row prime contributes two through \(2H\).  Prime by prime,
these coefficients are bounded by the valuations in
\[
 (q_k^2/q_{k_{\rm pow}})\,q_fq_{\mathfrak r_\rho}.
\]
Indeed, a powerful prime of row valuation \(e\ge2\) has valuation
\(e\) in the first factor, a squarefree row prime has valuation two,
and a prime belonging only to the squarefree \(f\) or to
\(\mathfrak r_\rho\) has valuation one.  Intersections only increase
the valuation of this upper bound.  Fixed excluded primes contribute
only a fixed factor \(C_S\ge1\).  Equivalently, summing over primes gives
the actual norm inequality
\[
 q_{k_{\rm res}}^2 Z^{2A_0-N_0-S_0-4B_0}
 \le C_S\frac{q_k^2}{q_{k_{\rm pow}}}\,q_fq_{\mathfrak r_\rho}.
\]
Choose a fixed \(C_f\ge1\) with \(q_f\le C_fZ^V\), and put
\(C_{\rm width}=C_H^2C_SC_k^2C_OC_f\).  The preceding inequality and
the definition of \(T_d\) imply
\begin{equation}\label{eq:common-width-charge}
 T_d-S_0-B_0
 \le 2M-O+Q+V-N_*+2z_a+
       \frac{\log C_{\rm width}}{\log Z}.
\end{equation}

For empty slot lists, \(z_a=u=0\).  Choose \(\eta>0\) and then take \(Z\)
so large that the two displayed constant errors are at most \(\eta\) and
\(e_\lambda\ge-\eta\); the latter follows from
\(\log Z\ge4\log q_\lambda/\eta\).  On a retained dyad,
Equation~\eqref{old-eq:4.6} and
\(v+3\ell_b+e_\lambda\le T_d+\tau_{\rm ref}\) give
\[
 E_{\rm ref}
 \le O/2+\max\{H,T_d-S_0-B_0\}+\tfrac53\eta+\tau_{\rm ref}.
\]
For the first branch, drop the nonpositive terms after using the lower
bound for \(e_\lambda\); for the second, use
\(v\le T_d-3\ell_b-e_\lambda+\tau_{\rm ref}\).
The maximum is increasing in each argument.  We may therefore use
Equation~\eqref{eq:actual-residual-row-dyad} for its first argument and
Equation~\eqref{eq:common-width-charge} for its second, then sum the
logarithmically many actual \(H\)-dyads.  Taking \(\eta,\tau_{\rm ref}\)
and the local small-power losses sufficiently small in terms of
\(\epsilon\) proves
\begin{equation}\label{eq:completed-moment}
 \sum_{0<q_k\ll Z^M}'|\mathcal T_1(Z^{N_*};k)|^2
 \ll Z^{O/2+\max\{M-O,\,2M-O+Q+V-N_*\}+\epsilon}.
\end{equation}
The prime on the sum restricts the powerful part to its \(O\)-dyad.
A bounded dual range is included in the first term; an empty range is
negligible.  No equality \(H=M-O\) has been used.

\section{The compensated low estimate}\label{sec:compensated-low}

We now bound the finite compensated probe defined in
Equation~\eqref{eq:compensated-probe-definition}.  Its exact low separation
has two factors.  We first bound the completed row after summing the marked
slots by applying Lemma~\ref{lem:reflected-energy} directly.  We then prove
a quantitative Gram bound for the additive polynomial from
Section~\ref{sec:probe}.  Cauchy--Schwarz will combine these estimates to
give the exponent $3/16$.

\subsection{The marked completed row}

For a fixed rescaled subset $J$, with marked subset $J^c$, put
\begin{equation}
 \begin{gathered}
 d=\sum_{i\in J}\ell_i,\qquad
 X'=X/q_{p_J}\asymp XZ^{-d},\qquad
 Y'=Y/q_{p_J}\asymp YZ^{-d},\\
 M'=M-2d,\qquad \ell'=\ell-d,\qquad 0\le d\le\ell=1/6.
 \end{gathered}
 \label{old-eq:5.2}
\end{equation}
For a squarefree product $D$ of slot primes, define the marked completed
series
\[
 T_D(s_0,\psi)=\sum_{\substack{c\ {\rm sf}\\n}}
 \gamma_2(c)\overline{\alpha(cn^3)}\psi(cn^3)
 1_{D\mid cn^3}q_c^{-s_0}q_n^{-3s_0+1/2}.
\]
The ideals $c,n$ are the original ideals prime to $S$, represented by their
primary generators, with $c$ squarefree;
they may share primes.  All character powers retain their original zero
extensions.  For the empty product $D=1$ this is the completed $T(s_0,\psi)$.
For this fixed $J$, let
\[
\begin{split}
 B^J_{m,\sigma}(Z)=
 \sum_{p_i\in\mathcal P_i(Z),\ i\notin J}
 \prod_{i\notin J}\overline{\eta(p_i)}W_i(q_{p_i}/P_i)
 \sum_{\theta\in\widehat T}a_\theta\frac1{2\pi i}
 \int_{(4)} &(Zq_{p_{J^c}})^t\Phi(t)\\
 &\cdot T_{p_{J^c}}(t+1/2,
             \nu_\sigma\theta\chi_\bullet(m))\,dt.
\end{split}
\]
Thus $B^J$ is exactly the completed-row factor for the $J$ summand
of Equation~\eqref{eq:compensated-probe-definition} after the
marked slots have been summed.  The fixed tuple $p_J$ affects
only $X',Y'$ in its low separation.

\begin{lemma}[The compensated completed-row norm]\label{lem:probe-row-norm}
For every $\epsilon>0$, every fixed rescaled subset $J$, and every
$\sigma\in T$, with $Q=q_{b_*}X'Y'\asymp Z^{M'}$,
\begin{equation}
 \sum_{0<q_m\ll Q}|B^J_{m,\sigma}(Z)|^2
       \ll Z^{M'+((d-1/6)/4)_++\epsilon}
       =Z^{M'+\epsilon}.
 \label{old-eq:5.4}
\end{equation}
The bound is uniform in the fixed rescaled tuple and has no
loss depending on the mesh of the surviving slots.
\end{lemma}

\begin{proof}
Put $A=cn^3$ and $\varrho_i=q_{p_i}/P_i$ for $i\notin J$.  Since
$q_A=q_cq_n^3$ even when $c,n$ share primes, and
$q_{p_{J^c}}=Z^{\ell'}\prod_{i\notin J}\varrho_i$, Mellin inversion
of the displayed integral defining $B^J$ gives the physical smooth factor
\[
 \begin{gathered}
 V\!\left(\frac{q_A}{Zq_{p_{J^c}}}\right)
 =W_{\mathrm G}\!\left(\frac{r}{\prod_{i\notin J}\varrho_i}\right),\\
 r=\frac{q_A}{Z^{1+\ell'}}.
 \end{gathered}
\]
Here $V=W_{\mathrm G}$ is the Gaussian of
Lemma~\ref{lem:gaussian-annular}.  Empty products are one.

Use the fixed translate partition $\chi$ from that lemma on $r$.
On the annulus $r=e^kx$, retain the single joint profile
\[
 w_{k,J}(x,\boldsymbol\varrho)
 =\chi(\log x)W_{\mathrm G}\!\left(
       \frac{e^kx}{\prod_{i\notin J}\varrho_i}\right)
       \prod_{i\notin J}W_i(\varrho_i).
\]
The logarithms of $x$ and of all the $\varrho_i$ lie in fixed compact
sets on this support.  Thus the logarithm of the Gaussian argument is
$k+O_{\rm fixed}(1)$.  The derivative estimate in
Lemma~\ref{lem:gaussian-annular}, with this bounded shift and the fixed
slot windows, gives for every fixed $\kappa_{\mathrm G}>0$,
$A_{\rm cnt},B_{\rm cnt},C_{\rm ann}\ge0$, $D_{\rm tail}>0$, and
fixed seminorm order $j$,
\[
 \begin{gathered}
 \sum_{k\in\mathbb Z}e^{A_{\rm cnt}|k|}p_j(w_{k,J})<\infty,
 \\
 Z^{B_{\rm cnt}}
 \sum_{|k|>\kappa_{\mathrm G}\log Z-C_{\rm ann}}
       e^{A_{\rm cnt}|k|}p_j(w_{k,J})
       \ll Z^{-D_{\rm tail}}.
 \end{gathered}
\]
The constants may depend on the fixed slot system, but not on its moving
prime labels.  On an annulus $\log r=k+O(1)$, an absolute bound for the
row $\ell^2$ norm is at most
$Z^{B_{\rm cnt}}e^{A_{\rm cnt}|k|}p_j(w_{k,J})$ for some fixed exponents,
by the elementary row, slot, and ideal counts.  Choose $C_{\rm ann}$ larger
than the fixed logarithmic radius of $\chi$ and discard only whole annuli
with $|k|>\kappa_{\mathrm G}\log Z+C_{\rm ann}$.  Summing their norm bounds
and then squaring gives less than any prescribed power of $Z^{-1}$.
Every annulus meeting $|\log_Z r|\le\kappa_{\mathrm G}$ is retained.

For each remaining annulus, keep fixed individual annular cutoffs equal to
one on the support of $w_{k,J}$ and apply
Lemma~\ref{lem:smooth-calculus} to this whole profile.  Its logarithmic
Fourier inversion uses one coefficient density common to every row and
every surviving slot tuple.  The normalized slot ratios remain variables
of the joint profile until separation; they do not index separately
chosen densities.  For fixed Fourier variables, each resulting slot
coefficient is a product of the
original arithmetic coefficient $\overline{\eta(p_i)}$ and an individual
annular cutoff and norm power.  It remains bounded and independent of the
row and of every other slot.  The completed factor is an annular test at
length
\[
 N_*=1+\ell'+\theta_N,\qquad
 \theta_N=\frac{k}{\log Z},\qquad
 |\theta_N|\le\kappa_{\mathrm G}+O_{\rm fixed}(1/\log Z).
\]
These retained lengths lie in a fixed bounded range.  The weighted Fourier
norms have a summable total by the displayed Gaussian estimate; the small
annular weight stays in this homogeneous single-profile Fourier norm, not
in an inhomogeneous tuple seminorm.  Minkowski's inequality passes the
separated row norm through this common density.  The existing $q_c^{-1/2}q_n^{-1}$
normalization stays unchanged when the test scale is recentered, so no
additional power of $e^k$ is introduced.  Choose $\kappa_{\mathrm G}$
within the final $\epsilon$ allowance.  For each fixed $\theta$,
Lemma~\ref{lem:reflection-row-sectors} supplies the row-sector reduction of
$\nu_\sigma\theta\chi_\bullet(m)$ with every original zero mask.  It covers
all nonzero element rows, including those meeting $S$; no factor $\xi(m)$ is
used in this row norm.  The row ball depends only on the fixed rescaled tuple
$p_J$, through $Q=q_{b_*}X'Y'$, and is independent of the surviving marked
slots and dual variables.  The physical slot sets remain disjoint and their
separated coefficients are product-form.  Finite triangle over $\theta$ and
Lemma~\ref{lem:reflected-energy} therefore give the exponent in
Equation~\eqref{old-eq:4.6} at this actual $N_*$, taking the retention
parameter $\tau_{\rm ref}$ and the independently prescribed output
and separation losses all at most a small $\epsilon_0>0$.  We use
the exponent $E_{\rm ref}$ after the fixed row parts are summed;
their count is already included in $E_{\rm ref}+\epsilon_0$, with
principal exponent $O/2$, and is not counted again.

Here $f=1$ and the puncture is $\rho=1$, so the only moving non-slot
primes are those of the row.  After the finite unit and reflection-sector
splits in Lemma~\ref{lem:reflected-energy}, freeze its powerful part of norm length
$O$ and its valuation-one part supported on $S$, and split its residual
squarefree good product into \emph{actual} dyads of norm $\asymp Z^H$.  Let
$\epsilon_Z=O_{\rm fixed}(1/\log Z)$ be nonnegative and large enough
to contain all fixed annular and fixed-conductor logarithmic offsets
in this calculation.  Then
\[
 H\le M'-O+\epsilon_Z,\qquad
 \Delta_H=M'-O-H\ge-\epsilon_Z,\qquad O\ge-\epsilon_Z.
\]
The actual row and dual annular cutoffs are kept in one joint
profile while its common Fourier density is separated.  The small
kernel amplitude is extracted from that homogeneous density before
the row norm is squared, and only then may the positive sieve sum be
enlarged, as in Equation~\eqref{eq:reflection-common-minkowski}.
In particular the kernel saving in
Equation~\eqref{old-eq:4.6} uses this $H$; it is not reevaluated on
any shorter rows added by the positive enlargement.

In the notation of that energy estimate, every moving non-slot prime
counted in $A_0$ divides the powerful row part to exponent at least
two.  The comparison of its exact norm with the powerful dyad, and
any fixed-conductor contribution, therefore gives
\[
 2A_0\le O+\epsilon_Z,\qquad N_0\le A_0,\qquad z_a\le\ell'.
\]
The available dual length at the actual completed scale is
\[
 T_d=2H+2A_0+2z_a-1-\ell'-\theta_N-N_0-3B_0.
\]
Since $M'+\ell'-1=-3d$, direct subtraction gives
\[
 T_d-(H-3d)
 =H-M'+2A_0+2(z_a-\ell')-N_0-3B_0-\theta_N
 \le 2\epsilon_Z+|\theta_N|.
\]
As in the proof of the reflected energy, dual ranges below a fixed
negative length are negligible after a positive Mellin shift, and
the bounded boundary range costs an arbitrarily small power.
The source dual ideals $n,b$ in these dyads retain the support of
Equation~\eqref{old-eq:4.3}: they may share primes and may contain
the permitted primes of $S$; no additional $S$-mask or coprimality
condition between $n$ and $b$ is imposed.
On each remaining dual dyad, put $y=v+3\ell_b+e_\lambda$.  Retention
gives $y\le T_d+\epsilon_0$, while the nonzero unit ranges give
$v,\ell_b,e_\lambda\ge-\epsilon_Z$ after increasing its fixed
constant.  It follows that
\[
 \max(H,v+\ell_b)=H+O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z).
\]

Write $s_{\rm hyb}=\min\{v,z_a,(v+z_a)/3\}$ for the quantity
called $u$ in Equation~\eqref{old-eq:4.7}.  If $s_{\rm hyb}=z_a$,
then $-S_0-B_0+z_a-s_{\rm hyb}\le0$, the kernel term is
nonpositive, and
\[
 -\ell_b-\frac{2e_\lambda}{3}\le\frac53\epsilon_Z.
\]
Here the standalone $O$ is the powerful-row logarithmic length.  It follows that
\[
 \begin{aligned}
 E_{\rm ref}&\le O/2+H+O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z)\\
 &=M'-O/2-\Delta_H+O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z).
 \end{aligned}
\]
Otherwise $s_{\rm hyb}\ge v/2-O(\epsilon_Z)$, and the bounded unit
ranges imply
\[
 s_{\rm hyb}+\ell_b+\frac{2e_\lambda}{3}
       +\frac{(T_d-y)_+}{2}
 \ge \frac y4+\frac{(T_d-y)_+}{2}-O(\epsilon_Z)
 \ge \frac{T_d}{4}-O(\epsilon_Z).
\]
The last inequality holds separately for $y\le T_d$ and $y\ge T_d$.
Here the squared kernel saving is obtained by first extracting
$\min\{1,Z^{(y-T_d)/4}\}$ from the common Fourier density and only
then squaring the row norm.  Relative to $M'$, the saving
before the remaining $+z_a$ is exactly
\begin{equation}
 \begin{split}
 O/2+\Delta_H+S_0+B_0+T_d/4
 ={}&\frac{2M'+2z_a-1-\ell'+2\Delta_H-\theta_N}{4}\\
    &+\frac{2A_0-N_0+B_0+4S_0}{4}.
 \end{split}
 \label{old-eq:5.3}
\end{equation}
The second numerator is nonnegative.  Since $z_a\le\ell'$, the
energy in this branch is at most
\[
 \begin{aligned}
 &M'+\frac{1+3\ell'-2M'-2\Delta_H+\theta_N}{4}
       +O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z)\\
 &\qquad=M'+\frac{d-1/6-2\Delta_H+\theta_N}{4}
       +O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z).
 \end{aligned}
\]
After using $O\ge-\epsilon_Z$ and absorbing $\theta_N$, the two
upper bounds are
\[
 \begin{gathered}
 M'-\Delta_H+O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z),\\
 M'+\frac{d-1/6-2\Delta_H}{4}
       +O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z).
 \end{gathered}
\]
They are nonincreasing in $\Delta_H$.  Since
$\Delta_H\ge-\epsilon_Z$, their formal $\Delta_H=0$ values bound
every actual dyad up to $O(\epsilon_Z)$, whether or not an endpoint
dyad occurs.  Their maximum is therefore at most
$M'+((d-1/6)/4)_++O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z)$.
At $d=1/6$ one has $\ell'=z_a=0$ and the clipped unit range falls
in the first branch; bounded negative unit dyads change only
$\epsilon_Z$.  Choose $\epsilon_0$ and $\kappa_{\mathrm G}$ within the prescribed
$\epsilon$ allowance, and then increase the fixed-data lower
threshold so that $\epsilon_Z$ also lies within it.  Summing the
logarithmically many row and dual dyads proves the lemma.
\end{proof}

\subsection{The additive Gram bound}

We next estimate the additive factor in the range $Y'^2/Q\ge1$.  The
full correlation of Lemma~\ref{lem:full-correlation} retains the collision
zeros that determine both its cancellation and its exceptional frequencies.

\begin{proposition}[A quantitative additive Gram bound]\label{prop:probe-gram}
Fix the arithmetic data $S,T,b_*,\xi$, the annular weight $W_1$,
and a ray class $\sigma\in T$.  Let $Q,Y'\ge1$ be in fixed
polynomial ranges in $Z$, and suppose
\[
 P_a=Y'^2/Q\ge1.
\]
For a real height $\nu$, put $A_m=A_{m,\sigma,\nu}(Y')$ as defined in
Section~\ref{sec:probe}.  For every fixed row ball $q_m\ll Q$ and every
$\epsilon>0$, one has
\begin{equation}
 \sum_{q_m\ll Q}|A_m|^2
 \ll (1+|\nu|)^{J_{\mathrm{Gr}}}\frac Q{Y'}
       \left(1+P_a^{1/6}+\frac{P_a^2}{Y'}\right)Z^\epsilon.
 \label{eq:gram-quantitative}
\end{equation}
Here $J_{\mathrm{Gr}}$ is a fixed seminorm order, independent of the moving
scales and of $\nu$.  The same assertion holds for any fixed finite
linear combination of the ray coefficients
$1_{s\in\sigma}\chi_s(b_*)/\xi(s)$.  No arbitrary
row-dependent arithmetic coefficient is asserted.
\end{proposition}

\begin{proof}
Majorize the row ball by a fixed nonnegative radial Schwartz function.
Put $r_s=q_s/Y'$ and, on the primary elements prime to $S$,
$c_\sigma(s)=1_{s\in\sigma}\chi_s(b_*)/(\tau\xi(s))$, extended by
zero elsewhere before the inverse character is evaluated.  The exact
annular rewriting of the polynomial is
\[
 A_m=Y'^{-3/2}\sum_s c_\sigma(s)W_1(r_s)r_s^{-1+i\nu}
                          g_{\chi_s}(s,-m).
\]
Thus the expanded square has the common normalization $Y'^{-3}$ and
the joint annular profile contains the factors
$W_1(r_{s_1})r_{s_1}^{-1+i\nu}W_1(r_{s_2})r_{s_2}^{-1-i\nu}$.
Write $s_1=Cn_1$, $s_2=Cn_2$, with $(n_1,n_2)=1$.  Poisson in the
row variable has factor $Q/(q_{s_1}q_{s_2})$, whereas the complete
transform of the two unnormalized Gauss sums is
$q_{s_1}q_{s_2}F(s_1,s_2;j)$.  It therefore gives the exact prefactor
$Q/Y'^3$ and the correlation
$F(Cn_1,Cn_2;Ck)$ of Equation~\eqref{eq:correlation-lift}.
Here $F$ again denotes the full correlation of Equation~\eqref{old-eq:2.11},
not the local coefficient of Equation~\eqref{eq:local-F}.
The first Fourier kernel has argument comparable to $q_Cq_k/P_a$.

On coprime residuals the factor $L_C$ in that correlation has
$|L_C|\le q_C$.  For the subsequent signed extension use its specified
bounded extension, periodic modulo $\operatorname{rad}C$ in both
columns, including when $C$ shares primes with a column.  Its value
is defined to be zero when a prime of $C$ divides both columns;
this is a definition of the extension, not a claim about the genuine
correlation there.  Before any factorwise estimate, insert the complete
M\"obius identity
\[
 1_{(n_1,n_2)=1}=\sum_{d\mid n_1,\ d\mid n_2}\mu(d)
\]
and write $n_i=dm_i$.  No coprimality condition on $m_1,m_2$ is then imposed.
Nonzero terms have $(d,Ck)=1$.  Throughout this step use the fixed
ray extension of $\mathcal R(n_1,n_2)$; a quotient of zero residue
symbols would not be defined.

We specify a common fixed modulus for that coefficient.  For $k\ne0$
write $k=\zeta k_Sk_{\mathrm{good}}$ using the unit convention and the
fixed $S$-prime generators of Lemma~\ref{lem:fixed-numerator-ray}, with
$k_S=\prod_{\mathfrak p\in S}\pi_{\mathfrak p}^{e_{\mathfrak p}}$
and $k_{\mathrm{good}}=\prod_{p\notin S}p^{e_p}$.  For a primary
residual ideal $m$ outside $S$, reciprocity gives
\[
 \begin{split}
 \chi_m(k)&=\kappa_{\zeta,\boldsymbol e}(m)
       \mathcal R(k_{\mathrm{good}},m)
       \prod_{p\mid k_{\mathrm{good}}}\chi_p(m)^{e_p},\\
 \kappa_{\zeta,\boldsymbol e}(m)
       &=\chi_m(\zeta)\prod_{\mathfrak p\in S}
             \chi_m(\pi_{\mathfrak p})^{e_{\mathfrak p}\bmod6}.
 \end{split}
\]
Every displayed good local power retains its zero on a shared prime,
including when $6\mid e_p$.  Lemma~\ref{lem:fixed-numerator-ray}
places the finitely many $\kappa_{\zeta,\boldsymbol e}$ in a fixed ray
group supported on $S$.  Also $\mathcal R(k_{\mathrm{good}},m)$
belongs to a fixed finite family in $m$.

Choose $\mathfrak l_{\rm fixed}$ once to contain the primary-class
modulus and the $S$ zero masks, the moduli of every fixed ray
coefficient in $A_m$, a period for $\mathcal R$ in each variable,
and the conductors of all the preceding
$\kappa_{\zeta,\boldsymbol e}$ characters.  The
extended coefficient is zero off the primary class or at a fixed
nonunit before any inverse character is evaluated.  This modulus is
independent of $C,d,k$, and may have higher powers at $S$ than
$\operatorname{rad}(k)$.  After reciprocity the joint arithmetic
coefficient in $m_1,m_2$ is periodic modulo
\[
 \mathfrak r=\operatorname{lcm}
   (\mathfrak l_{\rm fixed},\operatorname{rad}C,\operatorname{rad}(k)),
 \qquad q_{\mathfrak r}\ll_{\rm fixed}q_Cq_k.
\]
If $k$ has a valuation $e_p$ not divisible by six at a prime outside
$C$ and $\mathfrak l_{\rm fixed}$, call $k$ nonexceptional.  At that
prime the first residual column has the nonprincipal zero-extended
factor $\chi_p(m_1)^{e_p}$ up to a unit scalar.  The lift, the fixed
phases, and the other prime factors are independent of $m_1$ modulo
$p$.  Its complete mean, and hence the complete joint mean by the
Chinese remainder theorem, is zero.  The substitution by $d$ does
not change this conclusion because $(d,k)=1$.  Principal zero masks
at valuations divisible by six remain present in the exceptional
case.

The frequency $k=0$ contributes only the diagonal: the correlation
forces $n_1=n_2=1$ and equals $\varphi(C)$.  There are $O(Y')$
possible $C$ of norm comparable to $Y'$, each with
$\varphi(C)\le q_C\ll Y'$.  Its contribution is therefore
$O(Q/Y')$.

For $k\ne0$, split the Fourier argument into dyadic shells
$q_Cq_k/P_a\asymp R$, with $R=1,2,4,\ldots$ and the first shell
including all arguments at most two.  Choose dyad centers
$C_0,D_0\in\{1,2,4,\ldots\}$ for $q_C\asymp C_0$ and
$q_d\asymp D_0$, with the first dyads containing the bounded unit ranges.
Use the nominal positive residual scale
\[
 N=Y'/(C_0D_0),\qquad
 \mathcal Q=q_{\mathfrak r}\ll_{\rm fixed}RP_a.
\]
The actual residual norms lie in fixed multiples of $N$; the nominal value
$N$ is allowed to be below one.  Fix arbitrary $A>2$ and $B>2$.
For each fixed $C,d,k$, the one joint annular profile $W_R$ for the two
columns has the finitely many homogeneous single-profile seminorms used
below bounded by
$O_{B,J_{\mathrm{Gr}}}((1+|\nu|)^{J_{\mathrm{Gr}}} R^{-B})$, with one fixed
$J_{\mathrm{Gr}}$ chosen after these derivative orders and $A,B$.  This
follows by differentiating the
first Fourier kernel, whose Schwartz decay absorbs every resulting
power of $R$; derivatives of the norm powers cost a fixed power of
$1+|\nu|$.  This factor $R^{-B}$ remains in the single-profile
Fourier estimate and occurs once in the linear lattice sum below;
no small inhomogeneous tuple seminorm is used.

Let $a_{C,d,k}$ be this periodic arithmetic coefficient.  For a
nonexceptional frequency its normalized complete Fourier transform,
with normalization $\mathcal Q^{-2}$ on the two residue variables,
has absolute value at most $q_C$ and vanishes at $(0,0)$.  Since
every ideal of $\mathcal O$ is principal, the period lattice has
linear scale $\mathcal Q^{1/2}$.  Four-dimensional lattice Poisson
therefore gives, for this $A$,
\begin{equation}
 \left|\sum_{\mathbf m\in\mathcal O^2}
       a_{C,d,k}(\mathbf m)W_R(\mathbf m/\sqrt N)\right|
 \ll (1+|\nu|)^{J_{\mathrm{Gr}}} R^{-B}q_C N^2
             \min\{1,(\mathcal Q/N)^A\}.
 \label{eq:gram-periodic-poisson}
\end{equation}
For $N\ge\mathcal Q$, the zero frequency vanishes and the other
dual vectors have linear scale $(N/\mathcal Q)^{1/2}$.  Fourier
decay of order $2A>4$ bounds their sum by $(\mathcal Q/N)^A$.
For $N<\mathcal Q$, use the point count in the two annuli.  A nonempty
subunit annulus has $N$ bounded below by a fixed positive constant,
so it still contains $O(N)$ lattice points; below that constant it
is empty.  This proves Equation~\eqref{eq:gram-periodic-poisson}
without any primitivity or tensor-product assumption on $a_{C,d,k}$.

Put $K_R=RP_a/C_0$.  The fixed shell comparison constants give
$q_k\le C_{\rm sh}K_R$ for a fixed $C_{\rm sh}>0$.  Hence the
nonzero frequency range is empty when $K_R<C_{\rm sh}^{-1}$.  On a
nonempty range, elementary lattice counting gives
\[
 \#\{k:0<q_k\le C_{\rm sh}K_R\}
       \ll_{\rm fixed}K_R+\sqrt{K_R}+1\ll_{\rm fixed}K_R.
\]
There are therefore $O(C_0D_0K_R)$ choices of $C,d,k$ on these
dyads.  Multiplying
Equation~\eqref{eq:gram-periodic-poisson} by this count and by the
common prefactor gives at most
\[
 (1+|\nu|)^{J_{\mathrm{Gr}}} R^{-B}\frac Q{Y'}\frac{RP_a}{C_0D_0}
       \min\left\{1,\left(\frac{RP_aC_0D_0}{Y'}\right)^A\right\}.
\]
For $\Lambda=Y'/(RP_a)$ the required positive dyadic sum is
\[
 \sum_{i,j\ge0}2^{-i-j}\min\{1,(2^{i+j}/\Lambda)^A\}
 =\sum_{n\ge0}(n+1)2^{-n}\min\{1,(2^n/\Lambda)^A\}
 \ll_A\frac{1+\log(2+\Lambda)}{\Lambda}.
\]
For $\Lambda\ge1$, split the two geometric tails at
$2^n\asymp\Lambda$; for $\Lambda<1$, the left side is bounded and
the displayed right side is larger than a positive constant.
Thus the nonexceptional frequencies contribute
\[
 \ll (1+|\nu|)^{J_{\mathrm{Gr}}}\frac Q{Y'}\frac{P_a^2}{Y'}Z^\epsilon
\]
after the $R$ sum, on choosing a fixed $B>2$.

It remains to count the exceptional nonzero frequencies.  They have
$(k)=a_1^6e$, where $e$ is sixth-power-free and supported on the
primes of $C$ and the fixed support.  There are at most
$6^{\omega(C)+O_S(1)}\ll_\epsilon q_C^\epsilon$ possible $e$.
For each of them the shell bound $q_k\le C_{\rm sh}K_R$ permits only
$q_{a_1}\le(C_{\rm sh}K_R/q_e)^{1/6}$.  If this range is nonempty,
its upper bound is at least one, and the ideal count gives
$O_{\rm fixed}(K_R^{1/6})$ choices; if $q_e>C_{\rm sh}K_R$ it is
empty.  Unit factors cost only a fixed factor.  Hence there are
$O_\epsilon(K_R^{1/6}q_C^\epsilon)$ such elements $k$, including
the bounded subunit range of $K_R$.

Use $K_R=RP_a/C_0$, $|L_C|\le q_C$, and the trivial two-column
point count.  Before multiplication by $Q/Y'^3$, the contribution
on these dyads is
\[
 \ll (1+|\nu|)^{J_{\mathrm{Gr}}} R^{-B}Z^\epsilon
       Y'^2(RP_a)^{1/6}C_0^{-1/6}D_0^{-1}.
\]
The sums over $C_0,D_0$ converge after reducing the arbitrary small
power, and the $R$ sum converges for the same fixed $B>2$.  This
gives the term $(Q/Y')P_a^{1/6}Z^\epsilon$.  Adding the diagonal
and nonexceptional terms proves the proposition.
\end{proof}

\subsection{Completion of the low bound}

The two preceding estimates now apply to the two factors of the same
rescaled summand in the exact low separation.

\begin{proposition}[The compensated low estimate]\label{prop:probe-low}
For the probe in Equation~\eqref{eq:compensated-probe-definition},
the geometry in Equation~\eqref{old-eq:5.1}, and every
$\epsilon>0$,
\begin{equation}
 |I_{\eta,\mathrm{modified}}(Z)|
       \ll Z^{l_x/2+b/12+\epsilon}=Z^{3/16+\epsilon}.
 \label{old-eq:5.10}
\end{equation}
The exponent is independent of the target and the slot mesh; the
constant and lower threshold may depend on the fixed arithmetic
data, slot system, and smooth tests.
\end{proposition}

\begin{proof}
For a fixed rescaled subset and tuple, put $Q=q_{b_*}X'Y'$.
Its separation is Equation~\eqref{eq:low-separated} with
$A_{m,\sigma,v}(Y')$ and $B^J_{m,\sigma}(Z)$.  Put
$r_J=q_{p_J}/Z^d$; the fixed annular supports give
$0<c_J\le r_J\le C_J<\infty$ uniformly in the tuple.  Exactly,
\[
 \begin{gathered}
 X'=Z^{l_x-d}/r_J,\qquad Y'=Z^{l_y-d}/r_J,\\
 Q=q_{b_*}Z^{M'}/r_J^2,\qquad
 P_a=Y'^2/Q=q_{b_*}^{-1}Z^{1/8}.
 \end{gathered}
\]
In particular $P_a\ge1$ for $Z\ge q_{b_*}^8$.  Enlarge the allowed
fixed-data lower threshold so that $Q,Y'\ge1$ uniformly over the
tuple ratios as well; this is possible because $M'\ge1/2$ and
$l_y-d\ge5/16$.  The remaining length inequalities are
\[
 l_x-d\ge3/16,\qquad l_y-d\ge5/16,
 \qquad l_y-d-11b/6\ge1/12.
\]
The last one gives
\[
 \frac{Y'}{P_a^{11/6}}
 =q_{b_*}^{11/6}r_J^{-1}Z^{l_y-d-11b/6}
       \gg_{\rm fixed}Z^{1/12},
\]
and hence $P_a^2/Y'\ll P_a^{1/6}$.  Proposition~\ref{prop:probe-gram}
therefore gives
\begin{equation}
 \sum_{q_m\ll Q}|A_{m,\sigma,v}(Y')|^2
 \ll (1+|v|)^{J_{\mathrm{Gr}}}(Q/Y')P_a^{1/6}Z^\epsilon.
 \label{old-eq:5.6}
\end{equation}
Cauchy--Schwarz in the row sum, Lemma~\ref{lem:probe-row-norm},
and the integrability of $(1+|v|)^{J_{\mathrm{Gr}}/2}|\widehat W_0(iv)|$ now
bound this tuple's unscaled separation by $Z^\epsilon$ times
\[
 \begin{aligned}
 Q^{-1/2}\bigl[(Q/Y')P_a^{1/6}\bigr]^{1/2}
       \bigl[Z^{M'}\bigr]^{1/2}
 &=\bigl[Z^{M'}/Y'\bigr]^{1/2}P_a^{1/12}\\
 &=r_J(X')^{1/2}P_a^{1/12}.
 \end{aligned}
\]
Since $r_J$ is bounded, this is
$O((X')^{1/2}P_a^{1/12}Z^\epsilon)$.

For $d=\sum_{i\in J}\ell_i$ there are at most $Z^{d+\epsilon}$
rescaled tuples, the coefficient is $O(Z^{-3d/2})$, and
$(X')^{1/2}\asymp X^{1/2}Z^{-d/2}$.  The total additional exponent
is consequently
\begin{equation}
 f(d)=d-3d/2-d/2=-d\le0.
 \label{old-eq:5.5}
\end{equation}
Choose the small powers in the component estimates so that their sum lies
within the prescribed $\epsilon$.  Summing the finitely many subsets proves
Equation~\eqref{old-eq:5.10}.  For the already defined
$C=C_{\mathrm{II}}$ of Equation~\eqref{eq:part-II-mellin-target}, the
normalization used for the principal signal satisfies
\begin{equation}
 \begin{gathered}
 C_{\mathrm{II}}(s)=l_x/2+s-1+h/6=s-11/16,\\
 C_{\mathrm{II}}(7/8)=3/16=l_x/2+b/12.
 \end{gathered}
 \label{old-eq:5.11}
\end{equation}
Thus, under the contradiction assumed in Part~II, this low bound is smaller
than $Z^{C_{\mathrm{II}}(\beta_*)}$ by the exact power
$\Delta=\beta_*-7/8$, apart from the arbitrarily prescribed $\epsilon$.
\end{proof}

\section{The local compensation and its errors}
\label{sec:local-compensation}

The physical modification in Section~\ref{sec:compensation} has already
been estimated from its separated low representation.  We now identify its
full Euler correction and bound the local errors left by the two-term
operation.  Throughout, the local notation $P_p,P_p^*,H_p,V,W,D,R$ is that
of Lemma~\ref{lem:local-euler}; in the shared analytic estimates its variable
$x$ is called $s$.  Fix a nonzero sixth-power-free physical row $u$ with
$(u,S)=1$.  All slot primes belong to the sets $\mathcal P_i(Z)$ of
Section~\ref{sec:compensation}; for each such prime put $Q=q_p$.  As in
Lemma~\ref{lem:local-euler}, write $x_r=\Re x$, $w_r=\Re w$, and $z_r=\Re z$.
The factor $W_{\rm loc}=\rho Q^{-w}$ inside $P_p^*$ is distinct from
$W=\chi_p(u)Q^{-w}$: at $p\mid u$, the latter and $D$ vanish, while the
former retains its unit phase.

\subsection{The full holomorphic correction}

For a fixed slot prime $p$, restricting the completed index to
$p\mid A$ replaces $P_p$ by $P_p^*$ in the high series.  Changing
$Z$ to $ZQ$ multiplies its Mellin weight by $Q^{x+z-1}$.  The
marked term of Equation~\eqref{eq:compensated-probe-definition}
therefore has, at points where $P_p$ is defined and nonzero, multiplier
$\overline{\eta(p)}Q^{x+z-1}P_p^*/P_p$.  The rescaled term has
multiplier
\[
 Q^{-3/2}Q^{-(1/2-z)}Q^{-(w-1)}=Q^{z-w-1}.
\]
Thus, on the same locus, the exact local multiplier of the two-term operation is
\begin{equation}\label{old-eq:5.13}
 Q^{z-1}\mathcal B_p,\qquad
 \mathcal B_p=\overline{\eta(p)}Q^xP_p^*/P_p-Q^{-w}.
\end{equation}
Define the combined local replacement by
\begin{equation}\label{old-eq:5.13c}
 G_p=
 \frac{(\overline{\eta(p)}Q^x-Q^{-w})(1-V)(1-W)P_p^*
       -Q^{-w}(1-VW)}{1-D}.
\end{equation}
At points in the Euler regions where the raw quotient in
Equation~\eqref{old-eq:5.13} is defined, one has $G_p=H_p\mathcal B_p$.
The displayed formula defines
$G_p$ also where that quotient is not defined.  There is neither a $P_p$
nor a $1-W$ denominator on the right.  The bounds $|R|,|V|,|D|<1$ in
Lemma~\ref{lem:local-euler}, together with its formula for $P_p^*$, show
that $G_p$ is holomorphic in both stated Euler regions, including at zeros
of $P_p$ or $H_p$.

At points in the Euler regions where the selected quotients are defined,
write the full slot
multiplier
\[
 \mathcal B_i(u;x,w,z)=\sum_{p\in\mathcal P_i(Z)}
           W_i(q_p/P_i)q_p^{z-1}\mathcal B_p.
\]
The correction used for contour moves is defined without these quotients:
\begin{equation}\label{eq:holomorphic-selected-tuple}
 \mathfrak H_{\eta,u,Z}(x,w,z)=
 \sum_{(p_i)\in\prod_i\mathcal P_i(Z)}
 \prod_{i=1}^K\bigl[W_i(q_{p_i}/P_i)q_{p_i}^{z-1}G_{p_i}\bigr]
 \prod_{\substack{p\notin S\\p\notin\{p_1,\ldots,p_K\}}}H_p.
\end{equation}
This is the full correction after the scalar quotient in
Equation~\eqref{eq:probe-high} has been extracted.  To verify this assertion,
fix an allowed tuple and put $\mathcal P=\{p_1,\ldots,p_K\}$.  On the
absolute starting lines the selected operation replaces $P_p$ by
\begin{equation}\label{eq:selected-local-operation}
 \overline{\eta(p)}Q^{x+z-1}P_p^*-Q^{z-w-1}P_p
 =\frac{1-D}{(1-V)(1-W)}Q^{z-1}G_p.
\end{equation}
The equality follows by substituting
$P_p-P_p^*=(1-V)^{-1}+W/(1-W)$ into
Equation~\eqref{old-eq:5.13c}.  It extracts the same scalar local factor
$(1-V)^{-1}(1-W)^{-1}(1-D)$ as at an unselected prime.  Each selected
prime occurs exactly once, because the slot
supports are disjoint.  The complete factor for this tuple is therefore
the scalar quotient in Equation~\eqref{eq:probe-high} times its summand in
Equation~\eqref{eq:holomorphic-selected-tuple}.  This reasoning involves
no division by $P_p$ or $H_p$.  At points in the Euler regions where all
raw selected quotients are defined, the same finite sum also factors as
$\mathcal H_{\eta,u}\prod_i\mathcal B_i$.

The unselected product in each summand converges normally in both Euler
regions.  More quantitatively, in any fixed subregion of
Equation~\eqref{eq:local-region-one}, put
$\epsilon_H=\min(\epsilon_0,1/50)>0$.  The primewise defect bounds in
Lemma~\ref{lem:local-euler} give, uniformly in the selected tuple,
\begin{equation}\label{eq:unselected-local-product}
 \prod_{\substack{p\notin S\\p\notin\mathcal P}}|H_p|
 \le
 \prod_{\substack{p\notin S\\p\nmid u}}
       (1+Cq_p^{-1-\epsilon_H})
 \prod_{p\mid u}(1+Cq_p^{-\epsilon_H})
 \ll_\epsilon q_u^\epsilon.
\end{equation}
The first positive product converges by the ideal count, and the second
satisfies the divisor-product bound.  Omitting selected factors only removes
factors from these positive majorants.  In the second Euler region the same
argument uses the respective defect exponents $-363/200$ and $-33/40$.
Thus Equation~\eqref{eq:unselected-local-product} holds there as well,
with the corresponding positive majorants.  No nonvanishing of $H_p$ is
asserted by these upper bounds.

For each fixed $Z$, Equation~\eqref{eq:holomorphic-selected-tuple} is a
finite sum of products of holomorphic selected factors and normally
convergent unselected products.  It is therefore holomorphic on a
neighborhood of every point in both Euler regions.  It also satisfies
the all-height requirement in Definition~\ref{def:stage-high-data}.
Indeed, on a fixed real box in either region, the formula for $P_p^*$ and
the denominators bounded away from zero give $|G_p|\ll Q^B$ for some
fixed $B$, uniformly in all imaginary parts and unit phases.  There are
$O(P_i)$ ideals in a slot and $q_p\asymp P_i$ there.  The finite tuple
sum and Equation~\eqref{eq:unselected-local-product} consequently give
\begin{equation}\label{eq:compensated-all-height-majorant}
 |\mathfrak H_{\eta,u,Z}(x,w,z)|
 \ll_\epsilon q_u^\epsilon\prod_iP_i^{z_r+B}
 \ll_\epsilon q_u^\epsilon Z^{B'}.
\end{equation}
for a fixed $B'$ on that box.  For $q_u\le Z^D$ this is
Equation~\eqref{eq:stage-all-height-majorant}, even with height exponent
zero.  All constants here are fixed before any later order of integration
by parts.

Combining the exact finite operation with
Equations~\eqref{eq:probe-poisson-identity} and \eqref{eq:probe-high}
now gives
\begin{equation}\label{eq:compensated-poisson-identity}
\begin{split}
 I_{\eta,\mathrm{modified}}(Z)
 ={}&\frac1{(2\pi i)^3}\int_{(3)}\int_{(3)}\int_{(2)}
       \mathcal W(X,Y,Z;x,w,z)\\
 &\quad\cdot\sum_u^{(6)}q_u^{-z}\overline{\xi(u)}
       \frac{\zeta_F^S(6z)L^S(w,\chi_\bullet(u))}
       {L^S(x,\eta\overline{\chi_\bullet(u)})}
       \mathfrak H_{\eta,u,Z}(x,w,z)\,dz\,dw\,dx.
\end{split}
\end{equation}
This identity is absolutely convergent on the displayed lines, because for
fixed $Z$ it is obtained from finitely many absolutely convergent rescaled
base-probe identities.  It has no additional outside Euler factor.  It
verifies the exact high representation of Definition~\ref{def:stage-high-data}
for the physical expression $I_{\eta,\mathrm{modified}}$, with the geometry
in Equation~\eqref{old-eq:5.1}.  In particular,
$l_x/2-1+h/6=17/96-1+13/96=-11/16$, as required by
Equation~\eqref{eq:part-II-mellin-target}.  It is an identity for the very
probe bounded in Proposition~\ref{prop:probe-low}.  Every continuation
uses the full correction in Equation~\eqref{eq:holomorphic-selected-tuple},
not a globally defined product of individual quotients.

\subsection{Dynamic local errors}

At an identity-ray prime every fixed phase from $T$ is one, although
$\eta(p)$ may be any unit.  Suppose first that $p\nmid u$ and put
$v=\chi_p(u)$.  Then $D=\eta(p)v^{-1}Q^{-x}$, and
$\overline{\eta(p)}Q^xD=v^{-1}$.  Using
$v^{-1}W=Q^{-w}$ and $\mathcal E_p=P_p^*+D$ gives the exact
normalized cancellation
\begin{equation}\label{old-eq:5.13b}
\begin{split}
 G_p+v^{-1}H_p={}&
 \frac{(1-V)(1-W)}{1-D}
 \Bigl[(v^{-1}-Q^{-w})\{V/(1-V)-D\}\\
 &\hspace{17mm}+(\overline{\eta(p)}Q^x+v^{-1}-Q^{-w})
                    \mathcal E_p\Bigr].
\end{split}
\end{equation}
This is an identity of holomorphic local expressions in the Euler regions.
Within these regions, on the raw quotient locus its left side is
$H_p(\mathcal B_p+v^{-1})$.
To see the cancellation directly on that locus, substitute
$P_p^*=-D+\mathcal E_p$ and
$P_p=(1-V)^{-1}+W/(1-W)+P_p^*$ into the left side before
normalization.  The constants $-v^{-1}$, $v^{-1}-Q^{-w}$, and
$(v^{-1}-Q^{-w})W/(1-W)=Q^{-w}$ sum to zero.
The resulting rational identity has only the denominators $1-R$, $1-V$,
and $1-D$, which are nonzero in the Euler regions; hence it gives the
displayed holomorphic identity there.
For $p\mid u$, the main term is zero by the original zero extension,
and Equation~\eqref{old-eq:5.13c} instead becomes
\[
 G_p=
 \overline{\eta(p)}Q^x(1-V)\mathcal E_p-Q^{-w}H_p.
\]

We now give the bound on the remaining error slots.  Its analytic
input is stated explicitly: the reflected primitive numerator must
be small at the retained height.  Instantiate Section~\ref{sec:detector}
with the current fixed data and $\Theta=\langle\eta,\widehat T\rangle$,
using its zero-extended presentations $\nu(n)\chi_n(u)^\varsigma$ for
$\nu\in\Theta$ and $\varsigma\in\{-1,1\}$.  The
numerator $\chi_\bullet(u)$ and its conjugate belong to $\mathcal X_u$
with fixed multiplier $1\in\Theta$, while the denominator
$\eta\overline{\chi_\bullet(u)}$ belongs to $\mathcal X_u$ with fixed
multiplier $\eta\in\Theta$.  The buffered estimates there, together with
the inverse deleted-factor bound of Lemma~\ref{lem:deleted-euler-factors},
therefore supply the reflected-numerator hypothesis below at the retained
heights: for the points used below, $\Re(1-w)=a+6e$ and
$|\Im(1-w)|\le T_1$, within the buffered rectangle.  The identity-ray
restriction makes every fixed $T$ phase equal to one at a
slot prime, without changing the physical row or any nonunit zero.

\begin{proposition}[Dynamic local errors and conductor allocation]
\label{prop:probe-errors}
Let $51/100\le a\le1$, $0<e\le10^{-3}$, and take
\[
 x_r=a+16e,\qquad w_r=1-a-6e,\qquad z_r=17/50.
\]
Choose $P_0$ sufficiently large in terms of $e$ and the fixed data.
For $U,T_1\ge1$ and a sixth-power-free row $u$ with $(u,S)=1$
and $q_u\asymp U$, let $\psi_u^*$
be the primitive character inducing $\chi_\bullet(u)$.  Let
$\mathcal W_u$ be any subset of
$\{w:\Re w=w_r,\ |\Im w|\le T_1\}$.  Suppose $\psi_u^*$ is
nonprincipal and, uniformly for $w\in\mathcal W_u$,
\[
 |L(1-w,\overline{\psi_u^*})|\ll U^\epsilon
\]
with any prescribed small power.
For this row on the displayed dynamic region, define \(\mathcal B_i\) by
its preceding slot sum, interpreting each \(\mathcal B_p\) as \(G_p/H_p\),
with \(G_p\) from Equation~\eqref{old-eq:5.13c}.  The stated choice of
\(P_0\) makes \(H_p\ne0\) there for every slot prime, as proved at the start
of the proof.  This statement asserts individual continuation only on this
dynamic region.
Define the main and error parts of a slot by
\[
 \mathcal Q_i(u;z)=
       -\sum_{p\in\mathcal P_i(Z)}W_i(q_p/P_i)q_p^{z-1}\overline{\chi_p(u)},
 \qquad
 \mathcal D_i(u)=\mathcal B_i(u;x,w,z)-\mathcal Q_i(u;z).
\]
Then, for every subset $I$ of the slots and the same points $w\in\mathcal W_u$,
\begin{equation}\label{old-eq:5.13e}
 \left|L^S(w,\chi_\bullet(u))\prod_{i\in I}\mathcal D_i(u)\right|
 \ll U^{a-1/2+O(e)+\epsilon}(3+T_1)^C
       \prod_{i\in I}P_i^{z_r-1/2+O(e)+\epsilon}.
\end{equation}
The main parts retain the physical row, all zero masks, and the
fixed identity-ray restriction.  In particular, their central
normalization is
\[
 \mathcal Q_i(u;z)=-P_i^{z-1/2}
       \left(P_i^{-1/2}\sum_{p\in\mathcal P_i(Z)}
          \overline{\chi_p(u)}W_i(q_p/P_i)(q_p/P_i)^{z-1}\right).
\]
At the same points $w\in\mathcal W_u$, and on this dynamic region only, the
full correction has the finite decomposition
\begin{equation}\label{eq:dynamic-compensated-decomposition}
 \mathfrak H_{\eta,u,Z}
 =\mathcal H_{\eta,u}\prod_{i=1}^K(\mathcal Q_i+\mathcal D_i)
 =\mathcal H_{\eta,u}
   \sum_{I\subseteq\{1,\ldots,K\}}
       \prod_{i\in I}\mathcal D_i\prod_{i\notin I}\mathcal Q_i.
\end{equation}
\end{proposition}

\begin{proof}
In this region $\vartheta=(-w_r)_+\le6e$.  The estimates in the
proof of Lemma~\ref{lem:local-euler} sharpen to
$H_p-1=O(Q^{-1-10e})$ for $p\nmid u$ and
$H_p-1=O(Q^{-10e})$ for $p\mid u$.  Increase the fixed $P_0$ so
that $|H_p-1|<1/2$ for all these primes.  The holomorphic
factor $G_p$ in Equation~\eqref{old-eq:5.13c}, divided by $H_p$,
now defines $\mathcal B_p$ throughout this region.  It is $H_p$,
not the possibly singular raw factor $P_p$, that is bounded away
from zero when $w_r<0$.

Consequently the tuple sum in Equation~\eqref{eq:holomorphic-selected-tuple}
factors as $\mathcal H_{\eta,u}\prod_i\mathcal B_i$ on this dynamic
region.  Substituting $\mathcal B_i=\mathcal Q_i+\mathcal D_i$ proves
Equation~\eqref{eq:dynamic-compensated-decomposition}.  This factorization
is not used to continue the correction outside the present region.

For $p\nmid u$, Equation~\eqref{old-eq:5.13b} and the estimate for
$\mathcal E_p$ give
\[
 |H_p(\mathcal B_p+\overline{\chi_p(u)})|
 \ll Q^{-x_r+2\vartheta}+Q^{-6z_r+2\vartheta}
       +Q^{4-5x_r-6z_r+2\vartheta}
       +Q^{1-w_r-6z_r+\vartheta}
 \ll Q^{-51/100}.
\]
The four exponents are bounded respectively by
$-a-4e$, $-51/25+12e$, $49/25-5a-68e$, and
$a-51/25+12e$.  Division by the bounded $H_p$ preserves this
estimate.  The number of primes in a slot is $O(P_i)$ by the ideal
count; hence its total contribution from $p\nmid u$, including
$Q^{z-1}$, is $O(P_i^{z_r-51/100+\epsilon})$.

For $p\mid u$ there are only divisor-many possible labels.  A
monomial in $\mathcal E_p$ is multiplied by $Q^{z-1+x}$ in
$Q^{z-1}\mathcal B_p$, so after separating $Q^z$ its exponent is
$x_r-1$ plus the exponent of that monomial.  For the non-tail
boundary terms of the six-valuation table, these exponents at
$e=0$ are
\[
\begin{array}{c|ccccc}
 j&1&2&3&4&5\\\hline
 &a-2&1/2-2a&-a,\ 1-3a&1/2-2a&2-5a.
\end{array}
\]
At positive $e$ their changes are respectively
$+6e,-32e,(-26e,-48e),-42e,-80e$.  They are all strictly below
$-1/2$ in the stated range.  The common $R$ term is smaller still:
its exponent after separating $Q^z$ is at most $49/25-1-5a-80e<-1/2$.
Additional valuation pairs and $m$ values have ratios
$|R|\le Q^{-11/10}$ and $|V|=Q^{-51/25}$, so their geometric sums
preserve these bounds.  The strict second-family term with an
additional $V$ is also smaller than $Q^{-1/2}$ after this
normalization.

The sole remaining term is the strict
$(e_0,l,k,m)=(1,0,1,0)$ term, which occurs for $j\ge2$ and is
$-\eta(p)(Q-1)Q^{-x-w}$.  Its exponent after separating $Q^z$ is $-w_r$.
The explicit rescaling term $-Q^{-w}$ has exponent after separating $Q^z$
$-1-w_r<-1/2$, so it needs no further estimate.

Let $\mathfrak f_u$ be the conductor of $\psi_u^*$.  Write
$u=\epsilon_u\prod_{p\mid u}p^{j_p}$, with $\epsilon_u$ a unit.
For primary $n$ coprime to $uS$, reciprocity gives
\[
 \chi_n(u)=\epsilon_u^{(q_n-1)/6}
       \prod_{p\mid u}\mathcal R(p,n)^{j_p}\chi_p(n)^{j_p}.
\]
The unit exponent is read modulo six.  The prime formula extends to
composite $n$: when $A\equiv B\equiv1\pmod6$,
\[
 \frac{AB-1}{6}\equiv\frac{A-1}{6}+\frac{B-1}{6}\pmod6.
\]
Thus the unit factor depends only on $q_n$ modulo $36$, and the
$\mathcal R$ product depends only on $n$ modulo $4$.  Together with
the multiplicativity of $\mathcal R$ and the local symbols, the
displayed norm congruence makes the right side multiplicative on
integral ideals prime to $6u$; extend it to their fractional ideal
group.  A generator congruent to one modulo
$(36)\operatorname{rad}(u)$ is already primary and makes every
displayed factor one.  This character therefore has a ray
presentation with that fixed $2,3$ modulus times
$\operatorname{rad}(u)$.  It agrees with the Kummer character
$\chi_\bullet(u)$ of Lemma~\ref{lem:fixed-numerator-ray} on ideals
avoiding $S$.  These characters induce
the same primitive character: in the principal ideal ring
$\mathcal O$, the Chinese remainder theorem supplies a representative
avoiding the additional finite set $S$ in every ray class of a
common modulus.  At a good prime
$p\mid u$, hold the fixed class and all other good residues at one
and vary the residue modulo $p$ by the Chinese remainder theorem.
The remaining local character $\chi_p^{j_p}$ has exact order
$6/\gcd(6,j_p)>1$.  Its conductor exponent at $p$ is therefore
exactly one: the displayed ray modulus has only the first power of
$p$, and the character cannot descend to a modulus omitting $p$.
This is a local assertion; the ideal character may still need the
fixed normalization modulus at $2,3$.

It follows that, for any set $J_0$ of distinct selected ramified
labels,
\begin{equation}\label{old-eq:5.13d}
 q_{\mathfrak f_u}\ll_S q_{\operatorname{rad}(u)}
       \le q_u\prod_{p\in J_0}q_p^{-(j_p-1)}.
\end{equation}
The functional equation for the primitive numerator
\cite[Equation (1.1)]{GZ}, the assumed reflected bound, and Stirling
give a cost
$q_{\mathfrak f_u}^{A_*}U^\epsilon(3+T_1)^C$, where
\[
 A_*=1/2-w_r=a-1/2+6e>0.
\]
At a selected $p\mid u$ the primitive character already has value
zero, so restoring the original local factor there multiplies by
exactly one.  The remaining deleted Euler factors have radical
$O_S(U)$ and cost at most $U^{6e+\epsilon}$, because $w_r\ge-6e$.

Expand a product of error slots into their coprime-prime terms,
their already bounded ramified terms, and their strict ramified
terms.  For each fixed tuple in this triangle expansion, let $J_0$
be precisely its strict ramified labels.  They are distinct because
the slot supports are disjoint.  Apply the single inequality
Equation~\eqref{old-eq:5.13d} to this entire $J_0$ before summing
the labels.  The numerator together with these local factors,
including $q_p^{z-1}$ at each selected prime, is then bounded by
\[
 U^{A_*+6e+\epsilon}(3+T_1)^C
 \prod_{p\in J_0}q_p^{z_r-w_r-(j_p-1)A_*}
 \le U^{a-1/2+12e+\epsilon}(3+T_1)^C
       \prod_{p\in J_0}q_p^{z_r-1/2},
\]
because $j_p\ge2$ and $-w_r-A_*=-1/2$.  Thus different selected
labels use different factors of the conductor deficit, and the
bounds hold simultaneously.  Summing the divisor-many ramified labels costs
$U^\epsilon$, while the coprime labels have the stronger exponent
$z_r-51/100$.  This proves Equation~\eqref{old-eq:5.13e}.

This argument uses triangle only on error labels.  It never changes
the physical row or the coefficients and masks in any main slot.
The tuple-independent factor $\mathcal H_{\eta,u}$ separately costs
$U^\epsilon$ by Lemma~\ref{lem:local-euler}.  Relative to the
central scale $P_i^{z_r-1/2}$ of a main slot, an error slot thus
has no positive amplitude exponent in the later row count.
\end{proof}

\paragraph{The principal local factor.}
In the second Euler region of Lemma~\ref{lem:local-euler}, take
$u=1$ and $p\in1_T$.  Then $v=1$ in
Equation~\eqref{old-eq:5.13b}, and $H_p$ is bounded away from zero
after the same fixed enlargement of $P_0$.  For this principal row on
this region, define $\mathcal B_p=G_p/H_p$ and define $\mathcal B_i$
by the same slot sum as above.  The lower bound and the disjoint supports
give the separate principal factorization
\begin{equation}\label{eq:principal-slot-factorization}
 \mathfrak H_{\eta,1,Z}(x,w,z)
 =\mathcal H_{\eta,1}(x,w,z)\prod_{i=1}^K\mathcal B_i(1;x,w,z)
 \qquad\text{in the second Euler region}.
\end{equation}
This does not extend the central main/error decomposition beyond its stated
dynamic region.  The four error
exponents are now
\[
 -x_r,\qquad -6z_r,\qquad 4-5x_r-6z_r,
       \qquad 1-w_r-6z_r.
\]
Each is at most $-7/8$ in that region.  Hence the principal
multiplier satisfies
\begin{equation}\label{eq:principal-local-approximation}
 \mathcal B_p=-1+O(Q^{-7/8}),
\end{equation}
uniformly in the imaginary parts and the target unit phase.  This
is the local estimate used to normalize the principal residue.

It also gives the absolute principal correction required by the shared
residue estimate.  Indeed $G_p=H_p\mathcal B_p=O(1)$ here, and the
unselected product has a bounded positive majorant by the second-region
version of Equation~\eqref{eq:unselected-local-product}.  Ideal counting
in each annular slot therefore gives, on every fixed real box in the second
Euler region and uniformly in all imaginary parts,
\begin{equation}\label{eq:principal-local-tuple-bound}
 |\mathfrak H_{\eta,1,Z}(x,w,z)|
 \le\sum_{(p_i)\in\prod_i\mathcal P_i(Z)}
       \prod_i\bigl[|W_i(q_{p_i}/P_i)|q_{p_i}^{z_r-1}|G_{p_i}|\bigr]
       \prod_{\substack{p\notin S\\p\notin\{p_1,\ldots,p_K\}}}|H_p|
 \ll \prod_iP_i^{z_r}=Z^{\ell z_r}.
\end{equation}
This uses the separate principal-row lower bound only to obtain the local
approximation; it asserts no nonvanishing of a general $H_p$.

\subsection{Absolute bounds for the remaining contour lines}

The dynamic decomposition is not available on every contour used by the
shared analytic estimates.  The following bounds instead retain each
selected factor $G_p$ before taking absolute values.

\begin{lemma}[Absolute local tuple bounds]
\label{lem:compensated-absolute-local-bounds}
Let $Z\ge2$, and let $u$ be a nonzero sixth-power-free row with $(u,S)=1$ and
$q_u\asymp U\ge1$, with fixed comparison constants.  Retain the fixed
physical slot system of Section~\ref{sec:compensation}.  The following
bounds are uniform in all imaginary parts and unit phases.
\begin{enumerate}
\item On
\[
 x_r=\beta_*+e,\qquad w_r=1/2,\qquad z_r=17/50,
 \qquad 0<e\le10^{-3},
\]
one has $G_p=O(1)$ at selected primes $p\nmid u$ and
$G_p\ll Q^{1/2}$ at selected primes $p\mid u$.
\item For every fixed $z_\infty>2$, on
\[
 x_r=w_r=2,\qquad z_r=z_\infty,
\]
one has $G_p=O_{z_\infty}(1)$ at every selected prime.
\end{enumerate}
In either case the positive sum over the full selected tuples satisfies
\begin{equation}\label{eq:absolute-local-tuple-bound}
 \sum_{(p_i)\in\prod_i\mathcal P_i(Z)}
       \prod_i\bigl[|W_i(q_{p_i}/P_i)|q_{p_i}^{z_r-1}|G_{p_i}|\bigr]
       \prod_{\substack{p\notin S\\p\notin\{p_1,\ldots,p_K\}}}|H_p|
 \ll_\epsilon U^\epsilon\prod_iP_i^{z_r}.
\end{equation}
The constants may depend on the fixed data, $e$, and, in the second case,
$z_\infty$, but not on $Z,u$ or the imaginary parts.  These estimates remain
valid at zeros of $P_p$ or $H_p$.
\end{lemma}

\begin{proof}
Both sets of lines lie in the first Euler region with $\epsilon_0=3/8$.
Its primewise defect estimates give $H_p=O(1)$, while
Equation~\eqref{eq:unselected-local-product} bounds the positive product
of all unselected factors by $O_\epsilon(U^\epsilon)$.

Consider first $x_r=\beta_*+e$, $w_r=1/2$, and $z_r=17/50$.
For a selected $p\nmid u$, Equation~\eqref{old-eq:5.13b} gives
\[
 |G_p+\overline{\chi_p(u)}H_p|
 \ll Q^{-x_r}+Q^{-6z_r}
       +Q^{4-5x_r-6z_r}+Q^{1-w_r-6z_r}.
\]
The four exponents are at most
$-7/8-e,-51/25,-483/200-5e,-77/50$, respectively.
Consequently $G_p=O(1)$ off $u$, without division by $H_p$.

For a selected $p\mid u$, one has $W=D=0$ and
$\mathcal E_p=P_p^*$.  Equation~\eqref{old-eq:5.13c} becomes
\[
 G_p=\overline{\eta(p)}Q^x(1-V)\mathcal E_p-Q^{-w}H_p.
\]
After multiplication by $Q^{x_r}$, the strict term of the local table
has exponent at most $1-w_r=1/2$; for $j=1$ it also contains the factor
$V$.  The exponents of the boundary terms $J_j$, in order $j=1,\ldots,5$,
are
\[
 -\frac12,\quad \frac32-2x_r,\quad
 \left(\frac32-2x_r,\,2-3x_r\right),\quad
 2-3x_r,\quad 3-5x_r.
\]
They are bounded above by
$-1/2,-1/4-2e,(-1/4-2e,-5/8-3e),-5/8-3e,-11/8-5e$,
respectively.  The common $R$ term has exponent
$4-5x_r-6z_r\le-483/200-5e$.  All further terms in the geometric
families decrease these powers, since
$|R|\le Q^{-329/100-6e}$ and $|V|=Q^{-51/25}$.
Finally $Q^{-w}H_p=O(Q^{-1/2})$.  Hence $G_p\ll Q^{1/2}$ on $u$,
also at a zero of $H_p$.

The labels dividing $u$ are divisor-many, while a slot contains $O(P_i)$
ideals.  Its positive sum is therefore
\begin{equation}\label{eq:small-line-positive-slot}
 \sum_{p\in\mathcal P_i(Z)}
       |W_i(q_p/P_i)|q_p^{z_r-1}|G_p|
 \ll P_i^{z_r}+U^\epsilon P_i^{z_r-1/2}
 \ll U^\epsilon P_i^{z_r}.
\end{equation}
Using the positive majorant for the unselected product and distributing
the requested $\epsilon$ among the fixed number of slots proves
Equation~\eqref{eq:absolute-local-tuple-bound} on the first lines.

Now fix $x_r=w_r=2$ and $z_r=z_\infty>2$.  For a selected prime off
$u$, the four exponents in Equation~\eqref{old-eq:5.13b} are
$-2,-6z_\infty,-6-6z_\infty,-1-6z_\infty$, so
$G_p=O_{z_\infty}(1)$.  On $u$, the strict term after multiplication by
$Q^{x_r}$ has exponent at most $1-w_r=-1$.  The boundary exponents are
$-2,-5/2,(-4,-4),-11/2,-7$, the common $R$ exponent is
$-6-6z_\infty$, and $|R|=Q^{-8-6z_\infty}$, $|V|=Q^{-6z_\infty}$.
The rescaling term is $O(Q^{-2})$.  Thus $G_p=O_{z_\infty}(1)$ on $u$
as well.  A positive slot sum is consequently $O_{z_\infty}(P_i^{z_\infty})$.
Together with Equation~\eqref{eq:unselected-local-product}, this proves
Equation~\eqref{eq:absolute-local-tuple-bound} on the second lines.
\end{proof}


Equation~\eqref{eq:compensated-poisson-identity} supplies the full high
representation of the physical function just bounded on the low side.
Proposition~\ref{prop:probe-errors} controls the error factors jointly with
the numerator, and Lemma~\ref{lem:compensated-absolute-local-bounds} supplies
the bounds on the remaining contour lines. The main factors are the prime
polynomials displayed in Proposition~\ref{prop:probe-errors}. To count rows
on which those factors and a detector witness are large, we next prove
the inverse and fourth-moment estimates used in
Section~\ref{sec:refined-row-counts}.

\section{The inverse moment with prime factors}\label{sec:inverse}

We now bound an inverse Dirichlet polynomial multiplied by independently
weighted prime sums. The estimate applies to the inverse witness in
Section~\ref{sec:detector} together with selected main factors from the
local compensation. Its proof uses the completed row energy of
Section~\ref{sec:marked-energy}. A final amplification gives an unmarked
estimate on sixth-power-free rows at longer column lengths.

\subsection{Statement of the marked estimate}

Let \(\nu\) be a fixed finite-order ray character whose full
zero-extended defining modulus has prime support in \(S\).  This entire
presentation belongs to the fixed arithmetic datum; a locally frozen
moving zero support is instead retained as a puncture whose radical is
included in the explicit norm bound below.
Fix one sign \(\varepsilon_\chi\in\{1,-1\}\), and set
\[
 \psi_u(n)=\nu(n)\chi_n(u)^{\varepsilon_\chi}.
\]
The only masks in this definition are the fixed exclusions and the zero
extension of \(\chi_n(u)\).
For a smooth annular function \(W\), define
\[
 M_u(Z^r;W)=Z^{-r/2}\sum_n\mu(n)\psi_u(n)W(q_n/Z^r).
\]
For a fixed finite set \(I\) of slots, let \(\mathcal P_i\) be disjoint
sets of primes outside \(S\), with \(q_p\asymp Z^{z_i}\) for
\(p\in\mathcal P_i\).  The sets and the coefficients \(a_i(p)\) are
independent of \(u\), and \(|a_i(p)|\le1\).  Given fixed smooth annular
functions \(W_i\), put
\[
 Q_u=\prod_{i\in I}Z^{-z_i/2}
       \sum_{p\in\mathcal P_i}a_i(p)\psi_u(p)W_i(q_p/Z^{z_i}),
 \qquad z=\sum_{i\in I}z_i.
\]
The empty product is one.
A fixed finite sum of whole products \(M_uQ_u\) is also allowed by
triangle inequality, provided each summand uses one common \(\nu\)
in its inverse and all of its slots.

\begin{lemma}[Marked inverse moment]\label{lem:marked}
Fix bounded ranges for the nonnegative parameters \(m,r,z_i\), a bound
for \(|I|\), and positive constants \(c_1,c_2\).  Suppose
\[
 r+2z\le m-c_1,\qquad 2r+8z\le3m-c_2.
\]
Then, for every \(\epsilon>0\),
\begin{equation}\label{old-eq:4.1}
 \sum_{\substack{u\in\mathcal O\\0<q_u\ll Z^m}}
       |M_u(Z^r;W)Q_u|^2\ll Z^{m+\epsilon}.
\end{equation}
The implied constant may depend on the fixed arithmetic data, \(c_1,c_2\),
the bounded parameter ranges, and finitely many smooth seminorms of the
tests, but is uniform in the prime supports and their bounded coefficients.
The assertion holds for either common sign \(\varepsilon_\chi\), and for
every subcollection of the slots.  Norm twists of the tests have a fixed
polynomial cost in their heights.  There is no lower bound on the individual
slot lengths other than the existence of their stated prime supports, and
no mesh condition on those lengths.
\end{lemma}

The proof passes through the canonical family defined next, in which a
fourth-power residue symbol is averaged over an additional squarefree
ideal.  Lemma~\ref{lem:canonical-moment} is proved by a terminal application
of Lemma~\ref{lem:reflected-energy} and a recursive step using two masked
Poisson transformations.  After the principal contributions and tails have
been handled, the retained part is bounded in terms of new admissible sums
of that family with a shorter row range.  A separate initialization using
one masked Poisson transformation then deduces Lemma~\ref{lem:marked}, and
the final subsection gives its longer unmarked consequence for
sixth-power-free rows.

\subsection{The canonical estimate}

Use the fixed puncture, squarefree coefficient, row character, and
product-form mark defined in Section~\ref{sec:marked-energy}. The additional
squarefree label is now averaged, with a divisor-bounded weight depending
on that label alone.

The following is the statement closed by the two Poisson transformations.
Its puncture bound is part of the hypothesis, because a frozen moving
modulus cannot be treated as part of the fixed arithmetic data.
For the recursive Poisson large-sieve framework, compare
\cite[Section~2]{HBQuadratic}, \cite[Section~4]{HB}, and
\cite[Sections~4--7]{GL}; the present marked recursion is proved below.

\begin{lemma}[Canonical marked estimate]\label{lem:canonical-moment}
Fix bounded nonnegative ranges for \(M,N,V,z_0\), a bound for the number
of slots, and \(c_*>0\).  Let \(f\) range over squarefree ideals outside
\(S\) with \(q_f\asymp Z^V\).  Let \(w\) be a nonnegative function of
\(f\) alone satisfying \(w(f)\le C d_{\mathcal O}(f)^C\) for one fixed
\(C\ge1\).  Neither \(C\) nor the choice of \(w\) depends on \(Z\),
the current rows, or any other averaged variable.  The implied constant
below may depend on \(C\), but is uniform over all \(w\) satisfying this
fixed bound.  Let \(\nu\) be a fixed
multiplicative finite ray character whose full zero-extended defining
modulus has prime support in \(S\), and let
\(\rho(n)=1_{(n,\mathfrak r_\rho)=1}\) be a fixed puncture.  Both are
independent of \(k,f\).  Let \(\mathfrak d\) have the product form in
Equation~\eqref{old-eq:4.2}, with disjoint fixed prime lists of total
length at most \(z_0\) and bounded coefficients independent of \(k,f\).
No other row-dependent column coefficient or puncture is allowed.
There is no additional residual coefficient \(b(n)\), even one
independent of \(k,f\): arbitrary bounded weights are permitted only
as the stated product of individual prime-slot coefficients.

Put \(F_0=N+V\).  Suppose
\begin{equation}\label{eq:invariant}
 \begin{gathered}
 q_{\mathfrak r_\rho}\le Z^{F_0-M-z_0-c_*},\\
 3M+6z_0+c_*\le4F_0.
 \end{gathered}
\end{equation}
For every smooth annular \(W\) and every \(\epsilon>0\), define
\begin{equation}\label{eq:canonical-energy}
 \begin{split}
 \mathcal E={}&Z^{-V}\sum_f w(f)
 \sum_{\substack{k\in\mathcal O\\0<q_k\ll Z^M}}
 \Biggl|Z^{-N/2}\sum_{n\ {\rm sf}}
 \bar\alpha(n)\gamma_2(n)\nu(n)\rho(n)\\
 &\hspace{24mm}\times
 \chi_n(k)\chi_n(f)^4\mathfrak d(n)W(q_n/Z^N)\Biggr|^2.
 \end{split}
\end{equation}
Then \(\mathcal E\ll Z^{F_0+\epsilon}\).  The constant is uniform in
the moving moduli and the frozen outer ideals within the stated ranges.
It depends on finitely many smooth seminorms and has a fixed polynomial
dependence on separated norm-twist heights.  The same conclusion holds
for all subcollections of the slots with the original cap \(z_0\),
including the empty collection.
\end{lemma}

The reflected energy in Section~\ref{sec:marked-energy} supplies the
terminal estimate for this family. The remaining ranges will be reduced
to the same family with shorter rows. Before the two Poisson
transformations, we give the common analytic rule for propagating smooth
seminorm and height orders through the finite induction.

\subsection{Finite propagation of seminorm and height orders}\label{sec:moment-propagation}

We use the following abstract statement about a finite sequence of estimates. It
does not assert that any particular arithmetic reduction has its hypotheses.
Its purpose is to state exactly which uniform bounds on transformed profiles
and Fourier coefficient measures suffice to choose all internal derivative
orders before an external height cutoff.

Write $\langle\boldsymbol t\rangle=1+|\boldsymbol t|$.  Let a finite rooted
directed graph have no directed cycles, and let every directed path have at
most $D$ edges.  At each vertex $v$ let
$\mathcal N_v(Z,\boldsymbol w,\boldsymbol t)$ be a nonnegative quantity,
where $Z\ge1$, $\boldsymbol w$ is a fixed finite tuple of annular profiles,
and $\boldsymbol t$ is a finite-dimensional height vector.  The profile
dimensions, supports, and height dimensions may depend on $v$, but are fixed
throughout the graph.  Additional labels are allowed in these quantities;
every bound below is required uniformly in those labels.

\begin{lemma}[Finite seminorm propagation]\label{lem:finite-seminorm-propagation}
Suppose that at each vertex there is a terminal bound
\[
 \mathcal T_v(Z,\boldsymbol w,\boldsymbol t)
 \le C_v Z^{a_v}p_{j_v}(\boldsymbol w)^{b_v}
                   \langle\boldsymbol t\rangle^{h_v}
\]
with fixed real $a_v$ and finite nonnegative $j_v,b_v,h_v$.  Suppose also that
\[
 \begin{split}
 \mathcal N_v(Z,\boldsymbol w,\boldsymbol t)
 \le{}&\mathcal T_v(Z,\boldsymbol w,\boldsymbol t)\\
 &+\sum_{e:v\to v'}Z^{a_e}\int_{\mathbb R^{d_e}}
 |K_e(Z,\boldsymbol w,\boldsymbol t;\boldsymbol\xi)|
 \mathcal N_{v'}\bigl(Z,\boldsymbol W_e,
       A_e\boldsymbol t+B_e\boldsymbol\xi\bigr)\,d\boldsymbol\xi,
 \end{split}
\]
where $A_e,B_e$ are fixed bounded linear maps and
$\boldsymbol W_e=\boldsymbol W_e(Z,\boldsymbol w,\boldsymbol t,
\boldsymbol\xi)$ is the child profile tuple.  Empty edge sums are allowed.
Assume the following two bounds for each edge.

For every fixed $j$ there are finite nonnegative numbers
$m_e(j),b'_e(j),c'_e(j)$ and a constant $C_{e,j}$ such that
\[
 p_j(\boldsymbol W_e)
 \le C_{e,j}p_{m_e(j)}(\boldsymbol w)^{b'_e(j)}
       \langle\boldsymbol t\rangle^{c'_e(j)}
       \langle\boldsymbol\xi\rangle^{c'_e(j)}.
\]
For every fixed $H\ge0$ there are finite nonnegative numbers
$\ell_e(H),b_e(H),c_e(H)$ and a constant $C_{e,H}$ such that
\[
 \int_{\mathbb R^{d_e}}|K_e(Z,\boldsymbol w,\boldsymbol t;
            \boldsymbol\xi)|\langle\boldsymbol\xi\rangle^H
                         \,d\boldsymbol\xi
 \le C_{e,H}p_{\ell_e(H)}(\boldsymbol w)^{b_e(H)}
                 \langle\boldsymbol t\rangle^{c_e(H)}.
\]
The seminorm indices may be rounded up to integers.  All these constants and
indices are uniform in $Z$ and the additional labels; the exponents $a_v,a_e$
are fixed real numbers independent of the requested seminorm and height orders.

Then there are finite $J,B,H$ and $C$ such that, at the root $v_0$,
\[
 \mathcal N_{v_0}(Z,\boldsymbol w,\boldsymbol t)
 \le C Z^E p_J(\boldsymbol w)^B\langle\boldsymbol t\rangle^H,
 \qquad
 E=\max_{v_0\to\cdots\to v}
       \left(a_v+\sum_{e\text{ on the path}}a_e\right).
\]
The indices $J,B,H$ can be selected backward through the at most $D$ stages.
In particular they are fixed before any restriction
$|\boldsymbol t|\le T_1=Z^\tau$ is imposed.

The same conclusion holds for a finite product of children in an integral.
Precisely, an edge term may instead have the form
\[
 Z^{a_e}\int_{\mathbb R^{d_e}}|K_e|
 \prod_{r=1}^{r_e}
 \mathcal N_{v_{e,r}}\bigl(Z,\boldsymbol W_{e,r},
       A_{e,r}\boldsymbol t+B_{e,r}\boldsymbol\xi\bigr)^{\theta_{e,r}}
 \,d\boldsymbol\xi,
\]
where $r_e\ge1$ is a fixed integer and $\theta_{e,r}\ge0$ are fixed, all children have smaller
remaining depth, and each child profile satisfies its own version of the
stated profile bound.  The joint coefficient measure satisfies the same
weighted bound.  In this case define the norm exponent recursively by
\[
 E_v=\max\left\{a_v,\ \max_e
             \left(a_e+\sum_{r=1}^{r_e}\theta_{e,r}E_{v_{e,r}}\right)\right\}.
\]
The conclusion holds with $E=E_{v_0}$.  This form includes the square roots
of two nonnegative child estimates arising from Cauchy--Schwarz.
\end{lemma}

\begin{proof}
At a terminal vertex the assertion is its stated bound.  Suppose a child has
already been bounded by
$C Z^{E'}p_{J'}(\boldsymbol W_e)^{B'}
 \langle A_e\boldsymbol t+B_e\boldsymbol\xi\rangle^{H'}$.
Since the linear maps are bounded,
\[
 \langle A_e\boldsymbol t+B_e\boldsymbol\xi\rangle^{H'}
 \ll_e\langle\boldsymbol t\rangle^{H'}
              \langle\boldsymbol\xi\rangle^{H'}.
\]
The child-profile bound adds the power $B'c'_e(J')$ to each of these two
height exponents and replaces its profile factor by
$p_{m_e(J')}(\boldsymbol w)^{B'b'_e(J')}$.  Use the coefficient-measure
bound with
$H_e=H'+B'c'_e(J')$.  The contribution of this edge is at most
\[
 C_e Z^{a_e+E'}
 p_{\max\{m_e(J'),\ell_e(H_e)\}}(\boldsymbol w)^{B'b'_e(J')+b_e(H_e)}
 \langle\boldsymbol t\rangle^{H_e+c_e(H_e)}.
\]
Every displayed index is finite.  Take the maximum of these indices and of
the terminal indices over the finitely many outgoing edges, increasing the
profile exponent when necessary because $p_j(\boldsymbol w)\ge1$.
This proves the parent bound with exponent equal to the maximum path exponent
through that vertex.  Reverse induction on the acyclic graph completes the
proof.  None of these choices mentions an external cutoff or a late
derivative order.

For the product form, insert the already proved bound of each child and
raise it to $\theta_{e,r}$.  The total power of
$\langle\boldsymbol\xi\rangle$ that the coefficient measure must integrate is
\[
 H_e=\sum_{r=1}^{r_e}\theta_{e,r}
       \bigl(H_r+B_r c'_{e,r}(J_r)\bigr).
\]
The powers of the parent profile and of $\langle\boldsymbol t\rangle$ are
the corresponding finite sums, followed by the single coefficient-measure
bound at $H_e$.  The powers of $Z$ add as in the displayed recurrence for
$E_v$.  Taking maxima over the finitely many terms proves this extension by
the same reverse induction.
\end{proof}

\begin{corollary}[Indexed finite propagation]
\label{cor:indexed-seminorm-propagation}
In Lemma~\ref{lem:finite-seminorm-propagation}, let
$\boldsymbol\ell$ denote any admissible length and outer labels at a vertex
$v$, and let $\Phi_v(\boldsymbol\ell)$ be its desired norm exponent.  Suppose
the terminal bound, after division by $Z^{\Phi_v(\boldsymbol\ell)}$, has the
form in that lemma with a fixed exponent $\delta_v$.  An edge may be indexed
by a $Z$-dependent family $\Gamma_e$ with a nonnegative measure $\nu_e$, and
may have the form
\[
 \int_{\Gamma_e}\int_{\mathbb R^{d_e}}
 Z^{a_e(\boldsymbol\ell,\gamma)}|K_{e,\gamma}|
 \prod_{r=1}^{r_e}\mathcal N_{v_{e,r}}
  \bigl(Z,\boldsymbol\ell_{e,r},\boldsymbol W_{e,r},
        A_{e,r}\boldsymbol t+B_{e,r}\boldsymbol\xi\bigr)^{\theta_{e,r}}
 \,d\boldsymbol\xi\,d\nu_e(\gamma),
\]
where the child labels $\boldsymbol\ell_{e,r}$ may depend on
$\boldsymbol\ell,\gamma$.  Assume uniformly in those labels that
\begin{equation}\label{eq:indexed-exponent-defect}
 a_e(\boldsymbol\ell,\gamma)
   +\sum_r\theta_{e,r}\Phi_{v_{e,r}}(\boldsymbol\ell_{e,r})
   -\Phi_v(\boldsymbol\ell)\le\delta_e,
\end{equation}
with fixed $\delta_e$, and that the child-profile bounds of the lemma hold.
Assume also that for every fixed $H\ge0$,
\begin{equation}\label{eq:indexed-coefficient-measure}
 \int_{\Gamma_e}\int_{\mathbb R^{d_e}}|K_{e,\gamma}|
       \langle\boldsymbol\xi\rangle^H\,d\boldsymbol\xi\,d\nu_e(\gamma)
 \le C_{e,H}Z^{\delta_e^{\mathrm{mass}}}
       p_{\ell_e(H)}(\boldsymbol w)^{b_e(H)}
       \langle\boldsymbol t\rangle^{c_e(H)},
\end{equation}
where $\delta_e^{\mathrm{mass}}$ is fixed independently of $H$.  When the
quantities are squared Hilbert-space row norms obtained by separating a
profile, this hypothesis must use the coefficient measure common to those
rows.  There need only be finitely many edge types at each
of the finitely many depths; the cardinalities of the $\Gamma_e$ may vary
with $Z$.

Then $\mathcal N_v/Z^{\Phi_v(\boldsymbol\ell)}$ has the conclusion of
Lemma~\ref{lem:finite-seminorm-propagation}, uniformly in
$\boldsymbol\ell$, with terminal exponents $\delta_v$ and edge exponents
$\delta_e+\delta_e^{\mathrm{mass}}$.  In particular all required seminorm
and height orders are finite and independent of an external cutoff.
\end{corollary}

\begin{proof}
Set $\overline{\mathcal N}_v=Z^{-\Phi_v(\boldsymbol\ell)}\mathcal N_v$.
Substitution in an edge and Equation~\eqref{eq:indexed-exponent-defect}
bound its norm power by $Z^{\delta_e}$ times the product of the normalized
children.  Insert their inductive bounds.  As in the product proof of
Lemma~\ref{lem:finite-seminorm-propagation}, the required coefficient moment
is
\[
 H_e=\sum_r\theta_{e,r}
              \bigl(H_r+B_r c'_{e,r}(J_r)\bigr),
\]
which is finite and independent of $\gamma$.  Equation~\eqref{eq:indexed-coefficient-measure}
at this order contributes $Z^{\delta_e^{\mathrm{mass}}}$ and only finite
profile and height orders.  Backward induction over the finite edge types
therefore gives exactly the stated normalized recurrence.  A label averaged
inside a child remains inside that child throughout this argument; it is
not a second integration variable of $\nu_e$.
\end{proof}

We make explicit how discrete labels are normalized in this corollary.  After
the relevant weighted Cauchy inequality, suppose a nonnegative outer measure
satisfies
$\sum_{\gamma\in\Gamma}w_\gamma\le C Z^{c+\epsilon}$, with fixed $c$ and
$\epsilon$ selected before the height orders.  For a per-label prefactor
$Z^{a_{\mathrm{raw}}}$ common on the block, put
\[
 d\nu_\Gamma(\gamma)=Z^{-c}
             \sum_{\gamma\in\Gamma}w_\gamma\,\delta_\gamma,
 \qquad \nu_\Gamma(\Gamma)\le C Z^\epsilon.
\]
Here $\delta_\gamma$ denotes unit point mass.  For nonnegative $B_\gamma$
for which the displayed integrals are finite, the exact identity is
\begin{equation}\label{eq:normalized-label-measure}
 \begin{split}
 &Z^{a_{\mathrm{raw}}}\sum_{\gamma\in\Gamma}w_\gamma
              \int |K_\gamma(\boldsymbol\xi)|B_\gamma(\boldsymbol\xi)
                                      \,d\boldsymbol\xi\\
 &\qquad=Z^{a_{\mathrm{raw}}+c}
       \int_\Gamma\int |K_\gamma(\boldsymbol\xi)|B_\gamma(\boldsymbol\xi)
                                      \,d\boldsymbol\xi\,d\nu_\Gamma(\gamma).
 \end{split}
\end{equation}
Uniform weighted moments of $K_\gamma$ now give
Equation~\eqref{eq:indexed-coefficient-measure} with only the remaining
$Z^\epsilon$ mass.  Thus a displayed exponent that already includes the
count $c$ must use this normalized measure; retaining the unnormalized sum
would count the same labels twice.  The rule concerns weighted mass, not
cardinality in addition to that mass.  It is applied only after any
row-dependent eligibility remains within the child or has been removed by a
nonnegative inequality.  Labels still averaged in a child norm are not
included in $\Gamma$.

There is a separate normalization for a small kernel amplitude.  The
single-profile seminorm $p_j(w)$ is homogeneous, whereas
$p_j(\boldsymbol w)=1+\sum p_j(w_i)$ is not.  Suppose a joint annular profile
on a fixed block satisfies, for a scalar $0<m_R\le1$,
\[
 p_j(F_R)\le m_R C_j p_{\ell(j)}(\boldsymbol w)^{b(j)}
                         \langle\boldsymbol t\rangle^{h(j)}
       \qquad(j\ge0).
\]
One may keep $F_R$ in the Fourier coefficient measure, whose weighted
$L^1$ norm then contains the factor $m_R$.  Alternatively, write
$F_R=m_R\widetilde F_R$ and put $m_R$ outside that measure.  If a polynomial
or row vector $\mathcal P(F_R)$ is linear in this whole profile, then
\[
 \mathcal P(F_R)=m_R\mathcal P(\widetilde F_R),\qquad
 \|\mathcal P(F_R)\|_2^2=m_R^2\|\mathcal P(\widetilde F_R)\|_2^2.
\]
The tuple containing $\widetilde F_R$ has bounded, not small, seminorms.
A scalar may be removed from a centered difference only when it multiplies
the whole difference.  A kernel occurring once in an already expanded
quadratic expression contributes one factor $m_R$, not automatically its
square.  These conventions use either the small measure or the outside
scalar, never both.  If the positive row range will later be enlarged, the
scalar and the common profile are first fixed on the actual annuli, and the
norm inequality containing that scalar is obtained before the enlargement.

Here are sufficient analytic ways to verify the two edge hypotheses.
Let $m(\boldsymbol y)$ be a fixed norm monomial and let $\chi$ be a smooth
cutoff with compact logarithmic support in a product of fixed annuli.
Suppose a normalized kernel has Euler bounds
\[
 |(r\partial_r)^jK(r)|\le C_{A,j}\mathfrak S_{m(A,j)}
                                      \min(1,r^{-A}),
\]
where $\mathfrak S_k$ is an increasing family of specified finite input
seminorm and polynomial height bounds, and $m(A,j)$ is taken nondecreasing
in $j$.  The product and chain rules give, uniformly for $R>0$,
\begin{equation}\label{eq:normalized-kernel-profile}
 p_j\bigl(\chi(\boldsymbol y)K(Rm(\boldsymbol y))\bigr)
 \le C'_{A,j}\mathfrak S_{m(A,j)}\min(1,R^{-A}).
\end{equation}
Indeed $y_i\partial_{y_i}$ acting on the kernel is the fixed exponent of
$y_i$ in $m$ times $r\partial_r$, and $m$ is bounded above and below on the
cutoff support.  There is no extra factor $R$ from differentiation.  The
same argument preserves a bound with
$\min(R^{1/4},1,R^{-A})$, when those three kernel estimates are available.
The decay order $A$ is the same at every requested $j$; only the input order
and its finite height degree increase.  For an old window
$w(c m(\boldsymbol y))$ with $c>0$, Euler derivatives are bounded by the
global logarithmic seminorms of $w$ and introduce no power of $c$ depending
on the derivative order.  Normalized real powers such as $y^{-1/2}$ obey the
same rule.  Pure twists of normalized norms insert only fixed powers of
their heights.

Ordinary radial Fourier seminorms must be used only after the radial scale
has been normalized.  For a fixed annular $w$ in two real dimensions,
$f_R(x)=w(R|x|^2)$ satisfies
\[
 \|\partial^j f_R\|_1=R^{j/2-1}\|\partial^j f_1\|_1
\]
for any fixed directional derivative of order $j$, by $x\mapsto\sqrt R x$.
Thus an unnormalized shrinking radial test could introduce a scale power
depending on $j$.  Sufficient hypotheses are a fixed-shape radial Schwartz
test at its stated row scale, or a radial $C_c^\infty$ test constant near
zero at that scale.  After normalization, Lemma~\ref{lem:kernel-seminorms}
applies to its fixed transform, and moving scale ratios enter only as in
Equation~\eqref{eq:normalized-kernel-profile}.  For example
$q_u^{-1/2}=U_0^{-1/2}y^{-1/2}$ on $q_u=U_0y$; the central power belongs to
the norm exponent, and the annular factor has order-independent scale bounds.
Likewise $q_u^{it}=U_0^{it}y^{it}$, and the central unit phase is factored
as a scalar before differentiating the normalized profile.

Logarithmic Fourier separation uses a joint cutoff in a coordinate list
containing every normalized norm occurring in a coupled smooth factor.
Derived nonsmooth row and label masks remain outside that factor and need no
coordinate.  A nonzero norm used as such a coordinate and initially ranging in
a ball down to norm one must first be partitioned into common whole annuli
fixed for the current sector; the zero frequency is separate.
In bounded logarithmic ranges there
are $O((1+\log Z)^d)$ such blocks for a fixed number $d$ of variables, so
their count costs a preselected small power.  The cutoff and its full support
depend only on the block centers and fixed data, not on the individual values
of the labels inside a row norm.
Provided the coupled smooth factor is this one joint function with no
further dependence on the individual labels, its Fourier density is common
to those rows, as required by
Lemma~\ref{lem:smooth-calculus}; separate uniform bounds for row-dependent
densities would not imply the same Hilbert-space inequality.  Nonsmooth
masks are retained or resolved by exact arithmetic identities, never
differentiated or incorporated as label-dependent sharp Fourier selectors.

Finally, suppose an internal dyadic ratio $R>Z^\xi$, with fixed $\xi>0$,
has absolute arithmetic count at most $Z^B R^b$.  A coefficient bound with
the factor $R^{-A}$ gives, for $A>b$,
\[
 \sum_{\substack{R\ \mathrm{dyadic}\\R>Z^\xi}}Z^B R^{b-A}
       \ll Z^{B-\xi(A-b)}.
\]
Choose $A$ from the fixed $B,b,\xi$ and the desired internal saving.  The
actual bound may also have finite input seminorm and polynomial height
factors; these enter the finite propagation graph.  This tail estimate is
not a height-free assertion uniform in all heights.  Parameter Sobolev adds
only a fixed number of profile derivatives and height dimensions, and all
these internal orders are chosen before an external cutoff.

Finally suppose the graph and all of its kernel and annular orders have been
fixed, giving height degree $H$.  Suppose an additional, external separation
uses annular profiles with $p_j$ bounded uniformly in $Z$ for every fixed $j$,
or nonannular profiles with the weighted Mellin or annular norms in the
preceding lemmas uniformly bounded at every fixed order, and
its discarded integrand has bound $Z^{B_0}\langle\boldsymbol t\rangle^{J_0}$
with fixed $B_0,J_0$.  The external truncation is required to remain outside
the graph: it does not replace a profile $\boldsymbol w$ or a kernel $K_e$
by a cutoff-dependent profile.  Given a retained-height allowance $\delta>0$,
first choose $0<\tau<\delta/(1+H)$.  Then
$(1+T_1)^H\le2^H Z^\delta$ for $T_1=Z^\tau$.  After this choice,
Equation~\eqref{eq:late-fourier-tail} with weight $J_0$ and any fixed integer
$N>(B_0+M)/\tau$ makes an integrated external tail $O(Z^{-M})$ for any
prescribed $M>0$.
For a horizontal join use Equation~\eqref{eq:pointwise-mellin-tail} with the
additional fixed height weight, or Equation~\eqref{eq:joint-gaussian-slice}
for a joint Gaussian slice.  When other coordinates on such a slice are
extended to the whole real line, the accompanying arithmetic factors must
have global bounds on those coordinates; a coordinate entering a
denominator controlled only in a buffered region must remain restricted to
that region.  The external constants may depend on $N$ and on the fixed
data.  This late choice does not alter any internal profile or the internal
orders $J,H$, which is the asserted order of dependence.

\subsection{Finite Poisson summation with a mask}

We now prove Lemma~\ref{lem:marked}, using the arithmetic conventions of
Section~\ref{sec:arithmetic} and Lemma~\ref{lem:reflected-energy} for the
terminal case.  The recursive reduction uses two applications of the
following form of Poisson summation, and the initialization uses one.
Its explicit divisor variable retains every zero extension.

\begin{lemma}[Masked primitive Poisson]\label{lem:masked-poisson}
Let \(\psi\) be a primitive finite character modulo an ideal \(m\),
extended by zero on nonunits, and let \(R\) be any ideal.  Let
\(\Phi(q_z)\) be a radial Schwartz function on \(\mathbb C\), and
let \(\mathcal F\Phi(q_y)\) denote its Fourier transform for the
self-dual measure and the kernel \(e(-zy)\).  For \(K>0\),
\begin{equation}\label{eq:masked-poisson}
 \begin{split}
 &\sum_{k\in\mathcal O}\psi(k)1_{(k,R)=1}\Phi(q_k/K)\\
 &\quad=\frac{K\gamma(\psi;m)}{\sqrt{q_m}}
 \sum_{d\mid\operatorname{rad}R}\frac{\mu(d)\psi(d)}{q_d}
 \sum_{h\in\mathcal O}\bar\psi(h)
       \mathcal F\Phi\!\left(\frac{Kq_h}{q_dq_m}\right),
 \end{split}
\end{equation}
where
\(\gamma(\psi;m)=q_m^{-1/2}\sum_{x\bmod m}\psi(x)e(x/m)\).
For a nonprincipal primitive character, \(|\gamma(\psi;m)|=1\) and the
\(h=0\) term is zero.  For the primitive principal character the
conventions are \(m=1,\ \gamma(1;1)=1,\ \psi(h)=1\), including at
\(h=0\).
\end{lemma}

\begin{proof}
Expand the mask as
\(\sum_{d\mid\operatorname{rad}R,\ d\mid k}\mu(d)\) and write \(k=dk'\).
The scale becomes \(K/q_d\) and the character contributes \(\psi(d)\).
Poisson summation in residue classes modulo \(m\) has prefactor
\(K/(q_dq_m)\).  Its finite transform is
\[
 \sum_{x\bmod m}\psi(x)e(hx/m)
   =\sqrt{q_m}\gamma(\psi;m)\bar\psi(h).
\]
For unit \(h\) this follows by changing variables.  For nonunit \(h\)
the transform is zero by primitivity; for \(m=1\) the stated principal
convention applies.  This proves Equation~\eqref{eq:masked-poisson}.
For a nonprincipal primitive character, finite Parseval gives
\(\sum_h|\sum_x\psi(x)e(hx/m)|^2=q_m\sum_x|\psi(x)|^2\).
Writing \(\phi(m)=|(\mathcal O/m)^\times|\), there are \(\phi(m)\)
unit \(h\)'s, all with magnitude
\(\sqrt{q_m}|\gamma(\psi;m)|\), and
\(\sum_x|\psi(x)|^2=\phi(m)\).  Thus \(|\gamma(\psi;m)|=1\).

A fixed ray restriction can first be expanded into finitely many
characters.  Each is then replaced by its primitive inducing character,
with the removed local zero extensions kept in \(R\); the same formula
applies term by term.  For the principal modulus the absolute sum of
the nonzero frequencies is \(O_\Phi(d_{\mathcal O}(\operatorname{rad}R))\).
Indeed, for all \(A>0\),
\[
 A\sum_{h\ne0}|\mathcal F\Phi(Aq_h)|\ll_\Phi1.
\]
Lattice counting proves this when \(A\le1\), and Schwartz decay proves
it when \(A\ge1\).  Sum this bound with \(A=K/q_d\) over \(d\).
At every use below the relevant nonempty row scale has \(K\ge1\), so
this is also \(O(KZ^\epsilon)\).  When principal nonzero frequencies
are added or removed while separating a nonprincipal sum, they will
carry the same outer row mask as that sum; their absolute contribution
is no larger than this full principal bound.
\end{proof}

\subsection{The canonical reduction and its induction}

We now prove Lemma~\ref{lem:canonical-moment}.  The reflected energy
handles the short completion; the remaining blocks undergo two Poisson
transformations to produce admissible canonical sums with shorter rows.

\begin{proof}[Proof of Lemma~\ref{lem:canonical-moment}]
The order of the analytic choices is organized by
Corollary~\ref{cor:indexed-seminorm-propagation} in
Section~\ref{sec:moment-propagation}. Its indexed conclusion will be
used on each positive Cauchy-side sum separately; the proof below verifies
the required common coefficient measures, fixed norm losses, and finite depth.

At each node, every joint \((f,k)\) Hilbert space uses the current label
measure \(w(f)\), and that factor is retained exactly once in every
parent \(f\)-sum until Equation~\eqref{eq:inverse-old-label-fibre}
removes the old label.

At each node keep the parameters \(N,V,M\) fixed, and write
\[
 F=N+V,\qquad Q=\log_Zq_{\mathfrak r_\rho}.
\]
Here \(F\) is the real length sum \(N+V\), so \(F=F_0\) at the starting
node.  The two hypotheses at a node with margin \(c_{\rm node}\) are exactly
\begin{equation}\label{eq:inverse-fixed-invariant}
 F-M-Q-z_0\ge c_{\rm node},\qquad
 4F-3M-6z_0\ge c_{\rm node}.
\end{equation}
In particular \(M+z_0+c_{\rm node}\le F\).  The number \(Q\) is the
actual logarithmic norm of the already fixed puncture radical, not a
dyadic center.  The parameter \(M\) is the fixed logarithmic row length;
the first Poisson test has scale \(Z^M\).

Let \(M_{\max}\) bound the starting row lengths.  Choose
\(0<d\le c_*/200\) and put
\[
 D=\left\lceil\frac{M_{\max}+2}{d}\right\rceil+1.
\]
We use positive parameters \(\eta,\tau,\pi,\tau_{\rm ref},\pi_{\rm ref}\),
chosen in the quantified order at the end of the proof.  Here \(\eta\)
is a localization tolerance bounding only the logarithms of fixed annular
ratios, \(\tau\) is the common tolerance in the two Poisson tail comparisons,
and \(\pi\) bounds the aggregate freely chosen local small-power losses.  At depth
\(h\) set
\[
 c_h=c_*-7h\eta,\qquad c_{\rm node}=c_h,
\]
and impose the row cap \(M\le M_{\max}-hd\).  We prove the estimate by
backwards induction on \(h\le D\).  Every retained row parameter below
is nonnegative.  We shall show that a nonterminal passage decreases it
by at least \(d\), loses at most \(7\eta\) in either required margin,
and increases the energy exponent by at most \(40\eta+\tau+\pi\).
No invariant will be transferred from the fixed \(F=N+V\) to an
actual column norm.

\paragraph{The short completion.}
M\"obius inversion of the cube factor in
Equation~\eqref{eq:marked-completion} gives the exact identity
\begin{equation}\label{eq:marked-completion-inversion}
 \begin{split}
 &Z^{-N/2}\sum_{n\ {\rm sf}}a(n)\chi_n(k)\chi_n(f)^4
       \mathfrak d(n)W(q_n/Z^N)\\
 &\quad=
 \sum_h\frac{\mu(h)\bar\alpha(h)^3\Psi_k(h)^3}{q_h}
       \mathcal T_{\mathfrak d_h}(Z^N/q_h^3;k),
 \qquad \mathfrak d_h(A)=\mathfrak d(h^3A).
 \end{split}
\end{equation}
To check both the mark and the normalization, expand the right side and
put \(c=hb\).  The factor \(V_*(y)=y^{1/2}W(y)\) changes the absolute
coefficient to
\[
 Z^{-N/2}\bar\alpha(c)^3\Psi_k(c)^3q_c^{1/2}
       \sum_{h\mid c}\mu(h).
\]
The mark is \(\mathfrak d(nc^3)\), independent of the divisor \(h\).
The divisor sum vanishes unless \(c=1\), proving the identity.
Multiplicativity of \(\Psi_k\), including its punctures, is used here.

Use one fixed smooth dyadic partition with nonnegative ideal centers,
the unit dyad having center zero.  Suppose \(V<d\) and that the center
\(\ell_1\) of the \(h\)-dyad is less than \(d\).  Freeze \(h\), assign
to it the slots it divides, and retain the others on the completed
index.  A surviving slot then has the individual coefficient
\(a_i(p)1_{p\nmid h}\).  Since \(h\) is fixed, this is a bounded
coefficient independent of the current \(k,f\), on the same nominal
annulus and with the same cap.  It is not the unrestricted original
coefficient; the recursive branch below restores its original slot
lists separately by the child zeros.  For an active subcollection define \(z_a\) to be the sum of its
nominal slot lengths.  Assignments only delete slots, so
\(0\le z_a\le z_0\) exactly; an actual tuple norm is never used as this
cap.  Triangle inequality in the joint \((f,k)\) Hilbert space uses
\(\sum_{q_h\asymp Z^{\ell_1}}q_h^{-1}\ll1\) for the fixed annulus,
apart from the separately chosen divisor loss for assignments.  The
bounded factor \(\Psi_k(h)^3\) is a contraction in that space.

Write \(\widehat h=\log_Zq_h\).  The completion is at the actual scale
\(N_*=N-3\widehat h\).  Its fixed support gives
\(\widehat h\le\ell_1+\eta<d+\eta\) once the fixed annular threshold
specified below is imposed.  In Lemma~\ref{lem:reflected-energy}, keep
the actual residual-row dyad and choose the same threshold so that
\begin{equation}\label{eq:inverse-terminal-width}
 H\le M-O+\eta,\qquad
 T_d-S_0-B_0\le2M-O+Q+V-N_*+2z_a+\eta.
\end{equation}
These are Equations~\eqref{eq:actual-residual-row-dyad} and
\eqref{eq:common-width-charge}, not estimates for an enlarged row
range.  Substituting \(N_*=N-3\widehat h\) and the first inequality in
Equation~\eqref{eq:inverse-fixed-invariant} gives
\begin{equation}\label{old-eq:4.9}
 T_d-S_0-B_0
 \le M-O+2z_a-z_0-c_{\rm node}+5d+4\eta.
\end{equation}
Indeed, the additional terms are \(2V+3\widehat h+\eta<5d+4\eta\).

Take the threshold also to ensure \(e_\lambda\ge-\eta\), and discard
the whole reflected tail dyads as in Lemma~\ref{lem:reflected-energy}.
Thus every retained dyad satisfies
\(v+3\ell_b+e_\lambda\le T_d+\tau_{\rm ref}\).  The row branch of
Equation~\eqref{old-eq:4.6}, after dropping its nonpositive terms, obeys
\[
 E_{\rm ref}\le M+z_a+\tfrac53\eta
       \le F-c_{\rm node}+\tfrac53\eta.
\]
For the column branch with \(u=z_a\), the retained dual bound first gives
\(E_{\rm ref}\le O/2+(T_d-S_0-B_0)+5\eta/3+\tau_{\rm ref}\).
Equation~\eqref{old-eq:4.9} and the first invariant then give
\begin{equation}\label{old-eq:4.10}
 E_{\rm ref}\le F-2c_{\rm node}+5d+\tfrac{17}{3}\eta+\tau_{\rm ref}.
\end{equation}
If \(u\ne z_a\), its definition implies \(u\ge v/2\).  The retained
dual bound now first gives
\[
 E_{\rm ref}\le O/2+(T_d-S_0-B_0)/2+z_a
                    +\tfrac76\eta+\tfrac12\tau_{\rm ref}.
\]
Using Equation~\eqref{old-eq:4.9}, \(z_a\le z_0\), and then the second
invariant yields
\begin{equation}\label{old-eq:4.11}
 E_{\rm ref}\le F-M/4-3c_{\rm node}/4+(5/2)d
                    +\tfrac{19}{6}\eta+\tfrac12\tau_{\rm ref}.
\end{equation}
Use the reflected estimate with aggregate local loss
\(\pi_{\rm ref}\), including its freely chosen exponent, the label divisor
bound, and the finitely many logarithmic sums.  If
\[
 c_{\rm node}\ge c_*/2,\qquad
 d\le c_*/200,\qquad \eta,\tau_{\rm ref},\pi_{\rm ref}\le c_*/1000,
\]
all three bounds, after adding \(\pi_{\rm ref}\), are strictly below
\(F\).  In the last bound \(M\ge0\).  For \(z_0=0\) only \(u=z_a=0\) occurs.
The reflected estimate was applied for each fixed \(f\); the weighted
mass bound \(\sum_f w(f)\ll Z^{V+\pi_{\rm ref}}\) shows that summing its
squared bounds over \(f\) costs at most this amount, already included
in the aggregate loss, and cancels the outside \(Z^{-V}\).
Thus no moving fourth-power label entered the hybrid norm, and these
terminal terms satisfy the canonical estimate.

\paragraph{The remaining terms and their marks.}
At every factorization, assign a slot first to an extracted factor
which its prime divides, and otherwise retain it on the residual
column.  The basic identity for the completion is
\begin{equation}\label{old-eq:4.12}
 1_{p\mid nb^3}=1_{p\mid b}+1_{p\nmid b}1_{p\mid n}.
\end{equation}
More generally, for an ordered list of extracted factors
\(D_1,\ldots,D_h\), the exact priority identity is
\begin{equation}\label{eq:inverse-slot-priority}
 1_{p\mid D_1\cdots D_hn}
 =\sum_{j=1}^h1_{p\mid D_j}\prod_{a<j}1_{p\nmid D_a}
   +1_{p\mid n}\prod_{a\le h}1_{p\nmid D_a}.
\end{equation}
It holds even when extracted factors share primes.  Applying it to each
slot and each copy of a square gives a disjoint partition for each tuple
of primes; the two copies of one slot have independent prime choices.
There are at most \(2^{|I|}\) choices at one single-factor assignment,
and at most \((h+1)^{|I|}\) for the displayed ordered list.  The
assigned slots have divisor-bounded coefficients on the extracted
factor.  Because the original coefficient is
\(\prod_i a_i(p_i)\), removing assigned slots leaves exactly a
subcollection with its original product coefficients.  An ideal which
will remain an averaging variable is not fixed merely because a slot
was assigned to it.

If \(V<d\) and \(\ell_1\ge d\), reopen the completion on the right of
Equation~\eqref{eq:marked-completion-inversion}.  If its cube ideal is
\(h'\), first isolate its dyad with center \(\ell_2\ge0\) and the
\(h\)-dyad by triangle inequality in the joint Hilbert space, before
squaring.  Put \(b=hh'\) and define the fixed centers
\[
 \ell=\ell_1+\ell_2,\qquad r=N-3\ell.
\]
Both copies \(b_1,b_2\) in the resulting square have this same nominal
center \(\ell\); their actual norms need not agree.  The mark is
\(\mathfrak d(nb^3)\), independent of the
choice of the divisor \(h\) of \(b\).  Multiplicativity gives a factor
\(\chi_b(k)^3\); the fourth power in \(\Psi_k(b)^3\) is exactly the
mask \(1_{(b,f)=1}\).  The remaining sum over divisors \(h\) of \(b\)
has absolute value at most a divisor function.  After dyadic
decomposition and separation of \(q_nq_b^3/Z^N\), each squared block
is bounded by a small power times
\begin{equation}\tag{A}\label{eq:canonical-block}
 \begin{split}
 &Z^{-r-2\ell-V}\sum_f w(f)\sum_{k\in\mathcal O}\Phi(q_k/Z^M)
 \Biggl|\sum_{q_b\asymp Z^\ell}\beta(b,f)\chi_b(k)^3\\
 &\hspace{26mm}\times
       \sum_{n\ {\rm sf}}a(n)\chi_n(k)\chi_n(f)^4
       \mathfrak d(nb^3)W(q_n/Z^r)\Biggr|^2 .
 \end{split}
\end{equation}
Here \(\Phi\) is a fixed nonnegative radial Schwartz function
majorizing the original row ball, and
\(|\beta(b,f)|\ll Z^\epsilon\) independently of \(k,n\).  Its
\(f\)-dependence includes the original mask \(1_{(b,f)=1}\);
all other original cube masks are retained.  There is no condition
\((b,n)=1\).  The exponent in front is correct because the original
normalization is \(Z^{-N/2}q_b^{1/2}\asymp Z^{-r/2-\ell}\).
If \(V\ge d\), use the same block with \(\ell=0,\ b=1,\ r=N\).
Thus every remaining block has exactly \(V+\ell\ge d\), by the center
classification and \(\ell_2\ge0\).  The added row \(k=0\)
can only contribute in the principal cases counted below.

\paragraph{The two-transform reduction.}
We prove the following conditional estimate for every remaining block
in Equation~\eqref{eq:canonical-block}. Let \(\epsilon_{\rm child}\ge0\).
Suppose every energy \(\mathcal E'\) of the form
\eqref{eq:canonical-energy}, with all coefficient and support hypotheses
of Lemma~\ref{lem:canonical-moment}, the same fixed slot cap, parameters
\[
 0\le M'\le M-d,\qquad 0\le N',V',\qquad F'=N'+V'\le F+11\eta,
\]
and both margins at least \(c_{\rm node}-7\eta\), satisfies
\(\mathcal E'\ll Z^{F'+\epsilon_{\rm child}}\).
The hypothesis is uniform over the bounded ranges used in this induction,
with finite-seminorm and fixed polynomial-height dependence as in that
lemma. For
\(\eta\le d/16\), the remaining block satisfies
\begin{equation}\label{eq:canonical-reduction-contract}
 \text{block in Equation~\eqref{eq:canonical-block}}
 \ll Z^{F+40\eta+\tau+\pi+\epsilon_{\rm child}},
\end{equation}
with the same kind of uniform dependence. The freely chosen local losses
sum to \(\pi\); the principal terms and discarded tails are included
in this estimate. Thus the reduction supplies exactly the implication
needed for backwards induction. Its proof occupies the two transformations
below: the first produces a positive inverse-polynomial norm, and the second
constructs the smaller canonical energies to which the hypothesis applies.

We next specify the support and localization conventions used in both
transformations.  For an actual ideal or nonzero element \(a\), write
\(\widehat a=\log_Zq_a\).  For a named scalar center, a hat will denote
the actual logarithmic norm of its indicated ideal.  All ideal centers
introduced below are nonnegative centers from the fixed dyadic partition.
The already fixed puncture and the ideal \(q_0\) introduced below are used
at their actual logarithmic norms, without independently rounded centers.

Every individual dyad introduced in the two transformations has a fixed
compact normalized support. If \(a\) is its ideal and \(a_{\rm cen}\)
its named logarithmic center, we use
\begin{equation}\label{eq:inverse-ratio-bounds}
 |\log_Zq_a-a_{\rm cen}|\le\eta.
\end{equation}
This convention applies to the original \(n_i,b_i,f\) windows and to
each extracted-ideal dyad when it is introduced. Errors for products
and quotients are the sums of these individual errors; the fresh residual
windows below have the explicitly stated bounds \(4\eta\), \(6\eta\)
and \(4\eta\). These estimates concern fixed compact normalized-ratio
intervals, not annuli of ratio \(Z^\eta\).

For clarity, the intervals are fixed uniformly through the whole finite
depth as follows.  At each factorization choose fresh individual smooth
cutoffs equal to one on the quotient supports, and include the full
supports of these cutoffs, of the larger cutoffs used in Fourier
separation, and of the enlarged label windows.  Include also the bounded
clipping families used below.  Products and quotients of endpoints locate
the initial quotient supports, such as those of \(b=hh'\), the residual
\(n\), and the child column and label.  Iterating this construction
through \(D\) levels gives a finite collection \(\mathcal I_D\) of compact
normalized-ratio intervals.  If \(L_{\rm win}\) is the maximum absolute
logarithm of their endpoints, the threshold
\(\log Z\ge L_{\rm win}/\eta\) implies
Equation~\eqref{eq:inverse-ratio-bounds} and all the fresh-support bounds
specified below.  After a positive sum is enlarged, its newly added
columns or labels need not satisfy an old parent product identity.  Their
norm bounds at this and the next node come directly from the full
independent fresh windows in \(\mathcal I_D\).  Separated norm powers do
not change these supports.

We use the following specialization of the common joint Fourier calculus in
Lemma~\ref{lem:smooth-calculus}.
Every normalized norm occurring in a coupled smooth factor to be separated
from the columns is a coordinate \(y_1,\ldots,y_d\) of one ambient profile.
A derived outer mask or label cutoff retained in the weight remains
outside and needs no profile coordinate.  Keep the current
individual dyad cutoffs and fresh column cutoffs outside the inversion,
until the weighted Cauchy inequality where they are used.  Multiply the
coupled profile by larger individual cutoffs \(\Omega_j\) equal to one
on the full supports of those retained cutoffs.  For the resulting
compact profile \(\mathfrak H\), define before summing any current row or
label
\begin{equation}\label{eq:inverse-log-fourier}
 \begin{split}
 \widehat{\mathfrak H}(\boldsymbol t)
   &=\int_{\mathbb R^d}\mathfrak H(e^{u_1},\ldots,e^{u_d})
             e^{-i\boldsymbol t\cdot\boldsymbol u}\,d\boldsymbol u,\\
 \mathfrak H(\boldsymbol y)
   &=(2\pi)^{-d}\int_{\mathbb R^d}\widehat{\mathfrak H}(\boldsymbol t)
          \prod_{j=1}^dy_j^{it_j}\,d\boldsymbol t.
 \end{split}
\end{equation}
Arithmetic relations among the norms restrict evaluation points of this
identity, not its density.  Outer modes stay in the outer weight; for
two column coordinates the first test receives \(y_1^{it_1}\) and the
second receives \(y_2^{-it_2}\), whose conjugate supplies
\(y_2^{it_2}\).  All normalized real inverse-root powers are included
in the coupled profile, so the final column tests contain no second copy.

This convention also makes the uniformity quantitative.  Write
\(D_j=y_j\partial_{y_j}\).  On a fixed log rectangle, integration by
parts with \((1-\Delta)^{m_0}\), for \(2m_0>J+d\), gives
\begin{equation}\label{eq:inverse-fourier-moment}
 \int_{\mathbb R^d}|\widehat{\mathfrak H}(\boldsymbol t)|
        (1+|\boldsymbol t|)^J\,d\boldsymbol t
 \ll_{J,d,\mathrm{box}}\max_{|\boldsymbol a|\le2m_0}
             \|D^{\boldsymbol a}\mathfrak H\|_\infty.
\end{equation}
For a fixed radial Schwartz Fourier kernel \(K\), every
\(\sup_{x\ge0}|(x\partial_x)^jK(x)|\) is finite, and
\[
 D^{\boldsymbol a}K\!\left(A\prod_jy_j^{e_j}\right)
 =\left(\prod_je_j^{a_j}\right)
   (x\partial_x)^{|\boldsymbol a|}K(x)
       \big|_{x=A\prod_jy_j^{e_j}}.
\]
Thus the separating seminorms are uniform for every scalar \(A>0\),
with no derivative-order power of \(Z\).  Normalized real powers have
bounded Euler derivatives on the fixed boxes, and inherited norm twists
have only a fixed polynomial height cost.  No sharp cutoff in a
column-dependent kernel ratio is put inside such a profile.
For the radial Fourier kernels in the Poisson steps, these are the
normalized-kernel bounds of Lemma~\ref{lem:kernel-seminorms} and
Equation~\eqref{eq:normalized-kernel-profile}.

The raw tail estimate we shall use is, for \(Y\ge1\) and \(A>1\),
\begin{equation}\label{eq:inverse-raw-tail}
 \sum_{h\ne0:\,a q_h>Y}|K(aq_h)|
 \ll_A(1+a^{-1})Y^{1-A}\qquad(a>0).
\end{equation}
Indeed, the shell \(2^jY<a q_h\le2^{j+1}Y\) contains
\(O(1+2^jY/a)\) lattice points and the kernel is
\(O_A((2^jY)^{-A})\) there.  Summing the two geometric series proves the
display; the same argument applies to the fixed lattice
\(\lambda^{-4}\mathcal O\).  The proof uses only the large-argument
bound \(|K(x)|\ll_Ax^{-A}\) for \(x\ge1\), so it also applies to the
reflected kernel on such a tail.  Below an actual-ratio inequality is used only
to show that the complement of a sector-fixed outer row ball is contained
in this tail for each supported raw column pair.  That complement is
removed before off-coprime extension and before Fourier absolutization.
The full smooth kernel is retained inside the ball, whose nonsmooth mask
is kept outside Fourier inversion until the relevant weighted Cauchy
inequality.  The tail orders and the crude raw counts are fixed at the end.

\paragraph{The first Poisson transformation.}
Expand the square in Equation~\eqref{eq:canonical-block}.  Put
\[
 \mathfrak B=\operatorname{rad}(b_1b_2),\qquad
 n_i=\mathfrak A_i C u_i\quad(i=1,2),
\]
where \(\mathfrak A_i=(n_i,\mathfrak B)\), and \(C\) is the gcd of
\(n_1/\mathfrak A_1\) and \(n_2/\mathfrak A_2\).  Then
\(\mathfrak A_i\mid\mathfrak B\), while \(u_1,u_2\) are squarefree,
coprime to one another, and prime to \(C\mathfrak B\).  Let \(A_i,B\)
be the fixed centers of \(\mathfrak A_i,C\), and write
\(\widehat A_i=\log_Zq_{\mathfrak A_i}\), \(\widehat B=\log_Zq_C\).
For \(p\mid\mathfrak B\), define
\[
 a_{ip}=1_{p\mid\mathfrak A_i},\qquad
 \pi_p=v_p(b_1b_2)\bmod2,\qquad
 t_p=a_{1p}-a_{2p}+3\pi_p\bmod6,
\]
and put \(R_1=\prod_{t_p\ne0}p\).  Its center is \(R\), and
\(\widehat R=\log_Zq_{R_1}\).
The row character and its remaining zero mask are exactly
\begin{equation}\label{eq:first-masked-data}
 \psi_1=\chi_{u_1}\bar\chi_{u_2}
               \prod_{t_p\ne0}\chi_p^{t_p},\qquad
 m_1=u_1u_2R_1,\qquad
 R_{\rm mask}=C\mathfrak B/R_1.
\end{equation}
Indeed, the exponents at \(u_1,u_2\) are \(1,-1\); the common
off-\(\mathfrak B\) factor \(C\) has exponent zero but retains its
mask; and the exponent at \(p\mid\mathfrak B\) is \(t_p\).  Each
nonzero local power is nonprincipal and primitive modulo \(p\), with
disjoint supports.  Thus \(\psi_1\) is primitive unless \(m_1=1\).
Every original column mask and the two factors
\(\beta(b_1,f)\overline{\beta(b_2,f)}\) remain in the expression.

Apply Lemma~\ref{lem:masked-poisson} with \(K=Z^M\), and denote its
divisor by \(d_k\mid C\mathfrak B/R_1\), with center \(\delta\) and
\(\widehat\delta=\log_Zq_{d_k}\).  On this genuine coprime expression,
the actual conductor length is
\begin{equation}\label{eq:first-poisson-lengths}
 \widehat L_1=\widehat n_1+\widehat n_2-\widehat A_1-\widehat A_2
                    -2\widehat B+\widehat R.
\end{equation}
The actual kernel argument is \(Z^Mq_h/(q_{d_k}q_{m_1})\), and the
Poisson prefactor has exponent \(M-\widehat\delta-\widehat L_1/2\).
We have not yet truncated or Fourier-separated this kernel.

If \(m_1=1\), then \(u_1=u_2=1\) and \(t_p=0\) for every
\(p\mid\mathfrak B\).  The latter condition forces \(\pi_p=0\)
and \(a_{1p}=a_{2p}\).  Hence \(n_1=n_2\) and \(b_1b_2\) is a
square.  Even counting all \(O(Z^{2\ell+\epsilon})\) pairs \(b_1,b_2\),
all \(O(Z^{r+\epsilon})\) equal columns, all \(O(Z^{V+\epsilon})\)
labels, and \(O(Z^M)\) rows, the normalization in
Equation~\eqref{eq:canonical-block} gives \(O(Z^{M+\epsilon})\).
If \(r<0\) but its fixed annular column window is nonempty, then
\(Z^{-r}\) is bounded by its fixed upper endpoint, so its ideal count
is still \(O(Z^r)\) with a fixed constant.
Marks and the retained masks do not increase this bound beyond a
small power.  When the nonzero principal terms are restored below, they
will carry the same outer ball mask as the nonprincipal terms; the last
part of Lemma~\ref{lem:masked-poisson} bounds them at this same cost.

We next identify its column coefficients.  For coprime squarefree primary \(a,b\),
CRT and reciprocity give
\[
 \gamma_2(ab)=\gamma_2(a)\gamma_2(b)
       \chi_a(b)^2\chi_b(a)^2
 =\gamma_2(a)\gamma_2(b)\chi_b(a)^4.
\]
In the first Poisson root, the factors involving \(u_1\), including
the divisor and frequency, are
\[
 \gamma_1(u_1)\overline{\chi_{u_1}(h)}
 \chi_{u_1}(d_kR_1)\prod_{t_p\ne0}\chi_{u_1}(p^{t_p}),
\]
up to a fixed-ray factor.  This follows by writing the CRT factors
between \(u_1\) and \(p\) as
\(\chi_{u_1}(p)\chi_p(u_1)^{t_p}\) and using reciprocity.  On the
conjugated second side \(t_p\) is replaced by \(-t_p\).  The cross
factor between \(u_1,u_2\) is the reciprocity factor
\(\mathcal R(u_1,u_2)\), not a quotient of zero-extended symbols.

Put \(P_T=\prod_{t_p\ne0}p^{t_p}\), regarded in the fixed ray group.
Equation~\eqref{eq:signal} and
Equation~\eqref{eq:G-quadratic-refinement} show that the remaining
two-column ray factor is
\[
 G(u_1u_2^{-1})\mathcal R(u_1u_2^{-1},P_T).
\]
For example, before placing factors inside the conjugated second
polynomial, the raw second Gauss-signal factor is
\(\mu(u_2)\chi_{u_2}(-1)\overline{G(u_2)}\).  Its corresponding
factor written inside that polynomial is
\(\mu(u_2)\chi_{u_2}(-1)G(u_2)\).  Also
\(G(u_2^{-1})=\chi_{u_2}(-1)\overline{G(u_2)}\).
These identities give the displayed quotient.  Fix the ray class of
\(P_T\) and Fourier-expand the displayed function on the finite ray
group.  Its factors on each side are genuine multiplicative ray
characters, denoted by \(\nu_i\).  Their number and coefficient norm
depend only on the fixed ray group.

Define
\[
 E=\prod_{\pi_p=1}p^{a_{1p}+a_{2p}},\qquad
 \widetilde h=hf^2E.
\]
The original fourth power at \(f\) equals
\(\overline{\chi_{u_i}(f^2)}\), including zeros.  Assign the slots
first to \(b_i,\mathfrak A_i,C\).  The remaining local factor at
\(p\mid\mathfrak B\) on side \(i\) has exponent
\[
 4a_{ip}+1_{t_p\ne0}(1\mathbin{\pm}t_p)
          +\pi_p(a_{1p}+a_{2p})\pmod6.
\]
The plus sign belongs to side one.  Direct reduction gives the same
answer on both sides:
\[
 \begin{array}{c|cc|c|c}
 \pi_p&a_{1p}&a_{2p}&t_p&\text{exponent on both sides}\\ \hline
 0&0&0&0&0\\
 0&1&0&1&0\\
 0&0&1&5&0\\
 0&1&1&0&4\\
 1&0&0&3&4\\
 1&1&0&4&4\\
 1&0&1&2&4\\
 1&1&1&3&4
 \end{array}
\]
Define \(\xi(n)\) as the product over \(p\mid\mathfrak B\) of
\(\chi_n(p)\) raised to the corresponding exponent in the last column.
An exponent zero still denotes the puncture at \(p\).  The calculation
proves that \(\xi\) is common to both sides and independent of \(h,f,C\).
The nonprincipal coefficient on side \(i\), after these assignments, is
\begin{equation}\tag{C}\label{eq:first-poisson-column}
 \mu(u_i)\nu_i(u_i)\rho(u_i)
 \overline{\chi_{u_i}(\widetilde h)}
 \chi_{u_i}(C)^4\chi_{u_i}(d_k)\xi(u_i)\mathfrak d_i(u_i).
\end{equation}
It also shows that every character involving the moving
\(b_i,\mathfrak A_i\) is in either \(\xi\) or \(\widetilde h\);
their other factors are outer coefficients.

The first transform has produced the inverse-type column coefficient
in Equation~\eqref{eq:first-poisson-column}.  We next collect its
fourth-power factors into one squarefree label before forming the
positive row norm for the second transform.

\paragraph{Retaining the fourth-power label from the cubes.}
Write uniquely \(b_1b_2=q^2s\) with \(s\) squarefree, and define
\[
 J_2=\prod_{\substack{\pi_p=0\\a_{1p}=a_{2p}=1}}p,\qquad
 J=sJ_2,\qquad q=J_2q_0.
\]
Every prime of \(J_2\) has positive even valuation in \(b_1b_2\),
so \(J_2\mid q\).  The ideals \(s,J_2\) are coprime and squarefree;
hence \(J\) is squarefree.  Let \(j\ge0\) be the center of \(J\), and put
\[
 \widehat s=\log_Zq_s,\quad \widehat j_2=\log_Zq_{J_2},\quad
 \widehat j=\log_Zq_J=\widehat s+\widehat j_2,\quad
 \widehat\ell_{\rm pair}=(\widehat b_1+\widehat b_2)/2.
\]
The ideal identity \(b_1b_2=q_0^2J_2^2s\) gives exactly
\begin{equation}\label{eq:inverse-cube-actual}
 \widehat c:=\log_Zq_{q_0}
   =\widehat\ell_{\rm pair}-\widehat s/2-\widehat j_2\ge0,
 \qquad
 4\widehat\ell_{\rm pair}-2\widehat s+\widehat j_2
   =4\widehat c+5\widehat j_2\ge0.
\end{equation}
The table above now proves the exact zero-extended identity
\begin{equation}\label{eq:retained-cube-label}
 \xi(n)=\chi_n(J)^4\,1_{(n,\operatorname{rad}q_0)=1}.
\end{equation}
Indeed, the primes with exponent four are exactly those of \(sJ_2\);
every other prime of \(\mathfrak B\) divides \(q_0\).  A prime in
both \(J\) and \(q_0\) only repeats the zero already supplied by
\(\chi_n(J)^4\).  We will fix \(q_0\) but keep \(J\) in a later
average.  The modulus \(q_0\) contributes a puncture and is not part
of the fixed ray group.
For fixed \(J,q_0\), the choices of \(s,J_2\), the factorizations
\(b_1b_2=q_0^2J_2^2s\), and the ideals \(\mathfrak A_i\) have only
divisor multiplicity.

We now localize the genuine first Poisson expression, before extending its
coprime support or taking absolute Fourier integrals.  The local table gives
the exact actual-norm identity
\[
 \widehat R-\widehat A_1-\widehat A_2+\widehat E
       =\widehat s-2\widehat j_2.
\]
For \(y=hf^2E\), the implication
\(Z^Mq_h/(q_{d_k}q_{m_1})\le Z^\tau\) and
Equation~\eqref{eq:first-poisson-lengths} give
\[
 \begin{split}
 \widehat y
 &\le \widehat n_1+\widehat n_2-2\widehat B-M
       +\widehat s-2\widehat j_2+\widehat\delta+2\widehat f+\tau\\
 &\le \widehat n_1+\widehat n_2-2\widehat B-M
       +4\widehat\ell_{\rm pair}-\widehat j
       +\widehat\delta+2\widehat f+\tau.
 \end{split}
\]
The second inequality is Equation~\eqref{eq:inverse-cube-actual}.
Define the formal center and the enclosing outer row scale by
\begin{equation}\label{eq:enlarged-first-frequency}
 H_c=2r-2B-M+4\ell+2V+\delta-j,\qquad
 H_{\rm use}=H_c+12\eta+\tau.
\end{equation}
Relative to \(H_c\), the last actual upper bound has error
\[
 (\widehat n_1-r)+(\widehat n_2-r)-2(\widehat B-B)
 +2(\widehat b_1-\ell)+2(\widehat b_2-\ell)
 -(\widehat j-j)+(\widehat\delta-\delta)+2(\widehat f-V),
\]
whose positive maximum under Equation~\eqref{eq:inverse-ratio-bounds} is
\(12\eta\).  Thus the actual-ratio inequality implies
\(q_y\le Z^{H_{\rm use}}\) for every supported raw column pair.

On that raw expression insert only the outer mask
\[
 \mathcal B_1(h)=1_{0<q_{hf^2E}\le Z^{H_{\rm use}}}.
\]
For each supported pair, its complement is contained in the actual kernel
tail \(Z^Mq_h/(q_{d_k}q_{m_1})>Z^\tau\); apply
Equation~\eqref{eq:inverse-raw-tail} to discard that complement, with the
raw counts and order fixed below.  No indicator of the ratio inequality
is inserted.  Inside \(\mathcal B_1=1\) retain the full smooth kernel,
including pairs whose ratio is larger than \(Z^\tau\).  The mask depends
on \(h,f,E\) and the fixed sector, but, after the indicated extraction,
not on \(u_1,u_2\).  If \(H_{\rm use}<0\), its nonzero ball is empty and
the same tail comparison discards every nonzero frequency.  Any principal
nonzero terms now restored to unify the formula carry this identical mask.
Their absolute contribution is bounded by the full principal restoration
already estimated.

Remove the condition \((u_1,u_2)=1\) by
\[
 1_{(u_1,u_2)=1}=\sum_{t'\mid(u_1,u_2)}\mu(t'),\qquad u_i=t'x_i,
 \qquad \widehat t=\log_Zq_{t'}.
\]
Let \(t\) be its fixed nonnegative center.
This inversion is made after the cross phases have been replaced by
the fixed ray functions above.  Those functions, the unchanged outer
mask \(\mathcal B_1\), all remaining
zero-extended local factors, and the formal product
\(q_{u_1}q_{u_2}q_{R_1}\) in the smooth kernel define an expression
also for noncoprime \(u_1,u_2\).  It agrees with Poisson summation on
the coprime support; the displayed divisor identity then recovers
exactly that support.  No primitive Poisson formula is asserted for
the newly introduced noncoprime pairs.  Squarefreeness retains the
puncture \((x_i,t')=1\).  Assigned slots at \(t'\) are outer
coefficients, while the other slots form the mark on \(x_i\).

Insert dyads of \(q_h\) from one fixed partition common to all \(f,E\),
not partitions recentered for those labels.  The mask \(\mathcal B_1\)
implies \(q_h\le q_{hf^2E}\le Z^{H_{\rm use}}\), so there are only
logarithmically many such dyads in the bounded retained scale range.
Keep \(\mathcal B_1\) outside every Fourier inversion.  Define
\[
 s_i=r-A_i-B-t,\qquad
 \kappa_i=M-2r-2\ell-V-\delta+A_i+B-R/2.
\]
The identity \(n_i=\mathfrak A_iCt'x_i\) on the factorized expression
gives \(|\widehat x_i-s_i|\le4\eta\).  The full fresh \(x_i\)-support
is included in \(\mathcal I_D\), so this same bound holds directly on
that support after separation, not only on the original product support.
The reconstructed formal products used in the current prefactor are also
included there.

\paragraph{Separating the first transformed product.}
We record the actual prefactor before applying Cauchy.  It splits exactly
over the two column sides as
\[
 Z^{-r-2\ell-V}Z^{M-\widehat\delta-\widehat L_1/2}
       =Z^{\widehat\kappa_1/2}Z^{\widehat\kappa_2/2},
 \quad
 \widehat\kappa_i=M-r-2\ell-V-\widehat\delta-\widehat n_i
                  +\widehat A_i+\widehat B-\widehat R/2.
\]
Equation~\eqref{eq:inverse-ratio-bounds} gives on each side
\begin{equation}\label{eq:inverse-first-prefactor-error}
 \widehat\kappa_i\le\kappa_i+\tfrac92\eta.
\end{equation}
The five error weights are \(1,1,1,1,1/2\).  We now implement this
normalization with all real inverse roots in one joint profile.
Fix a first dyadic, ray, and slot pattern \(\sigma\), and let \(h_1\)
be the center of its bare-\(h\) dyad.  Use the nine independent
normalized coordinates
\[
 \boldsymbol y=(y_{A_1},y_{A_2},y_C,y_d,y_R,y_t,y_h,y_{x_1},y_{x_2})
 =\left(\frac{q_{\mathfrak A_1}}{Z^{A_1}},
 \frac{q_{\mathfrak A_2}}{Z^{A_2}},\frac{q_C}{Z^B},
 \frac{q_{d_k}}{Z^\delta},\frac{q_{R_1}}{Z^R},\frac{q_{t'}}{Z^t},
 \frac{q_h}{Z^{h_1}},\frac{q_{x_1}}{Z^{s_1}},\frac{q_{x_2}}{Z^{s_2}}\right).
\]
The exact full root and kernel argument on the formal \(u_a=t'x_a\)
expression are
\begin{equation}\label{eq:inverse-first-root-kernel}
 \begin{split}
 \frac{Z^{-r-2\ell-V}Z^M}
 {q_{d_k}\sqrt{q_{R_1}}q_{t'}\sqrt{q_{x_1}q_{x_2}}}
 &=Z^{(\kappa_1+\kappa_2)/2}
 y_d^{-1}y_R^{-1/2}y_t^{-1}(y_{x_1}y_{x_2})^{-1/2},\\
 \frac{Z^Mq_h}{q_{d_k}q_{R_1}q_{t'}^2q_{x_1}q_{x_2}}
 &=A_{{\rm ker},1}\frac{y_h}{y_dy_Ry_t^2y_{x_1}y_{x_2}},\\
 A_{{\rm ker},1}&=Z^{M+h_1-\delta-R-2t-s_1-s_2}.
 \end{split}
\end{equation}
Choose fresh cutoffs \(\omega_{x,a}\) equal to one on the old
\(W\)-support divided by the individual \(\mathfrak A_a,C,t'\)
supports, and retain them in the columns.  Retain every current outer
dyad cutoff outside inversion.  Let \(\Omega_{1,\alpha}\) be larger
cutoffs equal to one on the full supports of these retained cutoffs.
With \(\mathcal K_1=\mathcal F\Phi\), define the single compact profile
\begin{equation}\label{eq:inverse-first-profile}
 \begin{split}
 \mathfrak H_{1,\sigma}(\boldsymbol y)={}&Z^{-9\eta/2}
 \prod_\alpha\Omega_{1,\alpha}(y_\alpha)
 y_d^{-1}y_R^{-1/2}y_t^{-1}(y_{x_1}y_{x_2})^{-1/2}\\
 &\times W(y_{A_1}y_Cy_ty_{x_1})
       \overline{W(y_{A_2}y_Cy_ty_{x_2})}
 \mathcal K_1\!\left(A_{{\rm ker},1}
       \frac{y_h}{y_dy_Ry_t^2y_{x_1}y_{x_2}}\right).
 \end{split}
\end{equation}
The scalar outside this profile is exactly
\(\prod_{a=1}^2Z^{(\kappa_a+9\eta/2)/2}\).
In particular, the profile contains each real root once, also when the
actual column norms differ.  Its transform is defined by
Equation~\eqref{eq:inverse-log-fourier} on the ambient nine-dimensional
box before any actual \(f,h\) or outer ideal is summed.  The norms of
\(b_a,f,J,E,q_0\) occur only in outer coefficients, characters, masks
or individual cutoffs after Equation~\eqref{eq:first-poisson-column};
they require no additional coupled-profile coordinate.

To expose the arithmetic extraction at \(t'\), put \(y=hf^2E\) and
\[
 B_{1,a}(t';y)=\mu(t')\nu_a(t')\rho(t')\overline{\chi_{t'}(y)}
       \chi_{t'}(CJ)^4\chi_{t'}(d_k)1_{(t',\operatorname{rad}q_0)=1}.
\]
Let \(\varepsilon_1=1\) and \(\varepsilon_2=-1\).  For a fixed first
Fourier mode \(\boldsymbol t\), the remaining polynomials are exactly
\begin{equation}\label{eq:inverse-first-polynomial}
 \begin{split}
 P_a(y)={}&\sum_{x\ {\rm sf}}\mu(x)\nu_a(x)\rho(x)
       \overline{\chi_x(y)}\chi_x(CJ)^4\chi_x(d_k)
       1_{(x,\operatorname{rad}q_0t')=1}\\
 &\qquad\times\mathfrak d_{I_a}(x)\omega_{x,a}(q_x/Z^{s_a})
                  (q_x/Z^{s_a})^{\varepsilon_a it_{x_a}}.
 \end{split}
\end{equation}
Multiplicativity is used on squarefree coprime factors, then through
the displayed zero extensions; no character is divided at a zero.
The one additional \(\mu(t')\) from the mutual-gcd inversion is outer.
Apply Equation~\eqref{eq:inverse-slot-priority} with the ordered list
\((b_a,\mathfrak A_a,C,t';x)\).  A surviving prime dividing \(x\)
is already prime to \(b_a,\mathfrak A_a,C,t'\): every prime of
\(\mathfrak B\) is in \(J\operatorname{rad}q_0\), and the displayed
fourth powers and punctures supply these zeros.  Thus
\(\mathfrak d_{I_a}\) has the original individual coefficients and
lists, while assigned prime identities and priority masks stay outer.

Here is the full structure of the first separated component.  Let
\(\Omega_\sigma\) consist of
\(b_1,b_2,\mathfrak A_1,\mathfrak A_2,C,d_k,t',f,h\) and the assigned
slot primes, subject to their reconstruction relations and individual
supports.  In particular
\(\mathfrak A_a\mid\mathfrak B\), \((C,\mathfrak B)=1\), and
\(d_k\mid C\mathfrak B/R_1\), while \(R_1,E,J,q_0\) are derived,
not free indices.
Let \(\mathcal A_{1,\sigma}\) be the product of the assigned original
coefficients and priority masks, with conjugation on side two, and let
\(\psi_{1,\sigma}^{\rm out}\) be the product of all retained outer
dyad cutoffs.  Let \(e_\sigma\) be the product obtained in the preceding
quotient-free CRT extraction of the old multiplicity \(w(f)\), the two original
\(\beta\) factors, their cube masks including \((b_a,f)=1\), the
extracted zero masks, and the
finite Gauss, unit, reciprocity, and first-ray factors at the old outer
ideals, excluding the displayed \(\mu(d_k)\) and \(B_{1,a}\).
It is independent of \(x_1,x_2\); its definition is by that product,
not by division by Equation~\eqref{eq:inverse-first-polynomial}.
Writing \(\mathcal O_1=\{A_1,A_2,C,d,R,t,h\}\), set
\[
 w_{1,\sigma}(\omega;\boldsymbol t)
 =e_\sigma(\omega)\mu(d_k)\mu(t')B_{1,1}(t';y)
       \overline{B_{1,2}(t';y)}\mathcal B_1(h)
       \mathcal A_{1,\sigma}(\omega)\psi_{1,\sigma}^{\rm out}(\omega)
       \prod_{\alpha\in\mathcal O_1}y_\alpha^{it_\alpha}.
\]
For the masked, principal-restored formal first component, Fourier
inversion gives the exact identity
\begin{equation}\label{eq:inverse-first-separated}
 \begin{split}
 \mathcal T_{1,\sigma}=(2\pi)^{-9}\int_{\mathbb R^9}
 &\widehat{\mathfrak H}_{1,\sigma}(\boldsymbol t)
 \sum_{\omega\in\Omega_\sigma}w_{1,\sigma}(\omega;\boldsymbol t)\\
 &\times Z^{(\kappa_1+9\eta/2)/2}P_1(y)
       \overline{Z^{(\kappa_2+9\eta/2)/2}P_2(y)}\,d\boldsymbol t.
 \end{split}
\end{equation}
Expanding the two polynomials restores the two column modes; the seven
outer modes restore the other coordinates of
Equation~\eqref{eq:inverse-first-profile}.  Its larger cutoffs are one
on the retained supports, and the fresh cutoffs are one wherever the
old windows are nonzero.  This verifies the identity term by term.
The original \(\beta\) factors and every outer mask are still in the
complete sum.

\paragraph{The first positive majorant.}
For any finite or absolutely convergent weighted sum we use
\begin{equation}\tag{D}\label{eq:weighted-cauchy}
 \left|\sum_\omega w_\omega U_1(\omega)\overline{U_2(\omega)}\right|
 \le\prod_{i=1}^2
       \left(\sum_\omega|w_\omega||U_i(\omega)|^2\right)^{1/2}.
\end{equation}
Apply this first with \(U_i=Z^{(\kappa_i+9\eta/2)/2}P_i\) on the whole
\(\Omega_\sigma\), for each fixed mode.  Only now bound the outer
factors absolutely.  Their pure arithmetic magnitudes are at most a
separately allocated \(Z^{\pi_\beta}\) times the old multiplicity and
the absolute assigned-slot product; the retained support gives
\(\mathcal B_1(h)1_{(C,J)=1}\).  In particular no factorization of
\(\beta(b,f)\) has been assumed.

For each fixed old outer reconstruction and nonzero \(y=\widetilde h\),
the relation \(y=hf^2E\) implies \(f^2E\mid(y)\).  If its multiplicity
is at most \(C_0d_{\mathcal O}(f)^{C_0}\), then
\begin{equation}\label{eq:inverse-old-label-fibre}
 \sum_{f^2E\mid(y)}w(f)
 \le C_0d_{\mathcal O}((y))^{C_0+1}
 \le C_{\pi_{\rm old}}Z^{\pi_{\rm old}}
 \qquad(0<q_y\le Z^{H_{\rm use}}).
\end{equation}
The last bound is uniform because \(H_{\rm use}\) stays in a bounded
range; \(h=y/(f^2E)\) is then uniquely determined as an element.
This is an old-label fibre bound, not a multiplicity depending on a
later new label.
The polynomial \(P_i(y)\) has no remaining dependence on \(f\) or on
the individual \(b_i,\mathfrak A_i\) beyond \(J,q_0\), the fixed
dyadic length \(A_i\), and the fixed ray sector: this is precisely
Equations~\eqref{eq:first-poisson-column} and
\eqref{eq:retained-cube-label}, with the norm profiles separated.
After the first weighted Cauchy inequality, \(\mathcal B_1\) is simply
\(1_{0<q_y\le Z^{H_{\rm use}}}\).  At this positive-sum stage, and only
now, majorize it by a fixed nonnegative radial Schwartz function
\(\Phi_+(q_y/Z^{H_{\rm use}})\) which is at least one for arguments at
most one.  A nonempty ball has \(H_{\rm use}\ge0\), so the second Poisson
scale is exactly \(Z^{H_{\rm use}}\ge1\).  The majorant is not substituted
as an equality in the original signed expression.  More precisely, let
\(\mathfrak O_\sigma\) consist of
\[
 o=(b_1,b_2,\mathfrak A_1,\mathfrak A_2,C,d_k,t';
       \text{old assigned slot primes}),
\]
retaining their full individual supports, the source reconstruction
relations independent of \(f,h,x\), the assigned priority masks, and
\((C,J)=1\).  Let \(w_o^+\) be the product of the absolute original
coefficients of its assigned primes, and zero off this set.  It is
nonnegative and independent of \(y\).  Each first positive side is at
most \(C Z^{\pi_\beta+\pi_{\rm old}}\), times its scalar
\(Z^{\kappa_i+9\eta/2}\), times the explicit positive majorant
\begin{equation}\label{eq:inverse-first-majorant}
 \mathcal S_{i,\sigma,\boldsymbol t}
 =\sum_{o\in\mathfrak O_\sigma}w_o^+
    \sum_{y\in\mathcal O}\Phi_+(q_y/Z^{H_{\rm use}})|P_i(y)|^2.
\end{equation}
At this positive step the old \(h\) cutoff and mode, and the zeros of
\(B_{1,a}\) depending on \(y\), may be removed by their upper bounds.
The first weighted Cauchy inequality and the old-label fibre bound have
therefore removed the old \(f,h\) from this positive majorant and every
later outer set; their original \(-V\) normalization remains in
\(\kappa_i\).  After the following count, we apply the second Poisson
transformation to its \(y\)-sum.

Let \(\widehat r_{\rm diff}\ge0\) be the actual log-norm of the
even-parity primes with \(a_{1p}\ne a_{2p}\).  The table gives
\(\widehat R=\widehat s+\widehat r_{\rm diff}\), so
\begin{equation}\label{eq:fixed-cube-count}
 \widehat c=\widehat\ell_{\rm pair}+\widehat R/2-\widehat j
                  -\widehat r_{\rm diff}/2
 \le\ell+R/2-j+\tfrac52\eta.
\end{equation}
Ideal counting with this actual upper bound counts the fixed \(q_0\)
while retaining \(J\).  For a diagonal bound which also counts \(J\),
its full window has exponent at most \(j+\eta\), giving the combined
upper count \(Z^{\ell+R/2+7\eta/2+\epsilon}\).  The choices of the
\(\mathfrak A_i\) and of \(d_k\mid C\mathfrak B/R_1\) add only
divisor factors once \(b_1,b_2,C\) are specified.

Combining Equation~\eqref{eq:inverse-first-separated}, weighted Cauchy,
and the old-label fibre bound gives the explicit output of the first
transformation:
\begin{equation}\label{eq:inverse-first-transform-output}
 \begin{split}
 |\mathcal T_{1,\sigma}|\ll{}&Z^{\pi_\beta+\pi_{\rm old}}
 \int_{\mathbb R^9}|\widehat{\mathfrak H}_{1,\sigma}(\boldsymbol t)|\\
 &\quad\times\prod_{i=1}^2
 \left(Z^{\kappa_i+9\eta/2}\mathcal S_{i,\sigma,\boldsymbol t}\right)^{1/2}
 \,d\boldsymbol t.
 \end{split}
\end{equation}
Each \(\mathcal S_{i,\sigma,\boldsymbol t}\) is the positive sum in
Equation~\eqref{eq:inverse-first-majorant}: its row polynomial has the
M\"obius coefficient in Equation~\eqref{eq:inverse-first-polynomial},
the cube-derived label \(J\) is retained, and the old \(f,h\) no longer
occur among the averaged indices. The following transformation is applied
to these positive row norms.

\paragraph{The second Poisson transformation.}
Fix one of the two positive sums obtained from the first Cauchy
inequality.  Its polynomial has the form
\[
 P_i(y)=\sum_x c_i(x)\overline{\chi_x(y)}.
\]
The coefficient \(c_i(x)\) consists of the other factors of
Equation~\eqref{eq:first-poisson-column}, with \(u_i=t'x\), the
puncture at \(t'\), and the separated weight.  It is independent of
\(y\).  In particular \(x\) is squarefree and
\[
 c_i(x)=0\quad\text{unless}\quad
 (x,CJ)=1,\quad (x,\operatorname{rad}q_0t')=1.
\]
The puncture \(\rho\) and the zero of \(\chi_x(d_k)\) are retained
as well.  The \(y\)-sum uses the fixed-shape smooth positive ball at
scale \(Z^{H_{\rm use}}\).

In its expanded square put
\[
 g'=(x_1,x_2),\qquad x_j=g'z_j,\qquad
 \widehat g=\log_Zq_{g'}.
\]
Let \(g\ge0\) be its fixed center.
Then \(z_1,z_2\) are squarefree and coprime, and each is prime to
\(g'\).  The row data for Lemma~\ref{lem:masked-poisson} are
\begin{equation}\label{eq:second-masked-data}
 K=Z^{H_{\rm use}},\qquad
 \psi_2=\bar\chi_{z_1}\chi_{z_2},\qquad
 m_2=z_1z_2,\qquad R_{\rm mask}=g'.
\end{equation}
The common factor \(\bar\chi_{g'}(y)\chi_{g'}(y)\) is exactly this
mask.  The character is primitive away from \(m_2=1\), because its
two nonzero local powers have disjoint supports.  For \(d_2\mid g'\),
let \(\theta\ge0\) be its center and write
\(\widehat\theta=\log_Zq_{d_2}\).  The actual conductor length is
\[
 \widehat L_2=\widehat x_1+\widehat x_2-2\widehat g,
 \qquad |\widehat L_2-2(s_i-g)|\le10\eta.
\]
The last inequality uses the full fresh \(x\)-support bound
\(|\widehat x_a-s_i|\le4\eta\) for each copy and
\(|\widehat g-g|\le\eta\).  For \(k_{\rm new}=d_kd_2k''\), the implication
\(Z^{H_{\rm use}}q_{k''}/(q_{d_2}q_{m_2})\le Z^\tau\) gives
\[
 \begin{split}
 \widehat{k_{\rm new}}
 &\le2(s_i-g)-H_{\rm use}+\delta+2\theta+13\eta+\tau\\
 &=M_c+\eta,
 \end{split}
\]
where
\begin{equation}\label{eq:second-poisson-lengths}
 \begin{split}
 M_c&=2(s_i-g)-H_c+\delta+2\theta\\
    &=M-4\ell-2A_i-2t-2g+2\theta-2V+j,\qquad
 M_{\rm ch}=M_c+\eta.
 \end{split}
\end{equation}
The \(13\eta\) consists of \(10\eta\) from the conductor, one from
\(d_k\), and two from \(d_2\).  Subtracting
\(H_{\rm use}=H_c+12\eta+\tau\) cancels the same tolerance \(\tau\)
used in this second Poisson comparison.  Thus no separate row clipping
or unrecorded boundary error is needed.

Separate the original principal contribution for the direct count below.
Before any off-coprime extension or Fourier absolutization, keep on the
genuine nonprincipal second Poisson expression only the outer mask
\[
 \mathcal B_2(k'')=1_{0<q_{d_kd_2k''}\le Z^{M_{\rm ch}}}.
\]
For each supported pair its complement is contained in the actual ratio
tail just considered, so Equation~\eqref{eq:inverse-raw-tail} discards it
with the order fixed below.  Inside the mask retain the full smooth kernel.
The mask depends on \(d_k,d_2,k''\) and the fixed sector, not on the residual
\(z_1,z_2\).  If \(M_{\rm ch}<0\), the nonzero ball is empty and the same
tail comparison discards every nonzero frequency.  In a retained nonempty
passage \(M_{\rm ch}\ge0\).

The actual second prefactor has the exact side split
\[
 Z^{H_{\rm use}-\widehat\theta-\widehat L_2/2}
       =Z^{\widehat\lambda_1/2}Z^{\widehat\lambda_2/2},
 \qquad \widehat\lambda_a=H_{\rm use}-\widehat\theta-\widehat x_a+
                           \widehat g.
\]
For \(\lambda_c=H_c-\theta-(s_i-g)\), the same support bounds give
\begin{equation}\label{eq:inverse-second-prefactor-error}
 \widehat\lambda_a\le\lambda_c+18\eta+\tau.
\end{equation}
Here \(18=12+1+4+1\).  As in the first split, all normalized real
inverse roots will remain in the joint smooth profile until Fourier
inversion; they are not also appended to the final child tests.
All original coefficient masks remain.  In particular
\[
 (g',CJ)=1,\qquad (z_j,g'CJ)=1
\]
on their nonzero support.

The principal case is \(z_1=z_2=1\), equivalently \(x_1=x_2\).
The nominal identity for its count is
\begin{equation}\label{eq:second-principal-exponent}
 \kappa_i+H_c+s_i+B+t+\ell+R/2=F-B-j,
 \qquad F=N+V=r+3\ell+V.
\end{equation}
The actual upper count adds \(9\eta/2\) from the first prefactor,
\(12\eta+\tau\) from the row scale \(H_{\rm use}\), \(4\eta\) from the
diagonal \(x\)-window, \(\eta\) each from \(C,t'\), and \(7\eta/2\) from
the combined \(q_0,J\) count.  Their sum is \(26\eta+\tau\).  With the
aggregate local loss \(\pi\), this principal contribution is therefore
at most \(Z^{F-B-j+26\eta+\tau+\pi}\le Z^{F+26\eta+\tau+\pi}\).
Any nonzero principal frequencies restored to the formal expression carry
the same \(\mathcal B_2\) mask; their absolute sum is bounded by the full
principal restoration in Lemma~\ref{lem:masked-poisson}, since
\(Z^{H_{\rm use}}\ge1\).

For the nonprincipal terms, we now identify the new coefficient
class.  For squarefree \(z\), complex conjugation of a primitive
Gauss sum and Equation~\eqref{eq:signal} give
\[
 \gamma_{-1}(z)=\chi_z(-1)\overline{\gamma_1(z)},\qquad
 \mu(z)\gamma_{-1}(z)
 =\bar\alpha(z)\gamma_2(z)\chi_z(-1)\overline{G(z)}.
\]
The raw second factor is
\(\mu(z)\gamma_1(z)=\overline{\bar\alpha(z)\gamma_2(z)}G(z)\);
the corresponding factor written inside the conjugated second
polynomial has ray factor \(\overline{G(z)}\).  The CRT cross factor of the two primitive
roots is \(\mathcal R(z_1,z_2)\).  Thus their combined ray factor is
\[
 \mathfrak G(a,b)=\chi_a(-1)\overline{G(a)}G(b)\mathcal R(a,b)
       =G(ba^{-1}).
\]
The last equality follows from
Equation~\eqref{eq:G-quadratic-refinement} and
\(\mathcal R(a,a)=\chi_a(-1)\), first on primes by
Equation~\eqref{eq:fixed-gauss-phases} and then multiplicatively.
All quotients in this display are in the fixed finite ray group.
In particular,
\[
 \mathfrak G(v'n_1,v'n_2)=\mathfrak G(n_1,n_2).
\]
The common multiplicative character \(\nu_i(v')\) also cancels
between the two sides.

Insert dyads of \(q_{k''}\) from one partition common to all \(d_k,d_2\).
The mask \(\mathcal B_2\) implies \(q_{k''}\le Z^{M_{\rm ch}}\), so their
number is logarithmic in the bounded retained range.  Let \(h_2\) be
the center of one such bare-frequency dyad.  We now record the exact
component to be separated.  Fix the first mode and selected side \(i\),
write
\[
 H_o=CJ,\qquad r_0=\operatorname{rad}q_0t',\qquad d_0=d_k,
 \qquad a_0(z)=\bar\alpha(z)\gamma_2(z),\qquad
 \mathcal K_2=\mathcal F\Phi_+,
\]
and retain \(K=Z^{H_{\rm use}}\) from
Equation~\eqref{eq:second-masked-data}.  For squarefree arguments put
\begin{equation}\label{eq:inverse-second-outer-coefficients}
 \begin{split}
 B_o(g')&=\mu(g')\nu_i(g')\rho(g')1_{(g',r_0)=1}
              \chi_{g'}(H_o)^4\chi_{g'}(d_0),\\
 L_{o,d_2,k''}(z)&=a_0(z)\nu_i(z)\rho(z)1_{(z,r_0)=1}
       \chi_z(H_o)^4\chi_z(d_0)\overline{\chi_z(d_2)}\chi_z(k'').
 \end{split}
\end{equation}
The old separated test in \(P_i\) is denoted by \(W_i\), so
\(W_i(x)=\omega_{x,i}(x)x^{\varepsilon_iit_{x_i}}\).
Let \(\psi_g,\psi_{d_2},\psi_{k''}\) denote the retained individual
cutoffs on the fixed \(g',d_2,k''\) dyads, evaluated at their normalized
norms.  For this preliminary pattern \(\sigma_0\), the masked
principal-restored nonzero component of
Equation~\eqref{eq:inverse-first-majorant} is exactly
\begin{equation}\label{eq:inverse-second-formal-component}
 \begin{split}
 \mathcal T_{2,\sigma_0,i}={}&
 \sum_{\substack{o\in\mathfrak O_\sigma,\ g'\ {\rm sf}\\d_2\mid g',\ k''\ne0}}
 w_o^+\psi_g\psi_{d_2}\psi_{k''}\mu(d_2)\mathcal B_2(k'')|B_o(g')|^2\\
 &\times\sum_{\substack{z_1,z_2\ {\rm sf}\\(z_1z_2,g')=1}}
 1_{(z_1,z_2)=1}L_{o,d_2,k''}(z_1)\overline{L_{o,d_2,k''}(z_2)}
       G([z_2][z_1]^{-1})\\
 &\times\mathfrak d_{I_i}(g'z_1)\overline{\mathfrak d_{I_i}(g'z_2)}
       W_i(q_{g'}q_{z_1}/Z^{s_i})
       \overline{W_i(q_{g'}q_{z_2}/Z^{s_i})}\\
 &\times\frac{K}{q_{d_2}\sqrt{q_{z_1}q_{z_2}}}
       \mathcal K_2\!\left(\frac{Kq_{k''}}{q_{d_2}q_{z_1}q_{z_2}}\right).
 \end{split}
\end{equation}
Here brackets denote classes in the fixed finite ray group.  Indeed the
arithmetic part of \(c_i(g'z)\) factors into \(B_o(g')\) and its
residual \(\mu(z)\) coefficient on \((g',z)=1\).  The common row factor
is the mask at \(g'\), whose Poisson expansion gives the one divisor
\(d_2\), while the signal calculation above gives the displayed two
\(L\) factors and ray quotient.  The restored principal pair
\(z_1=z_2=1\) agrees by \(G(1)=1\).  Its separate cost, and the raw
complement removed before this formula, have already been bounded.
Thus this is an equality for the specified component of the positive
majorant, not for the original signed block.

Define the summand of Equation~\eqref{eq:inverse-second-formal-component}
on all individually admissible squarefree \(z_1,z_2\) by its displayed
quotient-free \(L\) factors, fixed-ray quotient, formal product norms,
full kernel, and unchanged \(\mathcal B_2\).  It agrees with the
genuine formula on coprime pairs.  Insert the complete identity
\[
 1_{(z_1,z_2)=1}=\sum_{v'\mid(z_1,z_2)}\mu(v'),\qquad z_a=v'n_a,
\]
before any factorwise estimate.  Let \(v\ge0\) be the center of the
squarefree \(v'\)-dyad and \(\widehat v=\log_Zq_{v'}\).  The
\(n_a\) are squarefree, with \((v',n_a)=1\); they need not be mutually
coprime.  No primitive Poisson formula is used on this extension.
For squarefree \(v'\) put
\[
 D_o(v';d_2,k'')=a_0(v')\nu_i(v')\rho(v')1_{(v',r_0)=1}
   \chi_{v'}(H_o)^4\chi_{v'}(d_0)\overline{\chi_{v'}(d_2)}\chi_{v'}(k'').
\]
On squarefree coprime \(v',n\), the exact extraction is
\begin{equation}\label{eq:inverse-second-extraction}
 \begin{split}
 L_{o,d_2,k''}(v'n)={}&D_o(v';d_2,k'')a_0(n)\nu_i(n)\rho(n)1_{(n,r_0)=1}\\
 &\times\chi_n(H_o)^4\chi_n(d_0)\overline{\chi_n(d_2)}
          \chi_n(k'')\chi_n(v')^4.
 \end{split}
\end{equation}
This uses \(a_0(v'n)=a_0(v')a_0(n)\chi_n(v')^4\).  On an overlap the
right side defines the extension to be zero by \(\chi_n(v')^4\),
without evaluating a nonsquarefree Gauss sum.  The two common factors
give \(|D_o(v';d_2,k'')|^2\), including the zero \((v',k'')=1\) and
its fixed-puncture, \(d_0\), and label zeros.  They remain in the outer
weight through Cauchy.  The old gcd also leaves \(1_{(v',g')=1}\).
Since \(D_o\) vanishes on \((v',CJ)>1\), we retain equivalently the
explicit outer factor \(1_{(v',g'CJ)=1}\), which repeats that label zero.

Set \(r_g=g'/d_2\), \(k_{\rm new}=d_kd_2k''\) and
\(f_{\rm new}=JCd_2v'\).  All moving factors on \(n\) satisfy the
complete zero-extended identity
\begin{equation}\label{eq:inverse-moving-characters}
 \chi_n(d_k)\chi_n(k'')\overline{\chi_n(d_2)}
       \chi_n(CJ)^4\chi_n(v')^4
 =\chi_n(k_{\rm new})\chi_n(f_{\rm new})^4.
\end{equation}
It uses \(\overline{\chi_n(d_2)}=\chi_n(d_2)\chi_n(d_2)^4\), also on
nonunits.  The residual gcd with \(g'=d_2r_g\) splits into the repeated
zero at \(d_2\) and the fixed puncture at \(r_g\).  The old punctures
at \(q_0,t'\) remain.  If \(\mathcal C_{\rm ray}\) denotes the fixed
group through which \(G\) factors, put
\[
 \widehat G(\vartheta)=|\mathcal C_{\rm ray}|^{-1}
       \sum_{c\in\mathcal C_{\rm ray}}G(c)\overline{\vartheta(c)}.
\]
Then
\[
 G([n_2][n_1]^{-1})=\sum_{\vartheta\in\widehat{\mathcal C}_{\rm ray}}
   \widehat G(\vartheta)\overline{\vartheta(n_1)}\vartheta(n_2).
\]
Thus both new polynomials in a fixed ray summand use the same
\(\nu_i\overline\vartheta\), and its one bounded coefficient
\(\widehat G(\vartheta)\) remains outer.

\paragraph{The new canonical data and their admissibility.}
The second transformation has returned the moving characters to canonical
form. We now identify which data will be fixed and which will remain
averaged. This distinction also determines the puncture of the child.
Put \(\nu_a'=\nu_i\overline\vartheta\) for both sides and define
\begin{equation}\label{eq:canonical-child-data}
 \begin{gathered}
 \gamma=(q_0,t',r_g),\qquad r_g=g'/d_2,\\
 k_{\rm new}=d_kd_2k'',\qquad f_{\rm new}=JCd_2v',\\
 \rho'(n)=\rho_\gamma(n)=\rho(n)1_{(n,\operatorname{rad}q_0t'r_g)=1}.
 \end{gathered}
\end{equation}
On the nonzero coefficient support, \(J,C,d_2,v'\) are squarefree
and pairwise coprime: \((C,J)=1\) belongs to \(\mathfrak O_\sigma\),
\(|B_o(g')|^2\) forces \((g',CJ)=1\), \(d_2\mid g'\), and the
retained zero of \(D_o\), together with the old gcd restriction, gives
\((v',g'CJ)=1\). Thus \(f_{\rm new}\) is squarefree on that support.
These common factors remain in the outer weight through Cauchy.

The triple \(\gamma\) will be fixed before the child row and label sums.
Its puncture \(\rho_\gamma\) is therefore independent of those two
averaging variables. The ideals \(J,C,d_2,v'\) remain in the new
averaged label; they are not counted as additional fixed labels.

Apply Equation~\eqref{eq:inverse-slot-priority} on each selected-side
copy with the ordered list \((d_2,r_g,v';n_a)\).  Let \(I_a'\) be the
subcollection of the selected side's slots retained on \(n_a\) in copy
\(a\).  Assigned primes and their priority masks remain outer.
A surviving prime dividing \(n\)
is already prime to \(JCd_2v'\) by the fourth-power zero and to
\(\operatorname{rad}q_0,t',r_g\) by the fixed puncture.  These also
supply every earlier survival exclusion because
\(\operatorname{supp}\mathfrak B\subset\operatorname{supp}(Jq_0)\).
The surviving mark is therefore an original product-form subcollection.
An old slot described as assigned to \(J\) is only a regrouping of an
old \(b_a,\mathfrak A_a\) assignment by this derived support, not a
new independent prime choice.

Keep \(\mathcal B_2\) outside the Fourier separation of the full kernel
and old windows. Let \(I_g,I_d,I_v,I_k\) be the full supports of the
current individual dyad cutoffs \(\psi_g,\psi_{d_2},\psi_v,\psi_{k''}\).
Choose fresh cutoffs \(\omega_{n,a}\) equal to one on the support of
\(W_i\) divided by \(I_gI_v\), with fixed full supports in an interval
\([a,b]\), where \(b\ge1\). Choose a nonnegative \(\omega_f\) equal to
one on \(I_JI_CI_dI_v\), with fixed full support \(I_f\). Include these
full supports, and the reconstructed formal product \(g'v'n\) on them,
in the finite window family \(\mathcal I_D\) specified above.
These full fresh child windows give the formal centers and support bounds
\begin{equation}\label{eq:canonical-child-lengths}
 \begin{gathered}
 N_c=r-A_i-B-t-g-v,\qquad V_c=B+\theta+v+j,\qquad F_c=N_c+V_c,\\
 |\widehat n_{\rm child}-N_c|\le6\eta,\qquad
 |\widehat f_{\rm new}-V_c|\le4\eta.
 \end{gathered}
\end{equation}
On the factorized expression the bounds follow from the exact products
\(n_i=\mathfrak A_iCt'g'v'n_{\rm child}\) and
\(f_{\rm new}=JCd_2v'\).  The full fresh child cutoff and the full enlarged
label window are included in \(\mathcal I_D\), so the same bounds hold
directly on their supports when independent terms are later added by
positivity.  Those added terms are not asserted to arise from a parent
factorization.  During the current separation, the reconstructed formal
product \(x=g'v'n\) on the full fresh cutoff is also included in
\(\mathcal I_D\); thus the \(4\eta\) bound used in the second prefactor
holds on the whole separated side.

Keep the nonnegative label center \(V_c\) without rounding it upward.
Proceed to a child only if the retained row component is nonzero; then
the outer ball has \(M_{\rm ch}\ge0\).  This gate is separate from any
structural outer index set, which may contain zero-weight tuples even
when \(\mathcal B_2\) is identically zero.  Similarly, if a full fresh
child column window contains no squarefree ideal, its polynomial is zero
and that component is omitted.  If \(N_c<0\) and a child column exists, its norm is at least one, and
Equation~\eqref{eq:canonical-child-lengths} gives \(-6\eta\le N_c<0\).
For a retained nonempty child define
\begin{equation}\label{eq:inverse-child-clipping}
 N_{\rm ch}=\max(0,N_c),\qquad \delta_N=N_{\rm ch}-N_c\in[0,6\eta],
 \qquad F_{\rm ch}=N_{\rm ch}+V_c=F_c+\delta_N.
\end{equation}
This is a bounded change of the test, not a \(Z^\eta\)-wide support
enlargement. Use the fixed enclosing support interval \([a,b]\)
chosen above. If
\(N_c<0\), nonemptiness gives \(Z^{\delta_N}\le b\); if \(N_c\ge0\),
then \(\delta_N=0\) and the same bound follows from \(b\ge1\).
Thus \(1\le Z^{\delta_N}\le b\) in both cases.  The fresh annular cutoffs
\(\omega_{n,a}(Z^{\delta_N}y)\) have support in \([a/b,b]\) and uniformly
bounded Euler seminorms. Their scale depends only on the fixed sector
centers, not on a current row or label.  The full union of these clipping
supports is included in \(\mathcal I_D\).

Fix a refinement \(\sigma\) of \(\sigma_0\) by its \(v'\)-dyad,
second ray summand and slot branches.
Let \(\Xi_\sigma\) consist of
\[
 \xi=(o,g'\ {\rm sf},d_2\mid g',v'\ {\rm sf},k''\ne0;
               \text{new assigned slot primes})
\]
with the retained individual supports and slot branches.  It contains
neither the old \(f,h\) nor the current column indices \(n_1,n_2\).
The identities defining \(R_1,E,J,q_0\) are inherited from \(o\),
and \(\gamma,k_{\rm new},f_{\rm new}\) are the displayed functions of
\(\xi\).  The set may be taken before imposing \(\mathcal B_2\) and
\((v',g'CJ)=1\), retaining those masks in the signed outer weight.  Let
\(\mathcal A_{2,\sigma}(\xi)\) be the product of all newly assigned
original slot coefficients and their priority masks, conjugated on
side two.

Let \(\mathfrak X_\sigma^{\rm src}\) be the structural outer set of
this transformed component, before adding any independent child rows
or labels.  It retains
\(o\in\mathfrak O_\sigma\), in particular the exact reconstruction
\(b_1b_2=q_0^2J_2^2s\), the derived \(J=sJ_2,q_0\), and the full
fixed \(b_1,b_2,t'\) dyad supports; it also retains the fixed
\(g',d_2\) supports, \(d_2\mid g'\), and its other outer arithmetic
and slot conditions.  Zero-weight tuples may be retained, and unit
Fourier phases are ignored in defining this structural set.  Define
the set of distinct triples, with no witness multiplicity, by
\begin{equation}\label{eq:inverse-source-triples}
 \Gamma_\sigma=
 \{(q_0,t',r_g=g'/d_2):\xi\in\mathfrak X_\sigma^{\rm src}\}.
\end{equation}
This projection is taken once over all possible current rows and
labels, before fixing any \(k_{\rm new},f_{\rm new}\) or Fourier
mode.  Every nonzero transformed term maps into it.  Every member
has a source witness, which gives
\(\widehat c\le\ell+\eta\), \(\widehat t\le t+\eta\), and
\(\log_Zq_{r_g}\le g-\theta+2\eta\).  The first of these bounds
will be needed for the child puncture; the containing norm balls
used for counting do not imply it.  The nonempty row and child-column gates stated above
remain separate from this structural projection.

Put
\[
 D_c=\ell+A_i+t+g-\theta+V.
\]
Since \(r=N-3\ell\) is an exact definition tied to the fixed parent,
substitution in Equations~\eqref{eq:canonical-child-lengths} and
\eqref{eq:second-poisson-lengths} gives the exact formal identities
\begin{equation}\label{eq:canonical-transition}
 \begin{gathered}
 F_c=F-D_c-2\ell+j,\qquad M_c=M-2D_c-2\ell+j,\\
 F_c-M_c=F-M+D_c,\\
 4F_c-3M_c=4F-3M+2(A_i+t+g-\theta+V)+j.
 \end{gathered}
\end{equation}
These identities involve the same fixed \(F=N+V\) as
Equation~\eqref{eq:inverse-fixed-invariant}, not an actual parent norm.

For every \(\gamma\in\Gamma_\sigma\), choose any source witness.
Its actual divisibility \(d_2\mid g'\) and
Equation~\eqref{eq:inverse-cube-actual} give
\[
 \widehat\theta\le\widehat g,\qquad
 \widehat j\le2\widehat\ell_{\rm pair},\qquad
 \widehat c\le\widehat\ell_{\rm pair}.
\]
Consequently the only actual-to-center inequalities needed here are
\begin{equation}\label{eq:inverse-center-inequalities}
 g-\theta\ge-2\eta,\qquad j\le2\ell+3\eta,\qquad
 \widehat c\le\ell+\eta.
\end{equation}
The first two are not asserted with zero error.  In particular the
third inequality holds for every fixed triple because of its source
witness, even though it need not hold throughout the containing count
ball.  The new radical in Equation~\eqref{eq:canonical-child-data}
has actual logarithmic norm \(Q_{\rm new}=Q_\gamma\), satisfying
\begin{equation}\label{eq:inverse-new-puncture}
 \begin{split}
 Q_{\rm new}&\le Q+\widehat c+\widehat t+\widehat g-\widehat\theta\\
 &\le Q+\ell+t+g-\theta+4\eta\le Q+D_c+4\eta.
 \end{split}
\end{equation}
This also holds when its factors share primes with the old puncture,
because taking the radical only decreases the norm.  The last inequality
uses \(A_i,V\ge0\).

Use \(F_{\rm ch}=F_c+\delta_N\), \(M_{\rm ch}=M_c+\eta\), and
\(2(A_i+t+g-\theta+V)+j\ge-4\eta\) in
Equation~\eqref{eq:canonical-transition}.  The two child margins satisfy
\begin{equation}\label{eq:inverse-child-margins}
 \begin{aligned}
 F_{\rm ch}-M_{\rm ch}-Q_{\rm new}-z_0
   &\ge c_{\rm node}+\delta_N-5\eta\ge c_{\rm node}-5\eta,\\
 4F_{\rm ch}-3M_{\rm ch}-6z_0
   &\ge c_{\rm node}+4\delta_N-7\eta\ge c_{\rm node}-7\eta.
 \end{aligned}
\end{equation}
The nominal cap \(z_0\) is unchanged because the surviving mark is a
subcollection.  Moreover \(D_c\ge\ell+V-2\eta\) and
\(2\ell-j\ge-3\eta\), whence
\begin{equation}\label{eq:inverse-row-decrease}
 M-M_{\rm ch}=2D_c+2\ell-j-\eta
       \ge2(\ell+V)-8\eta\ge2d-8\eta.
\end{equation}
For \(\eta\le d/16\) this is at least \(3d/2\), hence at least \(d\).
The same identities give \(F_c\le F-\ell-V+5\eta\), so
\(F_{\rm ch}\le F+11\eta\).  Because its two summands are nonnegative,
this bounds the child \(N_{\rm ch},V_c\) through the finite depth.

The arithmetic construction has thus produced admissible child ranges
with a smaller row length. We next express the transformed sum in their
canonical polynomials. The required Fourier density is common to all
triples \(\gamma\), rows and labels; only after that identity will we
apply Cauchy and dominate the reconstruction multiplicities.

\paragraph{A common density for the second transformed sum.}
Before any actual outer ideal is summed, use the six independent coordinates
\[
 (y_g,y_d,y_v,y_k,y_1,y_2)
 =\left(\frac{q_{g'}}{Z^g},\frac{q_{d_2}}{Z^\theta},
 \frac{q_{v'}}{Z^v},\frac{q_{k''}}{Z^{h_2}},
 \frac{q_{n_1}}{Z^{N_c}},\frac{q_{n_2}}{Z^{N_c}}\right).
\]
Use the fresh cutoffs \(\omega_{n,a},\omega_f\) already chosen.
Retain the four current dyad cutoffs
and \(\omega_f\) outside inversion, and \(\omega_{n,a}\) in the two
columns.  Let \(\Omega_{2,\alpha}\) be larger cutoffs equal to one on
the full supports of the corresponding four dyad and two column
cutoffs.  These full supports, \(I_f\), and the formal products
\(g'v'n\) on them are the windows already included in \(\mathcal I_D\).

The exact second root and kernel argument are
\begin{equation}\label{eq:inverse-second-root-kernel}
 \begin{split}
 \frac{K}{q_{d_2}q_{v'}\sqrt{q_{n_1}q_{n_2}}}
 &=Z^{\lambda_c+12\eta+\tau}y_d^{-1}y_v^{-1}(y_1y_2)^{-1/2},\\
 \frac{Kq_{k''}}{q_{d_2}q_{v'}^2q_{n_1}q_{n_2}}
 &=A_{{\rm ker},2}\frac{y_k}{y_dy_v^2y_1y_2},\\
 A_{{\rm ker},2}&=Z^{H_{\rm use}+h_2-\theta-2v-2N_c}.
 \end{split}
\end{equation}
Indeed \(N_c=s_i-g-v\) and
\(\lambda_c=H_c-\theta-(s_i-g)\).  Define the one ambient profile
\begin{equation}\label{eq:inverse-second-profile}
 \begin{split}
 \mathfrak H_{2,\sigma,\boldsymbol t}(\boldsymbol y)={}&Z^{-6\eta}
 \prod_\alpha\Omega_{2,\alpha}(y_\alpha)
 y_d^{-1}y_v^{-1}(y_1y_2)^{-1/2}\\
 &\times W_i(y_gy_vy_1)\overline{W_i(y_gy_vy_2)}
 \mathcal K_2\!\left(A_{{\rm ker},2}\frac{y_k}{y_dy_v^2y_1y_2}\right).
 \end{split}
\end{equation}
The scalar outside it is exactly \(Z^{\lambda_c+18\eta+\tau}\).
The full \(q_{d_2}\) denominator and both normalized real column roots
are in this profile, and no such root is appended to the final tests.

For \(\boldsymbol\zeta=(\zeta_g,\zeta_d,\zeta_v,\zeta_k,
\zeta_1,\zeta_2)\), put \(z_n=q_n/Z^{N_{\rm ch}}\) and define
\[
 W_a'(z)=\omega_{n,a}(Z^{\delta_N}z)z^{\varepsilon_a i\zeta_a},
 \qquad \nu_a'=\nu_i\overline\vartheta,\qquad
 \mathfrak d_a'=\mathfrak d_{I_a'}.
\]
The polynomial on either new Cauchy side is then exactly
\begin{equation}\label{eq:canonical-child-polynomial}
 \begin{split}
 Q_{a,\gamma,\boldsymbol\zeta}(k_{\rm new},f_{\rm new})={}&
 \sum_{n\ {\rm sf}}\bar\alpha(n)\gamma_2(n)\nu_a'(n)\rho'(n)
 \chi_n(k_{\rm new})\\
 &\times\chi_n(f_{\rm new})^4
 \mathfrak d_a'(n)W_a'(q_n/Z^{N_{\rm ch}}).
 \end{split}
\end{equation}
After \(q_0,t',g'/d_2\) and the finite ray sector are fixed,
\(\nu_a'\) and \(\rho'\) are independent of \(J,C,d_2,v'\) and
the new row.  The only new characters on \(n\) are the explicitly
displayed row and fourth-power factors; all other phases involving
the old \(b_i,\mathfrak A_i\) were placed in
Equation~\eqref{eq:retained-cube-label} or in the old row
\(\widetilde h\).  The separated annular weight \(W_a'\) is also
independent of the new row and label.
The complete remaining outer weight is
\begin{equation}\label{eq:inverse-second-outer-weight}
 \begin{split}
 V_\sigma(\xi;\boldsymbol\zeta)={}&w_o^+\mu(d_2)\mu(v')\widehat G(\vartheta)
 |B_o(g')|^2|D_o(v';d_2,k'')|^2
 1_{(v',g'CJ)=1}\mathcal B_2(k'')\\
 &\times\mathcal A_{2,\sigma}(\xi)
 \psi_g(y_g)\psi_{d_2}(y_d)\psi_v(y_v)\psi_{k''}(y_k)
 \omega_f(q_{f_{\rm new}}/Z^{V_c})\\
 &\times y_g^{i\zeta_g}y_d^{i\zeta_d}y_v^{i\zeta_v}y_k^{i\zeta_k}
       Z^{i\delta_N(\zeta_1+\zeta_2)}.
 \end{split}
\end{equation}
All old pure arithmetic and priority masks are retained in
\(\mathfrak O_\sigma\), with their nonnegative old coefficient weight
\(w_o^+\).  The displayed two common magnitudes retain every other
common coefficient zero, including the one depending on \(k''\).
There is one \(\mu(d_2)\), one \(\mu(v')\), and one ray coefficient.
The label cutoff is inserted as an equality on the original product
supports and remains outer.

Let \(\mathcal T_{\sigma,i}\) be the fixed \(v'\)-dyad, ray, and slot
summand obtained from Equation~\eqref{eq:inverse-second-formal-component}
by the complete M\"obius and slot expansions above.  The exact identity
needed for the child is
\begin{equation}\label{eq:inverse-xi-identity}
 \begin{split}
 \mathcal T_{\sigma,i}={}&Z^{\lambda_c+18\eta+\tau}(2\pi)^{-6}
 \int_{\mathbb R^6}\widehat{\mathfrak H}_{2,\sigma,\boldsymbol t}
       (\boldsymbol\zeta)\\
 &\quad\times\sum_{\xi\in\Xi_\sigma}V_\sigma(\xi;\boldsymbol\zeta)
 Q_{1,\gamma,\boldsymbol\zeta}(k_{\rm new},f_{\rm new})
 \overline{Q_{2,\gamma,\boldsymbol\zeta}(k_{\rm new},f_{\rm new})}
 \,d\boldsymbol\zeta.
 \end{split}
\end{equation}
To check it, expand the two \(Q\)'s.  Since
\(y_a=Z^{\delta_N}z_{n_a}\), their modes together with the last
factor of Equation~\eqref{eq:inverse-second-outer-weight} are exactly
\(y_1^{i\zeta_1}y_2^{i\zeta_2}\).  The other four modes are outer.
Equation~\eqref{eq:inverse-log-fourier} restores the full profile,
whose larger cutoffs are one on the retained supports.  The fresh
column cutoffs are one wherever the old \(W_i\) factors are nonzero,
and \(\omega_f=1\) on every original label product.  The scalar and
profile therefore restore the full root, full kernel, and old windows.
Equations~\eqref{eq:inverse-second-extraction},
\eqref{eq:inverse-moving-characters}, and
\eqref{eq:inverse-slot-priority} restore the arithmetic factors term
by term.  The fixed-box sums are finite and
Equation~\eqref{eq:inverse-fourier-moment} justifies the integral.
Extra combinations in the full fresh windows cancel through the old
windows in this complex Fourier identity before Cauchy; only later are
they kept independently in a positive sum.

The density in Equation~\eqref{eq:inverse-xi-identity} depends on the
fixed sector, \(Z\), the fixed tests and \(\boldsymbol t\), not on
actual \(\gamma,k_{\rm new},f_{\rm new}\).  The coupled kernel uses
the bare \(k''\) coordinate, not the derived row; the latter appears
only in characters and \(\mathcal B_2\).  The derived label appears
only in characters and its outer cutoff.  Relations such as
\(g'=d_2r_g\) restrict evaluation points, not the transform.
For every fixed \(B_{\rm ht},J_{\rm ht}\ge0\), the Euler calculation
in Equation~\eqref{eq:inverse-fourier-moment} gives some fixed \(b\) with
\begin{equation}\label{eq:inverse-joint-fourier-moments}
 \int_{\mathbb R^9}\int_{\mathbb R^6}
 |\widehat{\mathfrak H}_{1,\sigma}(\boldsymbol t)|
 |\widehat{\mathfrak H}_{2,\sigma,\boldsymbol t}(\boldsymbol\zeta)|
 (1+|\boldsymbol t|)^{B_{\rm ht}}(1+|\boldsymbol\zeta|)^{J_{\rm ht}}
 \,d\boldsymbol\zeta\,d\boldsymbol t\ll p_b(W)^2.
\end{equation}
Indeed the inner weighted integral is polynomial in \(\boldsymbol t\),
because \(W_i\) is a fixed cutoff times a unit norm mode; apply the
first transform's bound at that polynomial order plus \(B_{\rm ht}\).
The constants are uniform in every actual ideal and every positive
kernel scalar, with no height-order power of \(Z\).

\paragraph{The second Cauchy inequality and the outer counts.}
Apply Equation~\eqref{eq:weighted-cauchy} to the second transformed
sum with
\(U_a=Z^{(\lambda_c+18\eta+\tau)/2}Q_{a,\gamma,\boldsymbol\zeta}\)
on the complete set \(\Xi_\sigma\), for each fixed Fourier mode.
In particular \(\mathcal B_2\), \(|B_o|^2\), \(|D_o|^2\), and
\(1_{(v',g'CJ)=1}\) are still in its weight. They retain the
squarefreeness of \(f_{\rm new}\) proved above. All other factors in
the weight are bounded by a fixed constant,
including the bounded number of original assigned slot coefficients,
the fixed ray coefficient, the individual cutoffs, and the unit modes.
Consequently, only after Cauchy, the absolute weights obey
\[
 |V_\sigma(\xi;\boldsymbol\zeta)|
 \ll \mathcal B_2(k'')\mu^2(f_{\rm new}).
\]
This is the step which permits the new averaged ideal to remain in
the squarefree class.

We first record the exact reconstruction needed to dominate a fibre.
\begin{equation}\label{eq:poisson-divisor-reconstruction}
 d_k\mid C\mathfrak B/R_1
 \quad\Longrightarrow\quad
 d_k\mid f_{\rm new}\operatorname{rad}q_0.
\end{equation}
To prove this, a prime of \(\mathfrak B/R_1\) has \(t_p=0\).
The local table shows that it has even parity and equal \(a_{ip}\).
If both are one it is in \(J_2\), and if both are zero it is in
\(\operatorname{rad}q_0\).  Primes of \(C\) are already in
\(f_{\rm new}\).  This proves the implication.  Thus \(d_k\) also
has divisor multiplicity after \(f_{\rm new},q_0\) are specified.
Once these factors are fixed, \(k''\) is uniquely determined by
the element \(k_{\rm new}=d_kd_2k''\).

Let \(K_{\rm slot}\) bound the original number of slots.  At fixed
\(\gamma=(q_0,t',r_g)\), squarefree \(f=f_{\rm new}\), and element
\(k=k_{\rm new}\), the number of possible \(\xi\in\Xi_\sigma\) with
nonzero weight is at most a fixed constant times
\(D_{K_{\rm slot}}(f)C_{K_{\rm slot}}(\gamma)\), where
\begin{equation}\label{eq:inverse-new-fibre}
 \begin{split}
 D_{K_{\rm slot}}(f)&=d_{\mathcal O}(f)^{9+4K_{\rm slot}},\\
 C_{K_{\rm slot}}(\gamma)&=
 d_{\mathcal O}(q_0)^{5+2K_{\rm slot}}
 d_{\mathcal O}(t')^{2K_{\rm slot}}
 d_{\mathcal O}(r_g)^{2K_{\rm slot}},
 \end{split}
\end{equation}
Here and below \(d_{\mathcal O}\) denotes the ideal divisor count.
For completeness, the ordered allocation of \(f\) into
\(J,C,d_2,v'\) costs \(4^{\omega(f)}=d_{\mathcal O}(f)^2\), and
splitting \(J=sJ_2\) costs at most \(d_{\mathcal O}(f)\).
The choices of \(b_1,b_2\) with
\(b_1b_2=q_0^2J_2^2s\) cost at most
\(d_{\mathcal O}(q_0)^2d_{\mathcal O}(f)^3\); the two choices
\(\mathfrak A_a\mid\operatorname{rad}(b_1b_2)\) cost at most
\(d_{\mathcal O}(q_0)^2d_{\mathcal O}(f)^2\).  These estimates use
\(d_{\mathcal O}(IJ)\le d_{\mathcal O}(I)d_{\mathcal O}(J)\) and
\(d_{\mathcal O}(I^2)\le d_{\mathcal O}(I)^2\).
Equation~\eqref{eq:poisson-divisor-reconstruction} costs at most
\(d_{\mathcal O}(q_0)d_{\mathcal O}(f)\) choices for \(d_k\).
Then \(g'=d_2r_g\) is fixed, and \(k''=k/(d_kd_2)\) is the unique
element if it is integral.  The at most \(2K_{\rm slot}\) old assigned
primes divide \(q_0ft'\), so their choices cost at most
\([d_{\mathcal O}(q_0)d_{\mathcal O}(f)d_{\mathcal O}(t')]^{2K_{\rm slot}}\).
The at most \(2K_{\rm slot}\) new assigned primes divide \(fr_g\),
and cost at most
\([d_{\mathcal O}(f)d_{\mathcal O}(r_g)]^{2K_{\rm slot}}\).
The other first side's surviving slots disappeared on selecting the
first positive side; the selected surviving slots occur twice in its
new square, exactly as counted here.  Thus the displayed product
dominates the fibre even if its actual size depends on \(k\).
There is no count of the old \(f,h\), no independent count of
\(J,C,d_2,v'\), and no second frequency count.

Equation~\eqref{eq:fixed-cube-count} and the same source witnesses
place \(\Gamma_\sigma\) inside the product of the three norm balls
\[
 q_{q_0}\le Z^{\ell+R/2-j+5\eta/2},\qquad
 q_{t'}\le Z^{t+\eta},\qquad q_{r_g}\le Z^{g-\theta+2\eta}.
\]
We use these balls only to bound the number of distinct triples, not
as a replacement domain for the child estimate.  Ideal counting gives
\begin{equation}\label{eq:canonical-fixed-count}
 \#\Gamma_\sigma\le C Z^{C_c+11\eta/2+\pi_{\rm count}},\qquad
 C_c=\ell+R/2-j+t+g-\theta.
\end{equation}
Here \(\pi_{\rm count}\) is part of the separately chosen aggregate
\(\pi\).  In a nonempty sector each upper exponent of these three
balls is nonnegative, since it bounds the norm of an existing ideal;
the constant term in ideal counting therefore adds no boundary power.
In particular \(J\) is not counted here.  The ranges of all three
ideals are bounded, so the adjustable divisor bound gives
\(C_{K_{\rm slot}}(\gamma)\ll Z^{\pi_{\rm fib}}\) uniformly on
the containing balls.  Put \(c_\sigma=C_c+11\eta/2\) and introduce
the genuinely row- and label-independent measure
\begin{equation}\label{eq:inverse-normalized-triples}
 d\nu_\sigma(\gamma)=Z^{-c_\sigma}
 \sum_{\gamma\in\Gamma_\sigma}C_{K_{\rm slot}}(\gamma)\delta_\gamma,
 \qquad \|\nu_\sigma\|\ll Z^{\pi_{\rm count}+\pi_{\rm fib}}.
\end{equation}

\paragraph{The positive canonical children.}
For a fixed Fourier mode define the positive child sums
\begin{equation}\label{eq:inverse-positive-child}
 \begin{split}
 \mathcal H_{a,\gamma}={}&
 \sum_{\substack{f\ {\rm sf}\\q_f/Z^{V_c}\in I_f}}
 D_{K_{\rm slot}}(f)
 \sum_{\substack{k\in\mathcal O\\0<q_k\le Z^{M_{\rm ch}}}}
 |Q_{a,\gamma,\boldsymbol\zeta}(k,f)|^2
 =Z^{F_{\rm ch}}\mathcal E_{a,\gamma}.
 \end{split}
\end{equation}
The equality is the normalization of Equation~\eqref{eq:canonical-energy}
with this same row ball, the label weight \(D_{K_{\rm slot}}\), and
the test and puncture in Equations~\eqref{eq:canonical-child-polynomial}
and \eqref{eq:canonical-child-data}; if a larger fixed row ball is
used there, this equality is instead an inequality in the needed
upper-bound direction.  The weight depends on \(f\) alone.
After reindexing, \(\mathcal B_2\) is exactly the displayed row ball.
Only the now positive outer row and label sums have been enlarged to
their full fixed windows.  The columns use their full fixed fresh
windows, but their puncture, mark, and test have not been deleted or
changed.  The actual fibre domination just proved gives
\[
 \begin{split}
 \left|\sum_{\xi\in\Xi_\sigma}V_\sigma Q_1\overline{Q_2}\right|
 &\le\prod_{a=1}^2
       \left(\sum_{\xi\in\Xi_\sigma}|V_\sigma||Q_a|^2\right)^{1/2}\\
 &\ll Z^{c_\sigma}\prod_{a=1}^2
       \left(\int_{\Gamma_\sigma}\mathcal H_{a,\gamma}
                    \,d\nu_\sigma(\gamma)\right)^{1/2}.
 \end{split}
\]
Here \(Q_a=Q_{a,\gamma,\boldsymbol\zeta}(k_{\rm new},f_{\rm new})\).
The two square roots each supply \(Z^{c_\sigma/2}\), hence there is
one fixed-count factor.  A uniform child bound supplies one total
mass \(\|\nu_\sigma\|\), not its square.  The child bound will be
used only for \(\gamma\in\Gamma_\sigma\), using the hypotheses already verified in
Equation~\eqref{eq:inverse-child-margins}.

\paragraph{The energy exponent.}
The formal centers satisfy the exact identity
\begin{equation}\label{eq:canonical-prefactor}
 \kappa_i+\lambda_c+C_c+2F_c=F.
\end{equation}
This follows by substituting their definitions and
\(H_c=2r-2B-M+4\ell+2V+\delta-j\); all extracted lengths cancel and the
result is \(r+3\ell+V=N+V\).

We now use the child estimate only for
\(\gamma\in\Gamma_\sigma\), for which the preceding margins and row
decrease have been proved.  If it gives
\(\mathcal E_{a,\gamma}\ll C(\boldsymbol t,\boldsymbol\zeta)
Z^{F_{\rm ch}+\epsilon_{\rm child}}\) uniformly there, with its
fixed polynomial height dependence, then
Equation~\eqref{eq:inverse-positive-child} and the final weighted
Cauchy bound above give
\[
 \left|\sum_{\xi\in\Xi_\sigma}V_\sigma Q_1\overline{Q_2}\right|
 \ll C(\boldsymbol t,\boldsymbol\zeta)
 Z^{c_\sigma+2F_{\rm ch}+\epsilon_{\rm child}}\|\nu_\sigma\|.
\]
The normalized mass is the one in
Equation~\eqref{eq:inverse-normalized-triples}; witness multiplicity
is already in its fibre weights.  The common Fourier density is then
integrated once using Equation~\eqref{eq:inverse-joint-fourier-moments}.
It is not part of the discrete measure and is not chosen afresh for
any row or label.  Thus both the genuine outer count and its normalized
mass occur once.  The averaged \(J,C,d_2,v'\) stay inside the positive
child sum and are not also counted.

If a child is bounded by \(Z^{F_{\rm ch}+\epsilon_{\rm child}}\), the
first prefactor error, the second prefactor error, the fixed count, and
the restored normalization give on each positive second-Cauchy side
\begin{equation}\label{eq:inverse-step-loss}
 \begin{split}
 &\kappa_i+\lambda_c+C_c+2F_{\rm ch}
      +(\tfrac92+18+\tfrac{11}2)\eta+\tau+\pi+\epsilon_{\rm child}\\
 &\qquad=F+28\eta+2\delta_N+\tau+\pi+\epsilon_{\rm child}\\
 &\qquad\le F+40\eta+\tau+\pi+\epsilon_{\rm child}.
 \end{split}
\end{equation}
Both copies of \(\delta_N\) are present: one restores the child column
normalization and the other occurs in its asserted exponent.  Each of
the two weighted Cauchy inequalities takes a geometric mean of its
positive sides, so it does not double this loss.  The first principal
bound is \(Z^{M+\pi}\), and the second principal bound above is at most
\(Z^{F+26\eta+\tau+\pi}\); both fit the same allowance.

The parameter \(\pi\) has an independent quantifier.  Let \(J_{\rm step}\)
be the maximum number of uses of a free divisor or sieve exponent and
of logarithmic dyadic-count factors in one fixed two-Poisson, two-Cauchy
factorization tree, maximized over the finitely many assignments with at
most the fixed slot bound.  Once \(L_{\rm pow}\) bounds the log-length of
every scale product in such a use, choose each free input exponent at most
\(\pi/(2J_{\rm step}\max(1,L_{\rm pow}))\).  For sufficiently large \(Z\),
the product of the at most \(J_{\rm step}\) logarithmic factors is at most
\(Z^{\pi/2}\).  These choices make all non-center, non-frequency local
losses at most \(CZ^\pi\).  They include the retained common frequency
dyads and the reconstruction divisor bounds.  In particular the distinct
allocations \(\pi_\beta,\pi_{\rm old},\pi_{\rm count},\pi_{\rm fib}\)
and the finite-union dyadic allocation are parts of this single
aggregate, not repeated allowances of size \(\pi\).  Fixed ray sums and fixed
seminorm constants are constants.  The same convention defines the
aggregate terminal loss \(\pi_{\rm ref}\).  In particular \(\pi\) hides
no multiple of \(\eta\) or \(\tau\).

This proves the conditional reduction estimate
\eqref{eq:canonical-reduction-contract}, including its common-measure and
finite-order uniformity once the choices below are made. It remains to fix
those choices uniformly and apply the reduction through the finite depth.

\paragraph{Order of choices and termination.}
For the requested canonical exponent \(\epsilon_c=\epsilon\), first fix
the starting ranges, tests, slot bound, \(c_*\), and then \(d,D\) as at
the beginning of the proof.  Choose
\begin{equation}\label{eq:inverse-parameter-order}
 \begin{gathered}
 \eta\le\min\{d/16,c_*/(14D),\epsilon_c/(160D),c_*/1000,1/100\},\\
 \tau\le\min\{\epsilon_c/(4D),c_*/1000,1/100\},\qquad
 \pi\le\epsilon_c/(4D),\qquad
 \tau_{\rm ref},\pi_{\rm ref}\le c_*/1000.
 \end{gathered}
\end{equation}
Use this same \(\tau\) in both Poisson comparisons.  At this point \(\pi\)
is a target aggregate loss; its subsidiary input exponents are chosen
only after the bounded scale range is known.

Choose the individual cutoffs through depth \(D\), including every full
larger or fresh support and the bounded clipping families described
above, and form \(\mathcal I_D,L_{\rm win}\).  Include the product
intervals for the active slots and the terminal constants
\(C_{\rm res},C_{\rm width},C_{\rm ker}\), maximized through the depth.
This is finite because there are finitely many factorization types per
passage and finitely many passages.  Put
\[
 F_{\max}=N_{\max}+V_{\max}+11D\eta,\qquad
 L_{\rm all}=100(1+M_{\max}+F_{\max}+z_{0,\max}+c_*).
\]
These bound all retained scale lengths.  Indeed,
Equation~\eqref{eq:inverse-row-decrease} and the bound following it give
\(M\le M_{\max}\) and \(N,V\le F_{\max}\).  A nonempty column window has
\(r\ge-\eta\), so \(\ell\le(F_{\max}+\eta)/3\).  The displayed parent
products bound every extracted divisor length by \(2F_{\max}+6\eta\).
For example \(d_k\mid C\operatorname{rad}(b_1b_2)\) gives
\(\delta\le r+2\ell+4\eta\), and Equation~\eqref{eq:enlarged-first-frequency}
then gives \(H_{\rm use}\le7F_{\max}+18\eta+\tau\).  The terminal active
conductor is supported on the row, \(f\), the puncture, and slots; their
total lengths are at most \(M_{\max}+2F_{\max}+z_{0,\max}\) plus the
already displayed ratio allowances.  Equations~\eqref{eq:inverse-terminal-width}
and \eqref{old-eq:4.4} bound the retained terminal dual lengths by
\(4M_{\max}+4F_{\max}+2z_{0,\max}+3d+12\eta+\tau_{\rm ref}\).
Under Equation~\eqref{eq:inverse-parameter-order}, all these bounds and
the exact conductor product lengths are below \(L_{\rm all}\).
We may now take \(L_{\rm pow}=L_{\rm all}\) and choose the subsidiary
small exponents that realize \(\pi\) and \(\pi_{\rm ref}\).

Define \(B_{\rm crude}\) to be the maximum exponent obtained by replacing
every bounded nonfrequency ideal or element sum in either raw Poisson
expansion, including both column sums, by its lattice count at length
\(L_{\rm all}\), and every explicit norm factor, including \(1+a^{-1}\)
in Equation~\eqref{eq:inverse-raw-tail}, by its absolute upper bound.
Replace divisor-bounded coefficients and marks by their trivial polynomial
norm bounds as well.
Exclude only the frequency kernel itself.  The expansions contain a
fixed finite number of factors, so this defines a finite uniform number.

For the reflected tail define \(B_{\rm ref}\) analogously using its bounded
row, local, and active-product counts and \(1+a^{-1}\) with \(a=X/q_c^2\).
Do not count discarded dual indices at a retained length.  Instead,
Equation~\eqref{old-eq:4.3} gives on its support
\[
 \frac{|d(\mu)|}{\sqrt{q_\mu}}
 \le27q_\lambda^{-k/3}q_n^{-1/2}q_b^{-1}
 \le27q_\lambda^{4/3}\qquad(k\ge-4).
\]
Every other local factor is bounded by the product of \(q_p^{1/2}\) over
the bounded active primes, independently of \(\mu\) after its indicators
are dropped.  Thus \(B_{\rm ref}\) covers all non-kernel coefficients and
bounded counts, while the remaining \(\mu\)-sum is an unrestricted
lattice sum on \(\lambda^{-4}\mathcal O\).

Fix a tail saving \(T>1+B_{\rm crude}+B_{\rm ref}\).  For each Poisson
tail, choose a Schwartz order
\[
 A>1+(B_{\rm crude}+T)/\tau.
\]
Equation~\eqref{eq:inverse-raw-tail} with \(Y=Z^\tau\) then makes the
discarded raw outer-ball complement \(O(Z^{-T})\), up to finitely many
input seminorms and a fixed polynomial in any twist heights.  For the
reflected whole-dyad tail the actual argument is at least
\(Z^{\tau_{\rm ref}/2}\); use the same lattice-shell bound on
\(\lambda^{-4}\mathcal O\) and choose
\[
 A_{\rm ref}>1+2(B_{\rm ref}+T)/\tau_{\rm ref}.
\]
This proves the discarded-tail assertion without presuming any bound on
the discarded dual lengths.

Next choose the finitely many Fourier-height and smooth-seminorm orders
backwards through depth \(D\), above the chosen tail orders and the
terminal input orders.  Normalized inverse roots are fixed real powers
on annuli, and Euler differentiation of a full kernel at a norm monomial
introduces no scale power.  Equations~\eqref{eq:inverse-joint-fourier-moments}
and \eqref{eq:inverse-normalized-triples} supply the common coefficient
measure for each positive indexed Cauchy-side sum.  Apply
Corollary~\ref{cor:indexed-seminorm-propagation} separately to those sums,
using the child-profile bounds just proved from
Lemma~\ref{lem:smooth-calculus}, and then take the displayed geometric
means.  The two square roots retain one normalized discrete mass, as
already shown, so genuine outer labels are counted only once.
Any external height cutoff in a later application
is chosen after this internal finite propagation, not inserted into it.

Finally choose \(Z\) large enough for \(\log Z\ge L_{\rm win}/\eta\),
the thresholds for \(C_{\rm res},C_{\rm width}\) in
Equation~\eqref{eq:inverse-terminal-width},
\(\log Z\ge2\log C_{\rm ker}/\tau_{\rm ref}\),
\(\log Z\ge4\log q_\lambda/\eta\), and the fixed \(h\)-annulus threshold
used in the short completion.  Impose also the logarithmic bounds defining
the aggregate local losses and the thresholds of the input lemmas.
All choices precede this final threshold and depend only on the fixed
data.  Bounded smaller \(Z\) are handled by increasing the final constant.

Now \(c_h=c_*-7h\eta\ge c_*/2\) for \(h\le D\).
Equations~\eqref{eq:inverse-child-margins} and
\eqref{eq:inverse-row-decrease} send every retained nonterminal child to
the next depth and its row cap.  At depth \(D\) that cap is negative,
whereas every retained row parameter is nonnegative.  The terminal bounds
are at most \(Z^F\); principals and recursive terms cost at most
\(L_{\rm step}=40\eta+\tau+\pi\) in exponent.  Backwards induction gives
\(Z^{F+(D-h)L_{\rm step}}\) at depth \(h\), and
\[
 D L_{\rm step}\le\epsilon_c/4+\epsilon_c/4+\epsilon_c/4<\epsilon_c.
\]
This proves \(\mathcal E\ll Z^{F_0+\epsilon}\), with the claimed
uniform finite-seminorm and polynomial-height dependence.
\end{proof}

\subsection{Initialization of the marked moment}

We now convert the inverse polynomial in Lemma~\ref{lem:marked} to
the canonical family.  Only one Poisson transformation is required.

\begin{proof}[Proof of Lemma~\ref{lem:marked}]
In the product \(M_u(Z^r;W)Q_u\), let \(P\) be the product of its
slot primes and put \(j=(n,P)\).  Both \(n\) and \(P\) are
squarefree.  Fix the subset of slots occurring in \(j\), and write
\[
 n=jn_0,\qquad P=jP_0,\qquad (n_0,P_0)=1.
\]
The product \(c=n_0P_0\) is squarefree.  The identities
\[
 \mu(n_0)=\mu(c)\mu(P_0),\qquad
 \psi_u(n)\psi_u(P)=\psi_u(j)^2\psi_u(c)
\]
hold with all zero extensions.  The sign \(\mu(P_0)\) is a product
of signs on the surviving slots and can be incorporated into their
bounded coefficients.  The coefficients thus remain of the form
in Equation~\eqref{old-eq:4.2}.

For this fixed assigned subset define \(G\) from its nominal slot centers:
\[
 G=\sum_{i\ {\rm assigned}}z_i,\qquad
 z_0=z-G,\qquad D'=r+z-2G.
\]
Then \(G,z_0\ge0\), and \(z_0\) is exactly the surviving nominal cap.
The normalization satisfies
\[
 Z^{-(r+z)/2}=Z^{-G}Z^{-D'/2}.
\]
The actual \(j\) is the product of the assigned primes, so
\(q_j/Z^G\) lies in a fixed product interval.  Ideal counting gives at
most \(CZ^G\) choices, apart from the separately chosen divisor loss
for assignments.  Triangle inequality in the row Hilbert space uses
\(Z^{-G}\) to cancel this count.  The whole factor \(\psi_u(j)^2\) is a
bounded row scalar, including its zeros, and is a contraction in that
space.  The residual column is prime to \(j\), giving the fixed
puncture \(1_{(c,j)=1}\).

Here is the exact identity underlying this overlap reduction.  Let
\(I_0\) be the surviving subset and put
\(y_j=q_j/Z^G\), \(y_c=q_c/Z^{D'}\), and
\(y_i=q_{p_i}/Z^{z_i}\) for every slot.  On the original support,
\[
 \frac{q_n}{Z^r}=\frac{y_jy_c}{\prod_{i\in I_0}y_i}.
\]
For a fixed assigned tuple, its contribution \(\mathcal U_{u,j}\) to
\(M_u(Z^r;W)Q_u\) is exactly
\begin{equation}\label{eq:initial-overlap-polynomial}
 \begin{split}
 \mathcal U_{u,j}={}&\mu(j)
 \prod_{i\notin I_0}a_i(p_i)W_i(y_i)\,
 Z^{-G}\psi_u(j)^2 Z^{-D'/2}
 \sum_{\substack{c\ {\rm sf}\\(c,j)=1}}\mu(c)\psi_u(c)\\
 &\quad\times
 \sum_{\substack{(p_i)\in\prod_{i\in I_0}\mathcal P_i\\P_0\mid c}}
 \mu(P_0)\prod_{i\in I_0}a_i(p_i)W_i(y_i)
 W\!\left(\frac{y_jy_c}{\prod_{i\in I_0}y_i}\right).
 \end{split}
\end{equation}
Indeed a squarefree \(c\) prime to \(j\), together with a surviving
tuple with \(P_0\mid c\), reconstructs uniquely
\(n=j(c/P_0)\) and the prescribed overlap.  Squarefreeness of \(c\)
already gives \((c/P_0,P_0)=1\), so no further condition was lost.
The sign \(\mu(P_0)\) is a product of the individual prime signs.
This also shows that the \(j\) on both sides of every subsequent
square is one fixed ideal: the preceding triangle inequality was
taken before that square.

The identity \(c=nP/j^2\) puts \(y_c\) in a fixed compact interval.
This remains meaningful if \(D'\) is slightly negative in a bounded
nonempty window; \(D'\) is not yet a canonical parameter.  Choose a
fresh individual cutoff \(\omega_c\) equal to one on this full quotient
support.  For fixed \(j\), let \(\Omega_c\) be one on the full support
of \(\omega_c\), and let \(\Omega_i\) be one on each full surviving
individual slot support.  The single joint profile
\[
 \mathfrak H_{{\rm ov},j}(y_c,(y_i)_{i\in I_0})
 =\Omega_c(y_c)\prod_{i\in I_0}\Omega_i(y_i)
 W\!\left(\frac{y_jy_c}{\prod_{i\in I_0}y_i}\right)
\]
is compactly supported in these independent coordinates.  The current
cutoff \(\omega_c\) and the individual factors \(W_i(y_i)\) stay outside
its transform.  Equation~\eqref{eq:inverse-log-fourier}, in dimension
\(1+|I_0|\), therefore expresses Equation~\eqref{eq:initial-overlap-polynomial}
exactly as the same assigned scalar and \(Z^{-G}\psi_u(j)^2\), times
\[
 (2\pi)^{-1-|I_0|}\int_{\mathbb R^{1+|I_0|}}
 \widehat{\mathfrak H}_{{\rm ov},j}(\boldsymbol\upsilon)
 Z^{-D'/2}\sum_{c\ {\rm sf}}\mu(c)\psi_u(c)1_{(c,j)=1}
 \mathfrak d_{\boldsymbol\upsilon}(c)
 \omega_c(y_c)y_c^{i\upsilon_c}\,d\boldsymbol\upsilon,
\]
where \(\mathfrak d_{\boldsymbol\upsilon}\) has the original surviving
lists and individual coefficients
\(\mu(p)a_i(p)W_i(y_i)y_i^{i\upsilon_i}\).  Expanding the mark and
inverting the joint transform proves this equality term by term.
No measure is chosen for an actual surviving tuple.  The measure may
depend on the already fixed \(j\), but Equation~\eqref{eq:inverse-fourier-moment}
is uniform for its \(y_j\) in the fixed product interval.  Minkowski's
inequality is used in the whole row Hilbert space for this one measure.
Include the full support of \(\omega_c\), its larger cutoffs, the
surviving slot supports, and the \(j\)-product interval in the finite
collection \(\mathcal I_D\) before the final threshold is chosen.
With the same \(\eta\) convention as the canonical proof, we then have
\[
 |\widehat c-D'|\le\eta,\qquad |\widehat j-G|\le\eta
\]
on the full supports actually used below, where \(\widehat c=\log_Zq_c\).
For the sign
\(\varepsilon_\chi=1\), conjugate the whole residual polynomial,
including its finite character, mark, and test; its row norm is unchanged.
For the other sign leave it unchanged.  It therefore suffices to estimate
one polynomial of the exact form
\begin{equation}\label{eq:initial-residual-polynomial}
 \mathcal R_0(u)=Z^{-D'/2}\sum_{c\ {\rm sf}}
 \mu(c)\nu_0(c)1_{(c,j)=1}\overline{\chi_c(u)}
 \mathfrak d_0(c)W_0(q_c/Z^{D'}).
\end{equation}
Here \(\nu_0\) is a fixed finite ray character, \(\mathfrak d_0\)
is the product mark of cap \(z_0\) with those individual coefficients
or their conjugates, and \(W_0\) is \(\omega_c(y)y^{i\upsilon_c}\)
or its conjugate.  All are independent of the row.

Majorize the original row ball by a fixed nonnegative radial Schwartz
function \(\Phi_{\rm init}\) at scale \(Z^m\), and expand its square.  Let \(C\) be the gcd
of the two columns and write \(c_i=Cz_i\), with \(z_1,z_2\) squarefree
and coprime.  Put \(\widehat c_i=\log_Zq_{c_i}\), so the preceding
support bound applies to each \(i=1,2\).  Let \(B\ge0\) be the center of \(C\), and write
\(\widehat B=\log_Zq_C\).  The row character has primitive modulus
\(z_1z_2\) and remaining zero mask \(C\).  Apply
Lemma~\ref{lem:masked-poisson}, with \(d'\mid C\).  Let \(\theta\ge0\)
be its center and \(\widehat\theta=\log_Zq_{d'}\).  Set
\[
 P_1=B-\theta\ge-2\eta,
\]
where the inequality follows from \(\widehat\theta\le\widehat B\) and
the two \(\eta\)-ratio bounds.  The actual conductor length is
\[
 \widehat L_{\rm init}=\widehat c_1+\widehat c_2-2\widehat B,
 \qquad |\widehat L_{\rm init}-2(D'-B)|\le4\eta.
\]
For a positive tolerance \(\tau_{\rm init}\), the implication
\(Z^mq_{h'}/(q_{d'}q_{z_1z_2})\le Z^{\tau_{\rm init}}\) gives
\[
 \widehat{d'h'}\le2D'-m-2P_1+6\eta+\tau_{\rm init}.
\]
Define the fixed formal and enclosing row scales
\begin{equation}\label{eq:inverse-initial-row}
 M_c^{\rm init}=2D'-m-2P_1,\qquad
 M_{\rm init}=M_c^{\rm init}+6\eta+\tau_{\rm init}.
\end{equation}
The \(6\eta\) is \(4\eta\) from the conductor and \(2\eta\) from the
two copies of \(d'\) in the new row bound.

Separate the original principal contribution for the direct count below.
On the genuine nonprincipal coprime Poisson expression, before any
off-coprime extension or Fourier absolutization, keep only the outer mask
\[
 \mathcal B_{\rm init}(h')=1_{0<q_{d'h'}\le Z^{M_{\rm init}}}.
\]
Its complement is contained in the actual ratio tail above for every
supported pair.  Equation~\eqref{eq:inverse-raw-tail}, with the separate
initial raw count and order specified below, discards precisely that
complement.  Retain the full smooth kernel inside the mask, with no
column-dependent ratio selector.  The mask depends on \(d',h'\), the
fixed overlap and sector, but not on the residual columns.  If
\(M_{\rm init}<0\), its nonzero ball is empty and the same tail comparison
discards every nonzero frequency; otherwise the retained row parameter is
nonnegative.

The actual normalization and Poisson prefactor split over the two sides as
\[
 Z^{-D'}Z^{m-\widehat\theta-\widehat L_{\rm init}/2}
   =Z^{\widehat\kappa_1^{\rm init}/2}
      Z^{\widehat\kappa_2^{\rm init}/2},
 \quad
 \widehat\kappa_i^{\rm init}=m-D'-\widehat\theta-\widehat c_i+\widehat B.
\]
Thus the actual side exponents obey
\begin{equation}\label{eq:initial-poisson-prefactor}
 \widehat\kappa_i^{\rm init}\le\kappa_c^{\rm init}+3\eta,
 \qquad \kappa_c^{\rm init}=m-2D'+P_1.
\end{equation}
All normalized real inverse roots will be kept in the single joint
profile below, not also in the final child tests.
Principal columns have the direct diagonal bound \(O(Z^{m+\epsilon})\),
also for a bounded nonempty negative \(D'\)-window.  Any principal nonzero
frequencies restored to the formal formula carry the same
\(\mathcal B_{\rm init}\) mask and are bounded by the full principal
restoration in Lemma~\ref{lem:masked-poisson}, since \(Z^m\ge1\).

Insert frequency dyads of \(q_{h'}\) from one partition common to all
\(d'\), not a label-recentered partition.  The outer mask implies
\(q_{h'}\le Z^{M_{\rm init}}\), so there are only logarithmically many
dyads in the bounded retained range.  Let \(H\) be the center of one
such bare-frequency dyad, and let \(\psi_C,\psi_{d'},\psi_{h'}\)
be the retained individual cutoffs on the current \(C,d',h'\) dyads,
evaluated at their normalized norms.

The orientation in Equation~\eqref{eq:initial-residual-polynomial}
is now fixed.  The calculation in the second canonical Poisson
transformation replaces \(\mu(z)\gamma_{-1}(z)\) by
\(\bar\alpha(z)\gamma_2(z)\chi_z(-1)\overline{G(z)}\).
On the second side the factor inside conjugation has ray factor
\(\overline{G(z)}\); together with the CRT cross phase the relative
ray factor is \(G([z_2][z_1]^{-1})\).  Expand it by the fixed group
formula above, and in a fixed ray summand put
\[
 \nu_*=\nu_0\overline\vartheta,\qquad
 A_*(z)=\bar\alpha(z)\gamma_2(z)\nu_*(z),\qquad
 \mathcal K_{\rm init}=\mathcal F\Phi_{\rm init}.
\]
For a preliminary pattern \(\Sigma_0\) fixing the overlap, these
dyads and the ray summand, and with the earlier mode held fixed, the masked
principal-restored nonzero component is exactly
\begin{equation}\label{eq:initial-formal-component}
 \begin{split}
 \mathcal T^{\rm init}_{\Sigma_0}={}&
 \sum_{\substack{C\ {\rm sf},\ d'\mid C\\h'\ne0}}
 \psi_C\psi_{d'}\psi_{h'}\mu(d')\widehat G(\vartheta)
 \mathcal B_{\rm init}(h')1_{(C,j)=1}\\
 &\times\sum_{\substack{z_1,z_2\ {\rm sf}\\(z_1z_2,Cj)=1}}
 1_{(z_1,z_2)=1}
 A_*(z_1)\overline{A_*(z_2)}
 \overline{\chi_{z_1}(d')}\chi_{z_1}(h')
 \chi_{z_2}(d')\overline{\chi_{z_2}(h')}\\
 &\times\mathfrak d_0(Cz_1)\overline{\mathfrak d_0(Cz_2)}
 W_0(q_Cq_{z_1}/Z^{D'})\overline{W_0(q_Cq_{z_2}/Z^{D'})}\\
 &\times\frac{Z^{-D'}Z^m}{q_{d'}\sqrt{q_{z_1}q_{z_2}}}
 \mathcal K_{\rm init}\!\left(
       \frac{Z^mq_{h'}}{q_{d'}q_{z_1}q_{z_2}}\right).
 \end{split}
\end{equation}
Indeed \(|\mu(C)\nu_0(C)|^2=1\) for the squarefree \(C\) outside
\(S\), the common row factor is exactly \(1_{(u,C)=1}\), and its
Poisson expansion contributes the one \(\mu(d')\).  The displayed
two numerator factors are those of the primitive character
\(\overline{\chi_{z_1}}\chi_{z_2}\) and its Fourier transform.
The finite ray sum restores the principal pair by \(G(1)=1\).
The cost of that restoration and the raw complement removed before
this equality were bounded separately above.

Define the summand in Equation~\eqref{eq:initial-formal-component}
on all individually squarefree \(z_1,z_2\) prime to \(Cj\) by the
displayed separate \(A_*\) factors, characters, formal product norms,
full kernel, and unchanged outer ball.  It agrees on the coprime
domain.  Insert the complete identity
\[
 1_{(z_1,z_2)=1}=\sum_{s\mid(z_1,z_2)}\mu(s),\qquad z_a=sn_a,
\]
before any factorwise bound.  Let \(v\ge0\) be the center of a
squarefree \(s\)-dyad, \(\widehat v=\log_Zq_s\), and \(\psi_s\)
its retained individual cutoff.  The \(n_a\) are squarefree and
prime to \(s\), but need not be mutually coprime.  This formal
extension does not apply Poisson summation to a noncoprime conductor.
For squarefree coprime \(s,n\), CRT and the zero-preserving identity
\(\overline{\chi_n(d')}=\chi_n(d')\chi_n(d')^4\) give exactly
\begin{equation}\label{eq:initial-common-extraction}
 \begin{split}
 A_*(sn)\overline{\chi_{sn}(d')}\chi_{sn}(h')
 ={}&A_*(s)\overline{\chi_s(d')}\chi_s(h')\\
 &\times A_*(n)\chi_n(d'h')\chi_n(d's)^4.
 \end{split}
\end{equation}
The fourth power includes the CRT factor \(\chi_n(s)^4\).
On an overlap it defines the residual coefficient to be zero, without
evaluating \(\gamma_2\) on a nonsquarefree ideal.  Since \(|A_*(s)|=1\)
outside \(S\), the two extracted scalars give
\[
 |A_*(s)\overline{\chi_s(d')}\chi_s(h')|^2
 =1_{(s,d'h')=1}=1_{(s,h')=1}
 \quad\text{when }d'\mid C,\ (s,C)=1.
\]
The original residual and overlap exclusions also give
\(1_{(C,j)=1}1_{(s,Cj)=1}\).  Thus, apart from the assigned
coefficients, individual cutoffs, and unit phases, the outer arithmetic
weight is precisely
\begin{equation}\label{eq:initial-outer-arithmetic}
 \mu(d')\mu(s)\widehat G(\vartheta)\mathcal B_{\rm init}(h')
 1_{(C,j)=1}1_{(s,Cj)=1}1_{(s,h')=1}.
\end{equation}
All these factors remain through weighted Cauchy.  Set
\[
 t=C/d',\qquad k=d'h',\qquad f=d's,\qquad
 \rho_{t,j}(n)=1_{(n,\operatorname{rad}(tj))=1}.
\]
The factors on \(n\) in Equation~\eqref{eq:initial-common-extraction},
together with \(\rho_{t,j}\), supply exactly the original exclusions
at \(C,s,j\) and the numerator zeros at \(d',h'\).  On the nonzero
support \(t,d',s,n\) are pairwise coprime.  Apply
Equation~\eqref{eq:inverse-slot-priority} to each mark with this
ordered list.  Assigned primes and their priority masks stay outer.
A surviving prime is already prime to \(d's\) by \(\chi_n(f)^4\)
and to \(tj\) by \(\rho_{t,j}\), so its original individual list and
coefficient can be retained.  Its mark is an original subcollection
of cap \(z_0\), not a row- or label-dependent residual coefficient.

The substitutions \(k=d'h'\), \(f=d's\), and the puncture
\(\rho_{t,j}\) have therefore supplied the canonical character and
coefficient class.  We still need a common separated profile, the
outer multiplicity bound, and admissibility of the resulting ranges.

Keep \(\mathcal B_{\rm init}\) outside the common Fourier separation
of the full kernel and the old windows.  The exact
formal child centers and the full fresh-support bounds are
\begin{equation}\label{eq:initial-canonical-lengths}
 \begin{gathered}
 N_c^{\rm init}=D'-B-v,\qquad V_c^{\rm init}=\theta+v,\qquad
 F_c^{\rm init}=D'-P_1,\\
 |\widehat n-N_c^{\rm init}|\le3\eta,\qquad
 |\widehat f-V_c^{\rm init}|\le2\eta.
 \end{gathered}
\end{equation}
On factorized terms these bounds follow from \(c=Csn\) and \(f=d's\).
The full fresh column cutoff and enlarged label window are also included
in \(\mathcal I_D\), so the same bounds hold directly for independent
terms later added by positivity.  The reconstructed formal product
\(c=Csn\) on the fresh cutoff is included there as well, so the initial
prefactor bound holds on its entire separated side.

Fix a refinement \(\Sigma\) by the \(s\)-dyad and the slot branches,
with the earlier mode \(\boldsymbol\upsilon\) held fixed.  Before
summing any actual outer ideal, use the six independent coordinates
\[
 (x_C,x_d,x_s,x_h,x_1,x_2)
 =\left(\frac{q_C}{Z^B},\frac{q_{d'}}{Z^\theta},
 \frac{q_s}{Z^v},\frac{q_{h'}}{Z^H},
 \frac{q_{n_1}}{Z^{N_c^{\rm init}}},
 \frac{q_{n_2}}{Z^{N_c^{\rm init}}}\right).
\]
Let \(I_C,I_d,I_s,I_h\) be the full supports of the four current
individual cutoffs.  Choose \(\omega_{n,a}\) equal to one on the
support of \(W_0\) divided by \(I_CI_s\), with fixed full supports
in an enclosing interval \([a,b]\) whose upper endpoint satisfies
\(b\ge1\).  Choose a nonnegative \(\omega_f\) equal to one on
\(I_dI_s\), with fixed full support \(I_f\).  The current four dyad
cutoffs and \(\omega_f\) remain outside inversion, and
\(\omega_{n,a}\) remain in the columns.  Let
\(\Omega_{{\rm init},\alpha}\) be larger cutoffs equal to one on the
full supports of the corresponding four current and two fresh cutoffs.
These full supports, \(I_f\), and the formal products \(Csn\) on them
are the windows already included in \(\mathcal I_D\).  None of these
supports depends on a current outer tuple or on a Fourier mode.

Keep \(V_c^{\rm init}\ge0\).  Proceed to a child only if the retained
row component is nonzero, which implies \(M_{\rm init}\ge0\).
This is a separate gate from the structural outer set below, which
may include zero-weight tuples even when the row ball is empty.
Independently, if a full fresh child column window contains no
squarefree ideal, its polynomial is zero and that component is omitted.
For a retained nonempty child define
\[
 N_{\rm init}=\max(0,N_c^{\rm init}),\qquad
 \delta_{\rm init}=N_{\rm init}-N_c^{\rm init}\in[0,3\eta],\qquad
 F_{\rm init}=N_{\rm init}+V_c^{\rm init}=F_c^{\rm init}+\delta_{\rm init}.
\]
For a nonempty negative-center window, \(q_n\ge1\) and
Equation~\eqref{eq:initial-canonical-lengths} give
\(\delta_{\rm init}\le3\eta\), while its upper support endpoint gives
\(Z^{\delta_{\rm init}}\le b\).  If \(N_c^{\rm init}\ge0\), then
\(\delta_{\rm init}=0\) and the latter inequality follows from \(b\ge1\).
Thus \(1\le Z^{\delta_{\rm init}}\le b\) in every retained case.
The cutoff \(\omega_{n,a}(Z^{\delta_{\rm init}}y)\) is supported in
\([a/b,b]\) and has uniformly bounded Euler seminorms.  Its rescaling
depends only on fixed centers, not on a current row or label, and its
full clipping family is included in \(\mathcal I_D\).

On the formal factorization \(z_a=sn_a\), the complete root and
kernel argument are exactly
\begin{equation}\label{eq:initial-root-kernel}
 \begin{split}
 \frac{Z^{-D'}Z^m}{q_{d'}q_s\sqrt{q_{n_1}q_{n_2}}}
 &=Z^{\kappa_c^{\rm init}}x_d^{-1}x_s^{-1}(x_1x_2)^{-1/2},\\
 \frac{Z^mq_{h'}}{q_{d'}q_s^2q_{n_1}q_{n_2}}
 &=A_{{\rm ker},{\rm init}}\frac{x_h}{x_dx_s^2x_1x_2},\\
 A_{{\rm ker},{\rm init}}&=Z^{m+H-\theta-2(D'-B)}.
 \end{split}
\end{equation}
For the first equality use \(N_c^{\rm init}=D'-B-v\) and
\(\kappa_c^{\rm init}=m-2D'+B-\theta\).  Equivalently its root is
\(x_d^{-1}x_C(x_{c_1}x_{c_2})^{-1/2}\) with
\(x_{c_a}=x_Cx_sx_a\).  Define the one joint profile
\begin{equation}\label{eq:initial-joint-profile}
 \begin{split}
 \mathfrak H_{{\rm init},\Sigma,\boldsymbol\upsilon}(\boldsymbol x)
 ={}&Z^{-3\eta}\prod_\alpha\Omega_{{\rm init},\alpha}(x_\alpha)
 x_d^{-1}x_s^{-1}(x_1x_2)^{-1/2}\\
 &\times W_0(x_Cx_sx_1)\overline{W_0(x_Cx_sx_2)}
 \mathcal K_{\rm init}\!\left(
 A_{{\rm ker},{\rm init}}\frac{x_h}{x_dx_s^2x_1x_2}\right).
 \end{split}
\end{equation}
Its outside scalar is exactly \(Z^{\kappa_c^{\rm init}+3\eta}\).
The old \(c\)-windows are coupled and therefore are in this profile,
as are all normalized real inverse roots and the full kernel.  No
real root is also put in a child test.

For \(\boldsymbol\xi=(\xi_C,\xi_d,\xi_s,\xi_h,\xi_1,\xi_2)\), put
\(z_n=q_n/Z^{N_{\rm init}}\), set again
\(\varepsilon_1=1,\varepsilon_2=-1\), and define
\[
 W_a^{\rm init}(z)=\omega_{n,a}(Z^{\delta_{\rm init}}z)
                       z^{\varepsilon_a i\xi_a}.
\]
If \(\mathfrak d_a'\) is the surviving subcollection on side \(a\),
the precise unnormalized child polynomial is
\begin{equation}\label{eq:initial-child-polynomial}
 Q^{\rm init}_{a,t,j,\boldsymbol\xi}(k,f)
 =\sum_{n\ {\rm sf}}A_*(n)\rho_{t,j}(n)
 \chi_n(k)\chi_n(f)^4\mathfrak d_a'(n)
 W_a^{\rm init}(q_n/Z^{N_{\rm init}}).
\end{equation}
Its fixed finite character, puncture, product mark, and test are
independent of \(k,f\).  The two sides can have different tests and
subcollections, but both have the same canonical coefficient class.

Let \(\Omega_\Sigma\) consist of
\[
 \omega=(C\ {\rm sf},d'\mid C,s\ {\rm sf},h'\ne0;
                      \text{assigned slot primes})
\]
with the retained individual supports and slot branches.  It has no
current column index.  The functions \(t,k,f\) are the products
defined above; the set may be taken before imposing the displayed
outer zero masks, which will be in the weight.  Let
\(\mathcal A_\Sigma^{\rm init}(\omega)\) be the product of all
assigned coefficients of \(\mathfrak d_0\) and their priority masks,
conjugated on side two.  The complete remaining outer weight is
\begin{equation}\label{eq:initial-outer-weight}
 \begin{split}
 V_\Sigma^{\rm init}(\omega;\boldsymbol\xi)={}&
 \mu(d')\mu(s)\widehat G(\vartheta)\mathcal B_{\rm init}(h')
 1_{(C,j)=1}1_{(s,Cj)=1}1_{(s,h')=1}
 \mathcal A_\Sigma^{\rm init}(\omega)\\
 &\times\psi_C(x_C)\psi_{d'}(x_d)\psi_s(x_s)\psi_{h'}(x_h)
 \omega_f(q_f/Z^{V_c^{\rm init}})\\
 &\times x_C^{i\xi_C}x_d^{i\xi_d}x_s^{i\xi_s}x_h^{i\xi_h}
 Z^{i\delta_{\rm init}(\xi_1+\xi_2)}.
 \end{split}
\end{equation}
The cutoff \(\omega_f\) is one on every original label product, so
its insertion is an equality before separation.  There is one copy
of each M\"obius factor and one ray coefficient.  In particular the
common zero at \((s,h')\) is still present.

Let \(\mathcal T_\Sigma^{\rm init}\) be the fixed \(s\)-dyad and
slot summand obtained from Equation~\eqref{eq:initial-formal-component}
by the complete M\"obius and priority expansions.  Its exact separated
identity is
\begin{equation}\label{eq:initial-separated-identity}
 \begin{split}
 \mathcal T_\Sigma^{\rm init}={}&
 Z^{\kappa_c^{\rm init}+3\eta}(2\pi)^{-6}
 \int_{\mathbb R^6}
 \widehat{\mathfrak H}_{{\rm init},\Sigma,\boldsymbol\upsilon}
       (\boldsymbol\xi)\\
 &\quad\times\sum_{\omega\in\Omega_\Sigma}
 V_\Sigma^{\rm init}(\omega;\boldsymbol\xi)
 Q^{\rm init}_{1,t,j,\boldsymbol\xi}(k,f)
 \overline{Q^{\rm init}_{2,t,j,\boldsymbol\xi}(k,f)}
 \,d\boldsymbol\xi.
 \end{split}
\end{equation}
Indeed \(x_a=Z^{\delta_{\rm init}}z_{n_a}\).  The two column modes,
with the last factor of Equation~\eqref{eq:initial-outer-weight},
are exactly \(x_1^{i\xi_1}x_2^{i\xi_2}\), and the other four modes
are outer.  Fourier inversion restores the full profile; its larger
cutoffs are one on the retained supports, and the fresh column
cutoffs are one wherever the old \(W_0\) windows are nonzero.
Equation~\eqref{eq:initial-root-kernel} restores the full prefactor
and kernel.  Equation~\eqref{eq:initial-common-extraction} and the
priority expansion restore every arithmetic factor term by term.
Extra combinations in the full fresh windows cancel through the old
windows in this complex identity before Cauchy.

The transform in Equation~\eqref{eq:initial-separated-identity}
is chosen from the fixed ambient profile before any actual
\(C,d',s,h'\), and hence before any \(t,k,f\), is evaluated.  It
depends on \(\Sigma,Z\), the fixed tests, and the earlier mode
\(\boldsymbol\upsilon\), and may depend on the already frozen \(j\),
but not on any current outer ideal or reindexed row or label.  The bare
\(h'\) norm is a coordinate; the derived row and label occur only
in the outer masks or in the displayed characters.  The relation
\(C=td'\) only restricts evaluation points.  By
Equation~\eqref{eq:inverse-fourier-moment}, for each fixed \(J\)
there are fixed \(L,b_0\) such that
\[
 \int_{\mathbb R^6}
 |\widehat{\mathfrak H}_{{\rm init},\Sigma,\boldsymbol\upsilon}
       (\boldsymbol\xi)|(1+|\boldsymbol\xi|)^J\,d\boldsymbol\xi
 \ll (1+|\boldsymbol\upsilon|)^L,\qquad
 \int_{\mathbb R^{1+|I_0|}}|\widehat{\mathfrak H}_{{\rm ov},j}(\boldsymbol\upsilon)|
       (1+|\boldsymbol\upsilon|)^L\,d\boldsymbol\upsilon
 \ll p_{b_0}(W).
\]
The first bound uses the fixed normalized roots and the uniform Euler
bounds of the full kernel for every positive
\(A_{{\rm ker},{\rm init}}\); \(W_0\) has only polynomial
\(\boldsymbol\upsilon\)-dependence.  The second is uniform over the
fixed \(j\)-product interval.  Taking larger \(L\) if needed also
controls the earlier row-space Minkowski inequality.  No derivative
order introduces a power of \(Z\).

Equation~\eqref{eq:initial-separated-identity} now expresses the component
in polynomials of canonical shape with one common Fourier density chosen
before the current outer ideals are evaluated.  It remains to bound the
positive fibres and source measure, verify child admissibility from source
witnesses, and restore the normalization in Lemma~\ref{lem:marked}.

Apply Equation~\eqref{eq:weighted-cauchy} on the whole
\(\Omega_\Sigma\), including all quotients \(t\), before fixing one
for the child.  Only afterward take absolute weights.  The retained
\((s,C)=1\) and \(d'\mid C\) make \(f=d's\) squarefree, and all
other factors of the weight are bounded by a fixed constant.  Thus
\[
 |V_\Sigma^{\rm init}(\omega;\boldsymbol\xi)|
 \ll\mathcal B_{\rm init}(h')\mu^2(f).
\]
For fixed \(t,f,k\), the identities
\[
 d'\mid f,\qquad s=f/d',\qquad C=td',\qquad h'=k/d'
       \quad\text{if this quotient is an element}
\]
show that the reconstruction has at most \(d_{\mathcal O}(f)\)
choices before slots.  Here \(K_{\rm slot}\) again denotes the fixed
bound for the original number of slots.  The at most \(2K_{\rm slot}\) assigned
residual primes divide \(tf\).  Their choices cost at most
\([d_{\mathcal O}(t)d_{\mathcal O}(f)]^{2K_{\rm slot}}\).
The earlier overlap primes were already frozen and counted in the
overlap triangle.  Hence the fibre, even if its actual size depends
on \(k\), is bounded by
\begin{equation}\label{eq:initial-fibre}
 D_{\rm init}(f)C_{\rm init}(t),\qquad
 D_{\rm init}(f)=d_{\mathcal O}(f)^{1+2K_{\rm slot}},\qquad
 C_{\rm init}(t)=d_{\mathcal O}(t)^{2K_{\rm slot}}.
\end{equation}
There is no independent \(d',s\) count and no second frequency count.

Let \(\Omega_\Sigma^{\rm src}\) be the structural outer set just
used, before independent positive row or label additions and ignoring
unit Fourier phases, and define the set of distinct source quotients
\[
 \mathfrak T_\Sigma=\{t=C/d':\omega\in\Omega_\Sigma^{\rm src}\}.
\]
It is projected once over all current rows and labels, not redefined
for a fixed \(k,f\).  Every member has a source witness on the
\(C,d'\) dyads, and therefore
\(\log_Zq_t=\widehat B-\widehat\theta\le P_1+2\eta\).
The containing norm ball bounds its cardinality by
\(\#\mathfrak T_\Sigma\ll Z^{P_1+2\eta+\pi_{{\rm count},{\rm init}}}\);
it is not used as a replacement child domain.  In a nonempty sector
\(P_1+2\eta\ge0\), since it bounds the norm of an existing ideal.
The divisor bound on this bounded range gives the normalized measure
\begin{equation}\label{eq:initial-normalized-quotients}
 d\nu_\Sigma^{\rm init}(t)
 =Z^{-P_1-2\eta}\sum_{t\in\mathfrak T_\Sigma}C_{\rm init}(t)\delta_t,
 \qquad
 \|\nu_\Sigma^{\rm init}\|
 \ll Z^{\pi_{{\rm count},{\rm init}}+\pi_{{\rm fib},{\rm init}}}.
\end{equation}
This set and measure ignore the unit Fourier phases and are common
to the whole current row and label sum.  The nonempty gates remain
separate from this structural projection.

For a fixed mode let
\begin{equation}\label{eq:initial-positive-child}
 \mathcal H_{a,t}^{\rm init}
 =\sum_{\substack{f\ {\rm sf}\\q_f/Z^{V_c^{\rm init}}\in I_f}}
 D_{\rm init}(f)\sum_{\substack{k\in\mathcal O\\0<q_k\le Z^{M_{\rm init}}}}
 |Q^{\rm init}_{a,t,j,\boldsymbol\xi}(k,f)|^2
 =Z^{F_{\rm init}}\mathcal E_{a,t}^{\rm init}.
\end{equation}
This is Equation~\eqref{eq:canonical-energy} with the same row ball;
with a larger fixed canonical ball the equality is replaced by the
corresponding upper bound.  The fibre estimate and weighted Cauchy give
\begin{equation}\label{eq:initial-weighted-count}
 \left|\sum_{\omega\in\Omega_\Sigma}V_\Sigma^{\rm init}Q_1\overline{Q_2}\right|
 \ll Z^{P_1+2\eta}
 \prod_{a=1}^2
 \left(\int_{\mathfrak T_\Sigma}\mathcal H_{a,t}^{\rm init}
                         \,d\nu_\Sigma^{\rm init}(t)\right)^{1/2}.
\end{equation}
Here \(Q_a=Q^{\rm init}_{a,t,j,\boldsymbol\xi}(k,f)\).
Only the positive outer \(f,k\) sums were enlarged to their full
fixed windows; the inner \(\rho_{t,j}\), mark, and fresh test are
unchanged.  The weight \(D_{\rm init}\) depends on \(f\) alone.
The new outer ball is exactly \(1_{0<q_k\le Z^{M_{\rm init}}}\).
Thus use of the uniform child bound introduces the displayed quotient-count
exponent once and the normalized mass once.

The fixed puncture is exactly at \(\operatorname{rad}(tj)\).
For each \(t\in\mathfrak T_\Sigma\), its source bound and the
fixed overlap bound give
\begin{equation}\label{old-eq:4.13}
 Q_{\rm init}:=\log_Zq_{\mathfrak r_\rho}
 \le\widehat B-\widehat\theta+\widehat j\le P_1+G+3\eta.
\end{equation}
Shared puncture primes only decrease this bound.  It continues to hold
when positive new labels or rows are added, because those additions
do not change \(t,j\) or delete their inner puncture.

The fixed formal centers satisfy
\[
 \begin{aligned}
 F_c^{\rm init}-M_c^{\rm init}-(P_1+G)-z_0&=m-r-2z+2G,\\
 4F_c^{\rm init}-3M_c^{\rm init}-6z_0&=3m-2r-8z+10G+2P_1.
 \end{aligned}
\]
Use the marked premises, \(G\ge0\), \(P_1\ge-2\eta\),
Equations~\eqref{eq:inverse-initial-row} and \eqref{old-eq:4.13}, and
the favorable clipping signs.  The actual canonical margins are
\begin{equation}\label{eq:inverse-initial-margins}
 \begin{aligned}
 F_{\rm init}-M_{\rm init}-Q_{\rm init}-z_0
   &\ge c_1+\delta_{\rm init}-9\eta-\tau_{\rm init}
     \ge c_1-9\eta-\tau_{\rm init},\\
 4F_{\rm init}-3M_{\rm init}-6z_0
   &\ge c_2+4\delta_{\rm init}-22\eta-3\tau_{\rm init}
     \ge c_2-22\eta-3\tau_{\rm init}.
 \end{aligned}
\end{equation}
The \(22\eta\) is \(4\eta\) from the possible negative \(2P_1\) and
\(18\eta\) from three copies of the row enclosure.

There is also an explicit energy calculation.  Apply the canonical
bound only for \(t\in\mathfrak T_\Sigma\), where the preceding puncture
and margin checks hold.  Equations~\eqref{eq:initial-positive-child}
and \eqref{eq:initial-weighted-count} then bound the unscaled signed
outer sum by
\[
 C(\boldsymbol\upsilon,\boldsymbol\xi)
 Z^{P_1+2\eta+2F_{\rm init}+\epsilon_c}
 \|\nu_\Sigma^{\rm init}\|.
\]
The one mass is that of Equation~\eqref{eq:initial-normalized-quotients};
the label \(d's\) remains inside the child average.  The common
Fourier density is integrated separately, once, with the uniform
weighted bounds above.  This counts the distinct frozen quotient
once, with its witness multiplicity already in the fibre weights.
The formal identity
\[
 \kappa_c^{\rm init}+P_1+2F_c^{\rm init}=m
\]
then gives, after restoring the child normalization and applying the
canonical bound with loss \(\epsilon_c\), the per-side exponent
\begin{equation}\label{eq:inverse-initial-energy}
 \begin{split}
 &m+3\eta+2\eta+2\delta_{\rm init}+\pi_{\rm init}+\epsilon_c\\
 &\qquad\le m+11\eta+\pi_{\rm init}+\epsilon_c.
 \end{split}
\end{equation}
Here \(\pi_{\rm init}\) is the aggregate freely chosen local divisor,
dyadic, and separation loss, including the assigned-overlap divisor
loss and the distinct allocations
\(\pi_{{\rm count},{\rm init}},\pi_{{\rm fib},{\rm init}}\).
These are parts of one aggregate, not repeated allowances.
The two clipping copies come from the restored normalization and
the child exponent.  There is no \(\tau_{\rm init}\) energy factor: the
row enlargement occurs inside the canonical child and was accounted for
in its margin test.  The initial weighted Cauchy takes a geometric mean,
and the overlap triangle was already canceled by \(Z^{-G}\).

We give the parameter and tail order explicitly.  Let
\(r_{\max},z_{\max},m_{\max}\) bound the marked ranges, let
\(s_{\rm slot}\) bound the number of slots, and put
\(D_{\max}=r_{\max}+z_{\max}\).  Set
\(\bar c=\min(c_1,c_2)\) and choose the canonical margin
\(c_*=\bar c/2\).  For \(\eta,\tau_{\rm init}\le1/100\), valid a priori
starting bounds for that application are
\[
 M_{\max}=2D_{\max}+1,\qquad N_{\max}=V_{\max}=D_{\max}+1.
\]
Indeed, \(P_1\ge-2\eta\) gives
\(M_{\rm init}\le2D_{\max}+10\eta+\tau_{\rm init}\), and
\(F_{\rm init}\le D_{\max}+5\eta\) bounds both nonnegative child
parameters.  Make the choices in Equation~\eqref{eq:inverse-parameter-order}
with \(\epsilon_c=\epsilon/2\), imposing in addition
\[
 \eta,\tau_{\rm init}\le\bar c/100,\qquad
 \eta\le\epsilon/88,\qquad \pi_{\rm init}\le\epsilon/8.
\]
One may take \(\tau_{\rm init}=\tau\) after imposing both sets of bounds.
Equation~\eqref{eq:inverse-initial-margins} then gives both margins at
least \(3\bar c/4>c_*\), so Lemma~\ref{lem:canonical-moment} applies.
Equation~\eqref{eq:inverse-initial-energy} adds at most \(\epsilon/4\)
to its \(\epsilon_c=\epsilon/2\) loss.

With these bounded initial ranges fixed, choose the subsidiary input
exponents realizing the target \(\pi_{\rm init}\), before choosing the
initial tail and Fourier-height orders.

For the separate initial raw tail, the supported columns satisfy
\(q_{c_i}\le C Z^{D_{\max}}\), while \(D'\in[-z_{\max},D_{\max}]\).
On the genuine initial expansion put
\(a=Z^m/(q_{d'}q_{z_1z_2})\).  Since \(m\ge0\), \(d'\mid C\), and
\(c_i=Cz_i\),
\[
 a^{-1}\le q_{d'}q_{z_1z_2}
       \le q_{c_1}q_{c_2}/q_C\le C Z^{2D_{\max}}.
\]
The explicit crude exponent
\[
 B_{\rm crude,init}=10\{1+m_{\max}+(s_{\rm slot}+1)(D_{\max}+1)\}
\]
bounds all raw nonfrequency factors even without cancellation.  To verify
this, the normalization costs at most \(z_{\max}\), the two column counts
cost \(2D_{\max}\), the \(d'\)-divisor count at most \(D_{\max}\), and
\(1+a^{-1}\) at most \(2D_{\max}\).  The two residual marks cost at most
\(2s_{\rm slot}D_{\max}\), using \(d_{\mathcal O}(c)^{s_{\rm slot}}\le q_c^{s_{\rm slot}}\).
Even counting overlap pairs and their assigned coefficients without
their canceling weights costs at most
\(2z_{\max}+2s_{\rm slot}z_{\max}\).  The Poisson prefactor costs at most
\(m_{\max}\).  The sum is at most
\(m_{\max}+(8+4s_{\rm slot})D_{\max}<B_{\rm crude,init}\).

Choose \(T_{\rm init}>1+B_{\rm crude,init}\) and then
\[
 A_{\rm init}>1+(B_{\rm crude,init}+T_{\rm init})/\tau_{\rm init}.
\]
Equation~\eqref{eq:inverse-raw-tail} makes the raw
\(\mathcal B_{\rm init}=0\) complement \(O(Z^{-T_{\rm init}})\), with
fixed seminorm and polynomial-height factors.  This order is chosen after
the marked ranges and \(\tau_{\rm init}\), but before the initial
Fourier-height orders and before the final \(Z\) threshold.  If
\(\tau_{\rm init}=\tau\), take the maximum of the orders required by this
separate crude bound and the canonical tails.  The full kernel retained
inside the outer ball has the uniform Euler bounds already used in the
canonical separation.  The final threshold also enforces the full initial support
ratios in \(\mathcal I_D\).

The principal count has exponent \(m\), and every nonprincipal term has
now been bounded by \(Z^{m+\epsilon}\).  The argument keeps all element
rows and every original mask, applies to subcollections and either common
orientation, and preserves arbitrary bounded row-independent prime
coefficients.  The common separated measures give the stated finite-
seminorm and polynomial-height uniformity.  This proves
Equation~\eqref{old-eq:4.1}.
\end{proof}

\subsection{Amplification on sixth-power-free rows}

We first extract an unmarked consequence on all element rows.  In
Lemma~\ref{lem:marked}, take \(Z=H,\ m=1\), no slots, and
\(r=\log_H D\).  If \(H\ge D^{1+c}\) for a fixed \(c>0\), then
\(1-r\ge c/(1+c)\); the second required margin is also positive.
For \(D\le H^{\epsilon/4}\), the direct bound
\(|M_u(D;W)|^2\ll D\) suffices.  Otherwise the starting lengths
are in a fixed bounded range away from zero.  Bounded \(H,D\)
are handled directly.  We obtain, for \(H,D\ge1\),
\begin{equation}\label{eq:raw-moment}
 \sum_{\substack{u\in\mathcal O\\0<q_u\le H}}
 \left|D^{-1/2}\sum_n\mu(n)\nu(n)\chi_n(u)^{\varepsilon_\chi}
                         W(q_n/D)\right|^2
 \ll_{c,\epsilon,W} H(HD)^\epsilon,\qquad H\ge D^{1+c}.
\end{equation}
The fixed arithmetic data and the stated seminorms are included in
the dependence of the constant.

\begin{lemma}[Sixth-power amplification]\label{lem:inverse-amplification}
Let \(\nu,W,\varepsilon_\chi\) and the zero extensions be as in
Lemma~\ref{lem:marked}, with no prime slots.  Let \(U,D\ge1\), and
sum over elements \(u\) with \(q_u\asymp U\) such that every prime
valuation of the ideal \((u)\) is at most five.  For every fixed
\(c>0\) and \(\epsilon>0\), put
\[
 H=\max(2U,D^{1+c}),\qquad P=(H/U)^{1/6}.
\]
Then
\begin{equation}\label{eq:amplified-note}
 \sum_u|M_u(D;W)|^2
 \ll \frac HP(UD)^\epsilon
 \ll \max\{U,U^{1/6}D^{5(1+c)/6}\}(UD)^\epsilon.
\end{equation}
The implied constant depends only on \(c,\epsilon\), the fixed
arithmetic data, and finitely many seminorms of \(W\).  In particular,
for \(D=U^r\) with \(r\) in a fixed bounded nonnegative range, for
every \(\epsilon>0\) the exponent may be written
\[
 \sum_u|M_u(U^r;W)|^2\ll U^{e(r)+\epsilon},
 \qquad e(r)=\max\{1,(1+5r)/6\}.
\]
Both assertions permit a separate test
\(W_{\sigma_u,t_u}(y)=W(y)y^{-\sigma_u+it_u}\) in each row, with
\(\sigma_u\) in a fixed compact interval and \(|t_u|\le T_1\),
at a cost \((1+T_1)^A\) for a fixed \(A\).
\end{lemma}

\begin{proof}
First, Equation~\eqref{eq:raw-moment} and
Lemma~\ref{lem:smooth-calculus} imply the rowwise scale bound
\begin{equation}\label{eq:inverse-scale-supremum}
 \sum_{\substack{v\in\mathcal O\\0<q_v\le C H}}
       \sup_{0<D'\le D}|M_v(D';W)|^2
 \ll H(HD)^\epsilon,\qquad H\ge\max(2,D^{1+c}).
\end{equation}
Indeed, sufficiently small scales have no ideals in the annular
support.  Differentiation with respect to \(\log D'\) replaces
\(W(y)\) by \(-W(y)/2-yW'(y)\).  The one-dimensional Sobolev
inequality in Lemma~\ref{lem:smooth-calculus}, followed by
Equation~\eqref{eq:raw-moment} for these two tests, controls the
supremum on a unit logarithmic interval.  There are
\(O(\log(2+D))\) such intervals.  A fixed enlargement of \(H\)
handles their endpoints, and the direct small-scale bound handles
the initial intervals.  This proves
Equation~\eqref{eq:inverse-scale-supremum}.

Ideal counting outside the fixed set \(S\) gives at least \(c_S P\)
primary ideals \(a\) with \(q_a\le P\), for a fixed \(c_S>0\).
For bounded \(P\) this follows after reducing \(c_S\), since the
unit ideal is available; for large \(P\) it follows from the
positive-density ideal count with finitely many primes removed.

For squarefree \(n\), separate the primes dividing \(a\) by
\(n=dm\), where \(d\mid\operatorname{rad}a\) and \((m,a)=1\).
With the zero extensions,
\[
 \psi_{ua^6}(m)=\psi_u(m)1_{(m,a)=1}.
\]
This is true also when \(u,a\) share primes: both sides vanish
at a prime dividing \(m\) and either \(u\) or \(a\), and otherwise
the sixth power is one.  The separated column therefore gives the
exact identity
\begin{equation}\label{eq:sixth-power-column-identity}
 M_u(D;W)=\sum_{d\mid\operatorname{rad}a}
       \mu(d)\psi_u(d)q_d^{-1/2}M_{ua^6}(D/q_d;W).
\end{equation}
The coefficient mass is at most \(d_{\mathcal O}(a)\ll_\delta P^\delta\).
It follows that
\[
 |M_u(D;W)|^2\ll_\delta
 P^{2\delta}\sup_{0<D'\le D}|M_{ua^6}(D';W)|^2.
\]

Average this bound over the \(a\)'s and sum over \(u\).  The map
\((u,a)\mapsto ua^6\) is injective on these pairs.  In fact, at
each prime the valuation modulo six recovers the valuation of
the sixth-power-free ideal \((u)\), and its quotient by six
recovers the valuation of \(a\).  The primary generator of \(a\)
then fixes the element \(u\), including its unit.  This argument
does not require \((u,a)=1\).  Also
\(q_{ua^6}\ll UP^6\ll H\).  Thus
Equation~\eqref{eq:inverse-scale-supremum} gives
\[
 \sum_u|M_u(D;W)|^2
 \ll \frac HP(UD)^\epsilon
 =U^{1/6}H^{5/6}(UD)^\epsilon,
\]
after choosing the preliminary \(\delta\) and power losses small
enough.  Since \(H=\max(2U,D^{1+c})\), this is
Equation~\eqref{eq:amplified-note}.  For bounded \(r\), choose
\(c>0\) sufficiently small in terms of \(r\)'s bound and the
desired power loss; the stated formula for \(e(r)\) follows.

Finally, on the fixed annular support, derivatives in \(\sigma,t\)
of \(W_{\sigma,t}\) insert bounded powers of \(\log y\).  Apply
the parameter Sobolev inequality in
Lemma~\ref{lem:smooth-calculus} on a bounded cover in \(\sigma\)
and \(O(1+T_1)\) unit intervals in \(t\), summing the row moments
before integrating the derivatives.  The finite-seminorm bound
already proved has fixed polynomial height order.  The cover and
these derivatives therefore give a factor \((1+T_1)^A\) for a
fixed \(A\), uniformly for a separate \((\sigma_u,t_u)\) in every
row.  The same reasoning applies to the marked estimate with any
fixed finite number of test parameters, while its prime coefficients
remain fixed across rows.
\end{proof}

In the later application, the positive exponent in
\(T_1=Z^{\tau_\eta}\) is chosen after the fixed order \(A\).
It may therefore be chosen small enough to fit the reserved power
loss.  This use of rowwise test parameters does not permit arbitrary
row-dependent prime coefficients.

\section{Fourth moments with short prime factors}\label{sec:plain}

The refined row count requires a fourth-moment estimate for two plain
character polynomials and a product of short prime polynomials.  We prove
that estimate here.  The two plain polynomials have no length restriction
when no prime polynomial is present; otherwise the permitted lengths lie
in an affine region.  Two finite Fourier transforms return products of the
same kind at a smaller effective width.  Some transformed rows, however,
induce characters in a fixed
finite family.  Their plain products can have volume-sized main terms.
For the longer inputs we therefore subtract a comparison product with the
same product of scales.  Preserving its common character, mask, and norm
power through the transforms makes those main terms cancel.  The estimate
itself concerns the original, uncentered product.

We use the notation of Section~\ref{sec:arithmetic} and the coefficient
conventions of Section~\ref{sec:additional-arithmetic}.  In particular, the
rows are elements \(k\in\mathcal O\) with \(0<q_k\ll Z^m\), and
\[
 \psi_k(n)=\tau(n)\chi_n(k),\qquad M=m+q.
\]
The character \(\tau\) is fixed within a row sum.  The union of the
prime supports of all its displayed moving residue-symbol factors,
before canceling factors or reducing exponents modulo six, has norm at
most \(Z^q\).  Every displayed factor retains its zero extension,
including a canceled or six-divisible factor.  A redundant zero may
instead be represented by an additional puncture mask only when an exact
factorization retains every surviving local and fixed-ray phase.  This
mask is the indicator of coprimality to one squarefree ideal of norm at
most \(Z^B\), for a bounded \(B\), fixed within the current row sum;
the same mask is used in both plain factors and in every prime factor.
It may depend on previously frozen labels, but not on the varying row.
All ideals in the polynomials are outside the fixed set \(\mathcal S\).

Fix a finite group \(\Theta\) of finite-order ray characters whose
conductor primes belong to \(\mathcal S\).  It contains the fixed
twists and all characters used to separate the fixed reciprocity
phases.  Thus it also contains the supplementary character
\(n\mapsto\chi_n(-1)\): by
Equation~\eqref{eq:fixed-gauss-phases} and the fixed bicharacter
table, this character is the diagonal character
\(n\mapsto\mathcal R(n,n)\).  Membership of an inducing character in
\(\Theta\) means equality with the primitive inducing character of some
member of \(\Theta\), not equality of the chosen zero-extended
presentations.  All zeros of those presentations remain in the polynomials.
Let \(\mathcal R_0\) be the rows for which
\(\psi_k\) induces a nonprincipal character.  For \(z>0\), let
\(\mathcal R_z\) be the rows for which the inducing character of
\(\psi_k\) does not belong to \(\Theta\).  Write
\begin{equation}
 A=n_1+n_2+z.
 \label{old-eq:3.1}
\end{equation}

\begin{lemma}[Fourth moment with short prime factors]\label{lem:plain}
Let \(3/4\le\kappa\le1\).  Let \(n_1,n_2\ge0\), and let
\(Q=\prod_{i\in\mathcal I}Q_{\psi_k,i}\) be a finite product of the
prime polynomials in Equation~\eqref{old-eq:1.2}, of lengths
\(z_i\ge0\) and total length \(z=\sum_{i\in\mathcal I}z_i\).
Their underlying prime supports are pairwise disjoint before common
masks and row zero extensions are imposed.  Their coefficients are
fixed finite linear combinations of finite-ray characters.  For every
positive-length slot the expansion has the form
\[
 \nu_i(n)=\sum_{j=1}^{J_i}c_{ij}\vartheta_{ij}(n),
 \qquad \vartheta_{ij}\in\Theta,
\]
where the finite list and its coefficients are fixed independently of
\(Z\) and of the row.  The row and mask hypotheses are those just stated.

For every \(\epsilon>0\), there is a slot mesh \(\eta>0\) such that,
if \(z_i\le\eta\) for every \(i\), then
\begin{equation}
 \sum_{k\in\mathcal R_z}
 \left|S_{\psi_k}(n_1;W_1)S_{\psi_k}(n_2;W_2)Q\right|^2
 \ll Z^{M+\epsilon}
 \label{old-eq:2.1}
\end{equation}
in either of the following cases:
\begin{enumerate}
\item \(z=0\).  There is no restriction on the bounded nonnegative
      lengths \(n_1,n_2\), and no lower bound on the conductor.
\item \(z>0\) and
\begin{equation}
 n_1+n_2+6\kappa z=A+(6\kappa-1)z\le M.
 \label{old-eq:3.9}
\end{equation}
      If \(\kappa<1\), assume in addition that
      \(\beta_*\le(1+\kappa)/2\).  If \(\kappa=1\), no zero-free
      hypothesis is required.
\end{enumerate}
For an empty slot list, \(Q=1\).  If \(z=0\) and the list contains
zero-length slots, their scales are bounded and absolute counting
absorbs them into the constant; the zero-slot assertion therefore
has the same strength.  All real length parameters range over prescribed bounded
sets.  The mesh depends only on those sets and \(\epsilon\), uniformly
for \(\kappa\in[3/4,1]\).  For each fixed \(Z\)-independent arithmetic
datum \(\mathcal A\) and fixed slot count, the bound uses finitely many
smooth seminorms and a fixed polynomial in the separated norm-twist
heights.  Their orders, and the bound itself, are uniform over all
moving moduli, admissible masks, and frozen outer labels in the stated
ranges.
\end{lemma}

The application to the \(7/8\) bound sets \(\kappa=2\beta_*-1\), so
its zero-free hypothesis in this lemma holds by equality.  We retain this
dynamic value rather than replace it by the value \(5/6\) supplied by the
\(11/12\) bound, since the refined row count uses the resulting affine
capacity.  The proof below uses the
finite Gauss identities of Section~\ref{sec:finite-correlations},
Lemma~\ref{lem:smooth-calculus}, and Lemma~\ref{lem:logarithmic-control}.
Absorb any zero-length slots by absolute counting at their bounded
scales; henceforth \(z=0\) means that no live slot remains.  We first
make two reductions that preserve the row family.

The later induction is on the effective width \(M=m+q\), with all zero-slot
bands completed before the positive-slot bands.  Within each band the range
\(A\le5M/6\) is proved first.  This threshold comes from the
transformed rows whose inducing characters lie in \(\Theta\): counting
their sixth-power form and
bounding their plain products by volume leaves an excess
\(A-5M/6\), before the common-support savings.  Reflection handles
some longer inputs; the others are reduced to an equal-product-scale
difference, whose cancellation removes that excess.

\subsection{Fixed masks and reflection}

Call a character \emph{natural} when its zeros consist exactly of its
primitive conductor primes, its redundant row and declared moving
radical primes, and the fixed primes in \(\mathcal S\).  Here a
redundant prime is one at which the inducing character is unramified
but the displayed zero-extended product still vanishes.  A displayed
six-divisible power still vanishes on nonunits.  If a redundant factor
of the fixed twist is not included in the declared moving radical,
first factor it exactly into its remaining phase and a coprimality
indicator, and put that indicator in a squarefree ideal
\(\mathfrak R\), with \(q_{\mathfrak R}\le Z^B\), fixed throughout
the row sum.  In particular, a cancellation between the fixed twist
and the varying factor \(\chi_n(k)\) is not reclassified as a
separately chosen extra puncture for each row.

Let \(\psi_k^0\) be the natural character obtained by deleting these
additional punctures, and let \(T_k(X)\) denote its centrally
normalized plain sum at scale \(X\).  Write \(S_{k,\mathfrak R}(X)\)
for the sum with the extra mask.  Multiplicativity, including zero
extensions, gives the exact identities
\begin{align}
 S_{k,\mathfrak R}(X)
 &=\sum_{d\mid\mathfrak R}\mu(d)\psi_k^0(d)q_d^{-1/2}
                  T_k(X/q_d),
 \label{old-eq:2.1a}\\
 Q_{k,i,\mathfrak R}
 &=Q_{k,i}^{\,0}
   -P_i^{-1/2}\sum_{\substack{p\mid\mathfrak R\\p\ {\rm prime}}}
       \psi_k^0(p)\nu_i(p)W_i^{\rm slot}(q_p/P_i),
 \qquad P_i=Z^{z_i}.
 \label{eq:prime-mask-erasure}
\end{align}
For example, the first identity follows by inserting
\(\sum_{d\mid(n,\mathfrak R)}\mu(d)\) and writing \(n=dl\).
The quotient \(l\) is unrestricted at primes of \(d\); the original
natural zero extension remains on it.

Apply these identities simultaneously to both plain factors and all
slots.  If the frozen slots form \(J\subseteq\mathcal I\), and the
plain divisors are \(d_1,d_2\), the scalar depending on the row has
modulus at most one.  The remaining coefficient has absolute value at
most a fixed profile constant times
\[
 q_{d_1d_2}^{-1/2}\prod_{i\in J}P_i^{-1/2}.
\]
The resulting natural product has lengths
\[
 A'=A-\log_Zq_{d_1d_2}-\sum_{i\in J}z_i,\qquad
 z'=z-\sum_{i\in J}z_i.
\]
Its inducing character is unchanged.  If \(z'>0\), the left side of
Equation~\eqref{old-eq:3.9} has decreased; if \(z'=0\), the zero-slot
assertion has no length restriction and
\(\mathcal R_z\subseteq\mathcal R_0\).

For every fixed \(\epsilon_1>0\),
\begin{equation}
 \sum_{d\mid\mathfrak R}q_d^{-1/2}
 \le \prod_{p\mid\mathfrak R}(1-q_p^{-1/2})^{-1}
 \ll_{\epsilon_1,B} Z^{\epsilon_1}.
 \label{old-eq:2.1b}
\end{equation}
This follows from Lemma~\ref{lem:deleted-euler-factors}.  The corresponding
mass for a frozen slot is also \(Z^{\epsilon_1}\): on its annular support,
\(P_i^{-1/2}\ll q_p^{-1/2}\), and the same product bounds the sum over
\(p\mid\mathfrak R\).  The number of slots is fixed.  Minkowski's
inequality therefore reduces Equation~\eqref{old-eq:2.1} to the
natural assertion at the same width, with an arbitrarily small power
loss.  A nonempty annular scale below one is bounded below by a
positive profile-dependent constant and may be rescaled to one.
After this rescaling its new nonnegative length is
\(\max\{n_i-\log_Zq_{d_i},0\}\le n_i\), so the asserted
nonincrease of the affine expression remains valid.

We next record precisely the reflection used for natural characters.
Let \(\psi_k^*\) be the primitive character inducing \(\psi_k^0\),
and let \(\mathfrak R_{0,k}\) be its redundant natural radical.
The primes of \(\mathfrak R_{0,k}\) are disjoint from the primitive
conductor.  If \(Q_k\) is that conductor norm, set
\(C_k=3Q_k/(2\pi)^2\), the conductor scale in the functional
equation.  Tameness at primes outside \(\mathcal S\) gives
\begin{equation}
 C_kq_{\mathfrak R_{0,k}}
 \ll_{\mathcal S,\tau_{\rm fixed}} Z^M.
 \label{old-eq:2.1c}
\end{equation}
Indeed, each good prime contributes at most once, either to the
primitive conductor or to the redundant radical.  Their union is
contained in the union of the row radical and the declared moving
radical, whose norm is at most \(q_kZ^q\).  By
Lemma~\ref{lem:fixed-numerator-ray}, the unit and
\(\mathcal S\)-supported parts of a row, with valuations reduced
modulo six when evaluated on primary elements prime to \(\mathcal S\),
range over a fixed finite ray
family.  Thus all primes in \(\mathcal S\) contribute only a fixed
factor.

Apply the primitive Hecke functional equation recorded in the proof of
Lemma~\ref{lem:hecke-strip-growth} to \(\psi_k^*\), using the entireness of
\(L(s,\psi_k^*)\) stated there.  Let \(\varepsilon_k\) be its root number,
so \(|\varepsilon_k|=1\).  Mellin inversion and a contour shift show that
the normalized plain sum of \(\psi_k^*\) at \(X\) equals the root number times the conjugate
character sum at \(C_k/X\), with transformed profile \(W^\sharp\)
defined by
\[
 \mathcal M W^\sharp(s)
 =\mathcal M W(1-s)\frac{\Gamma(s)}{\Gamma(1-s)}.
\]
The Mellin transform convention is the one in
Lemma~\ref{lem:smooth-calculus}.  The quotient of gamma functions has
poles only at the nonpositive integers and zeros at the positive
integers.  Shifting the inverse Mellin contour to the left gives an
expansion in nonnegative integral powers at zero; shifting it to the
right gives arbitrary decay at infinity.  Thus \(W^\sharp\) and its
Euler derivatives are bounded at zero and rapidly decreasing at
infinity.  The height assertion is also quantitative.  For
\(W_\omega(y)=W(y)y^{i\omega}\), the exact identity
\(\mathcal M W_\omega(s)=\mathcal M W(s+i\omega)\) puts all
height dependence in a translate of the rapidly decreasing Mellin
transform.  On each fixed vertical line the gamma quotient and
each fixed number of Euler derivatives have polynomial growth in
the integration height.  Integrating the translated Mellin decay
therefore gives a fixed polynomial in \(1+|\omega|\), with its
degree depending only on the fixed contour and derivative orders.
The pointwise Mellin integration-by-parts estimate on a compact real
strip in Lemma~\ref{lem:smooth-calculus} justifies the horizontal
joins in these shifts; an integrated Fourier tail is not used to
bound a fixed horizontal trace.  This proves the required
finite-seminorm and polynomial-height bounds for reflection.

Deleting the redundant Euler factors before reflection and restoring
them geometrically afterward gives
\[
 T_k(X;W)
 =\varepsilon_k
 \sum_{\substack{d_0\mid\mathfrak R_{0,k}\\
                  \operatorname{rad}(h_0)\mid\mathfrak R_{0,k}}}
 \frac{\mu(d_0)\psi_k^*(d_0)\overline{\psi_k^*(h_0)}}
      {\sqrt{q_{d_0}q_{h_0}}}\,
 T_{\overline{\psi_k^0}}
   \left(\frac{C_kq_{d_0}}{Xq_{h_0}};W^\sharp\right),
 \qquad |\varepsilon_k|=1.
\]
The geometric series is absolutely bounded by
\(\prod_{p\mid\mathfrak R_{0,k}}(1-q_p^{-1/2})^{-1}\).
Consequently its total coefficient mass, together with the \(d_0\)
sum, is \(Z^{\epsilon_1}\).  The scales are
\begin{equation}
 Y=\frac{C_kq_{d_0}}{Xq_{h_0}},\qquad
 d_0\mid\mathfrak R_{0,k},\quad
 \operatorname{rad}(h_0)\mid\mathfrak R_{0,k}.
 \label{old-eq:2.1d}
\end{equation}
By Equation~\eqref{old-eq:2.1c}, there is a fixed \(C_{\mathcal A}\ge1\)
such that
\[
 C_kq_{\mathfrak R_{0,k}}\le C_{\mathcal A}Z^M,\qquad
 Y\le C_{\mathcal A}Z^{M-\log_ZX}.
\]

The profile \(W^\sharp\) can be partitioned into smooth annuli.
Below its main scale, central normalization gives summable
coefficients \(O(2^{-j/2})\) on the annuli of relative scale \(2^{-j}\).
Above that scale, rapid decay gives an arbitrary summable power.
Fix a small length tolerance \(\xi>0\).  Truncate the upper annuli
after an enlargement \(Z^{\xi/2}\); sufficiently many fixed derivatives
make the omitted part negligible by absolute counting.  A main scale
below \(Z^{-\xi}\) is likewise negligible.  If a retained annular
profile has fixed upper support endpoint \(B\), a scale \(S\) that
can contain a nonzero integral ideal satisfies \(SB\ge1\).  On such
a scale,
\[
 \max(S,1)\le\max(1,B)S.
\]
Thus replacing a retained subunit scale by scale one costs this fixed
multiplicative factor.  All these assertions are uniform in the moving labels.

The row-dependent choices \(d_0,h_0\) are handled by their coefficient
mass followed by a rowwise supremum in the resulting scales.
Lemma~\ref{lem:smooth-calculus} bounds a supremum over two
polynomial-range scales using \(O((\log Z)^2)\) unit boxes and
derivatives of the profiles.  The row character and its natural zeros
remain unchanged: after reflection, conjugating that whole factor
inside its absolute value replaces the conjugate character by the
original character and conjugates its profile.  This operation is
valid because the absolute value of a product is unchanged by
conjugating one factor.

Let \(C_{\rm ref}\ge1\) include \(C_{\mathcal A}\), the fixed annular
endpoint multipliers, the logarithmic box multipliers for a reflected
factor and any paired unreflected factor, and \(\max(1,B)\) for the
retained profile types.  This constant is fixed after the data and
profile types, independently of \(Z\) and the moving labels.  For
\(n=\log_ZX\), every retained scale satisfying the preceding support
test, after the permitted clipping and box enlargement, has declared
nonnegative length
\begin{equation}
 n_{\rm ref}\le M-n+\frac\xi2+\frac{\log C_{\rm ref}}{\log Z}
 \le M-n+\xi
 \qquad\left(\frac{\log C_{\rm ref}}{\log Z}\le\frac\xi2\right).
 \label{eq:centered-reflection-length}
\end{equation}
This is a linear bound also when \(M-n\) is slightly negative; if its
right side is negative, no such retained scale exists.  The smaller
upper-annulus range changes only the fixed decay, seminorm, and height
orders used for the discarded tail.

For \(z=0\), reflect each original factor whose length exceeds \(M/2\)
once, and leave the other factor unchanged.  Equation~\eqref{eq:centered-reflection-length}
bounds every retained reflected factor; the fixed box multiplier for
an unreflected factor is included in \(C_{\rm ref}\).  After the preceding
annular truncation and scale-box enlargement, the resulting lengths
satisfy
\begin{equation}
 n_i^*\le M/2+\xi,\qquad
 A^*=n_1^*+n_2^*\le M+2\xi\le M+\delta,
 \label{old-eq:2.1h}
\end{equation}
provided \(2\xi\le\delta\).  We call this the padded zero-slot core.
Here \(\delta\) is the positive padding parameter chosen below after the
width floor and step.
There is no repeated reflection at the boundary \(M/2\).  Reflection
and scale suprema are taken before centering; on a later centered term
whose primitive inducing character lies outside \(\Theta\), one first
applies the triangle inequality to its two rectangles.  No such supremum
is applied to a centered difference whose primitive inducing character
belongs to \(\Theta\).

\subsection{Prime estimates and the induction order}\label{sec:weighted}

For \(\kappa<1\), put \(s_\kappa=(1+\kappa)/2\).  Under the hypothesis
\(s_\kappa\ge\beta_*\), the global part of
Lemma~\ref{lem:logarithmic-control} gives, for every fixed \(e>0\),
\begin{equation}
 \frac{L'}{L}(s_\kappa+e+it,\psi)
 \ll_e \log\!\bigl(2Q_\psi(3+|t|)^2\bigr)
 \label{old-eq:3.4}
\end{equation}
for a primitive nonprincipal inducing character \(\psi\).
To estimate a prime annulus of scale \(P\), apply Mellin inversion to
a smooth von Mangoldt sum and move its contour to
\(\Re s=s_\kappa+e\).  There are no poles in the region of the shift,
and the Mellin transform has arbitrary decay, so
Equation~\eqref{old-eq:3.4} bounds the new integral by
\(P^{s_\kappa+e}\) times a logarithm of the conductor and a fixed
polynomial in the height.  Prime powers of exponent at least two
contribute \(O(P^{1/2+\epsilon_1})\).  For large \(P\), dividing the
annular weight by \(\log(Py)\) replaces the von Mangoldt weight by
the prime weight; its smooth seminorms are bounded on the annulus.
Bounded \(P\) are estimated by absolute counting.  Expand each
positive-length coefficient in the characters \(\vartheta_{ij}\)
from the statement.  If the primitive inducing row character
\(\psi\notin\Theta\) made \(\psi\vartheta_{ij}\) principal, then
\(\psi=\vartheta_{ij}^{-1}\in\Theta\), a contradiction.  Thus
every resulting slot character is nonprincipal on \(\mathcal R_z\).

After extra-mask erasure, fix a positive slot list and put
\(\boldsymbol W=(W_i^{\rm slot})_{i\in\mathcal I}\).  For normalized
twist heights \(\boldsymbol\omega=(\omega_i)_{i\in\mathcal I}\), define
\[
 Q_{\boldsymbol\omega}
 =\prod_{i\in\mathcal I}\left\{P_i^{-1/2}
   \sum_{p\ {\rm prime}}\psi_k^0(p)\nu_i(p)W_i^{\rm slot}(q_p/P_i)
                       (q_p/P_i)^{i\omega_i}\right\}.
\]
For every \(\epsilon_1>0\), central normalization and multiplication over this
fixed list, with \(e\) and all subsidiary losses sufficiently small,
give finite \(j,b,h\ge0\) and a constant \(C\) such that
\begin{equation}
 |Q_{\boldsymbol\omega}|^2
 \le C Z^{\kappa z+\epsilon_1}p_j(\boldsymbol W)^b
                 (1+|\boldsymbol\omega|)^h
 \qquad(k\in\mathcal R_z,\ z>0).
 \label{old-eq:3.5}
\end{equation}
Here \(j\) may be rounded up to an integer and \(2s_\kappa-1=\kappa\).
The orders and \(C\) may depend on the fixed data, slot count, loss,
and any fixed requested internal logarithmic or normalized-twist
derivatives, but not on \(Z\), moving labels, rows, or the numerical
heights.  The exponent \(\kappa z+\epsilon_1\) is independent of
these derivative and height orders.  Indeed, translate each pure twist
in its Mellin variable before integration by parts.  The weighted
integral of the untwisted Mellin transform then contributes a finite
seminorm and a fixed polynomial in \(\boldsymbol\omega\), while the
contour displacement contributes exactly \(2ez\) to the squared
exponent.  Logarithmic derivatives remain annular, and normalized-twist
derivatives insert only powers of the logarithmic profile variable.
The redundant natural radical contains
only \(O(\log Z)\) primes, so deleting it changes a normalized slot
by \(O(P_i^{-1/2}\log Z)\) times its fixed profile factor; this is
within the stated power loss because \(P_i\ge1\).  For \(\kappa=1\),
absolute prime counting gives Equation~\eqref{old-eq:3.5} without a
zero-free assumption and with \(h=0\) for undifferentiated pure twists,
whose modulus is one.  For \(\kappa<1\) the hypothesis remains
\(\beta_*\le(1+\kappa)/2\).  The choice of losses is uniform in
\(\kappa\in[3/4,1]\), since the length ranges are bounded.  In every
later use of Equation~\eqref{old-eq:3.5}, its fixed seminorm and height
factor is retained in the weighted Fourier estimates; it is not included
in the exponent of \(Z\).

We now specify the induction, including the estimates at its smallest
widths.  Choose a width floor \(\rho>0\), a width step \(\sigma>0\),
and then \(\delta>0\), all small in terms of the final
\(\epsilon\), with \(\delta\ll\min(\rho,\sigma)\).  Precise loss
choices will be made after the depth is bounded.  Divide the bounded
range of \(M\) into consecutive bands of length \(\sigma/4\).
Prove all zero-slot bands first, in increasing order of width, and
then all positive-slot bands.  Within a zero-slot band, first prove
the uncentered assertion for \(A\le5M/6\), and then the padded core
in Equation~\eqref{old-eq:2.1h}.  Reflection then supplies all
zero-slot lengths in that band.  Within a positive-slot band, first
prove the uncentered assertion for \(A\le5M/6\) subject to
Equation~\eqref{old-eq:3.9}, and then the remaining part of that
region by centering.  A completed earlier band therefore includes
the unrestricted zero-slot assertion.  Let \(M_{\max}\) bound the
initial width range, and define
\[
 D=2+\left\lceil 2M_{\max}/\sigma\right\rceil.
\]
The strict width decrease proved below will show that at most \(D-2\)
nonterminal calls occur on a branch.

At \(M\le\rho\), the padded zero-slot core follows from absolute
counting: there are \(O(Z^m)\) rows and the squared product is
\(O(Z^{A+\epsilon_1})\).  Its exponent above \(M\) is at most
\(A-q\le\rho+\delta\).  For positive slots, combine the completed
zero-slot estimate with Equation~\eqref{old-eq:3.5}.  In the region
of Equation~\eqref{old-eq:3.9}, \(A\ge z\), whence
\begin{equation}
 z\le\frac{M}{6\kappa}\le\frac{2M}{9},
 \qquad \kappa z\le\frac M6.
 \label{old-eq:3.6}
\end{equation}
Thus the extra terminal exponent for positive slots is at most
\(\rho/6\).  These terminal exponents can be made smaller than the
reserved final loss by choosing \(\rho,\delta\) sufficiently small.

For a width above the floor, the two-transform estimate below will
first prove the uncentered range \(A\le5M/6\).  We explain now why
the remaining inputs can be compared with that range.  Set
\begin{equation}
 L=M/4.
 \label{old-eq:2.2}
\end{equation}
For \(A>5M/6\), the comparison lengths \(L,A-z-L\) are both at
least \(L\).  In the positive-slot case,
\((6\kappa-1)z\le M-A<M/6\), and \(6\kappa-1\ge7/2\); hence
\(z<M/21<M/20\).  This also shows \(A-z>2L\).

If one original plain length is \(b<L\), reflect the other factor.
If both are at least \(L\), introduce the comparison with lengths
\(L,A-z-L\) and the same two profiles; reflect its longer factor
only when estimating the comparison separately.  In both cases the
total length after that reflection is at most
\[
 b+\{M-(A-z-b)\}+z+\xi
 \le 3M/2-A+2z+\xi,
\]
where \(b\le L\).  Equation~\eqref{eq:centered-reflection-length}
includes the clipped reflected scale and the paired unreflected box
multiplier in this same \(\xi\); the discarded tails have the stated
fixed-order bounds.  Thus
\begin{equation}
 A_{\rm comp}\le3M/2-A+2z+\xi.
 \label{old-eq:3.11}
\end{equation}
For \(z=0\), this gives
\(5M/6-A_{\rm comp}\ge M/6-\xi\).  For \(z>0\), use
\((6\kappa-1)z\le M-A\), \(A>5M/6\), and \(z<M/20\) to get
\begin{equation}
 \begin{aligned}
 A_{\rm comp}&\le\frac{23}{30}M+\xi,\\
 A_{\rm comp}+(6\kappa-1)z
 &\le\frac52M-2A+2z+\xi\le\frac{14}{15}M+\xi.
 \end{aligned}
 \label{eq:comparison-uncentered-margin}
\end{equation}
Each required boundary has a margin of at least \(M/15\) before
the \(\xi\) term.  Choose \(\xi\le\rho/30\).  After the threshold in
Equation~\eqref{eq:centered-reflection-length}, the unit-box multipliers
are already included in \(\xi\), so every retained reflected
comparison calls the previously proved uncentered assertion at this
same width.  This also completes the original case with a factor
shorter than \(L\).

In the remaining case put \(X_i=Z^{n_i}\),
\(Y_1=Z^L\), and \(Y_2=X_1X_2/Y_1\).  Then
\(X_1X_2=Y_1Y_2\), and all four plain lengths are at least \(L\).
Subtract the unreflected comparison from the original product,
leaving the common product \(Q\) of slots; call the result
\(\Delta_k\).  Multiplicativity and the equal product normalization
give the explicit formula
\[
 \begin{split}
 \Delta_k
 =\frac{Q}{\sqrt{X_1X_2}}\sum_{l_1,l_2}\psi_k(l_1l_2)
 \bigl\{
 &W_1(q_{l_1}/X_1)W_2(q_{l_2}/X_2)\\
 -{}&W_1(q_{l_1}/Y_1)W_2(q_{l_2}/Y_2)
 \bigr\}.
 \end{split}
\]
The comparison is bounded on the original permissible
rows.  The squared norm of the whole \(\Delta_k\) is nonnegative,
so it can then be enlarged to all rows in the smooth row ball.  In
the uncentered case, enlarge the squared norm of the original
product instead.  The Poisson transforms below are therefore never
applied to an indicator selecting exceptional or nonexceptional rows.

\subsection{The centered coefficient and its support}

We keep the subtraction as one coefficient while transforming its row
norm.  For fixed ideals
\(\mathbf b=(\mathfrak b_1,\mathfrak b_2)\), define
\begin{equation}
 D_{\mathbf b}(l_1,l_2)
 =
 \prod_{i=1}^2W_i(q_{\mathfrak b_i}q_{l_i}/X_i)
 -
 \prod_{i=1}^2W_i(q_{\mathfrak b_i}q_{l_i}/Y_i),
 \qquad X_1X_2=Y_1Y_2.
 \label{old-eq:2.18a}
\end{equation}
An allocation \(\mathbf b\) records prime powers already extracted
from the two plain variables.  If its conditions are impossible for
one rectangle, that rectangle's profile is zero on the corresponding
sum; the formal difference in Equation~\eqref{old-eq:2.18a} is
nevertheless retained.

The reason for retaining these common data is already visible in the
exceptional case.  For a fixed \(\vartheta\in\Theta\), a common mask
\(\mathfrak R_*\), and a common norm power \(q_l^{it}\), the lattice
estimate proved below has leading term
\[
 \sum_l\vartheta(l)1_{(l,\mathfrak R_*)=1}q_l^{it}W_i(q_l/T)
 =c_{\vartheta,\mathfrak R_*}T^{1+it}I_i(t)+\text{error},
\]
where \(c_{\vartheta,\mathfrak R_*}\) does not depend on \(T\) or \(t\).
Here \(I_i(t)\) is a fixed measure constant times
\(\int_0^\infty W_i(y)y^{it}\,dy\).
The product main term is therefore
\(c_{\vartheta,\mathfrak R_*}^2T_1^{1+it}T_2^{1+it}I_1(t)I_2(t)\),
which agrees for the two rectangles when \(T_1T_2\) agrees.
Lemma~\ref{lem:centered-lattice-cancellation} will quantify the remaining
error.  This cancellation is used only on the transformed rows inducing
characters in \(\Theta\); the common coefficient below is retained on
all rows until that later division into cases.

For a remaining slot set \(\mathcal I'\), write
\(\Pi=\prod_{i\in\mathcal I'}p_i\).  A coefficient with a common
character and mask means a coefficient of the form
\begin{equation}
 \begin{split}
 &\left(\prod_{i\in\mathcal I'}
       \nu_i(p_i)W_i^{\rm slot}(q_{p_i}/P_i)\right)
 D_{\mathbf b}(l_1,l_2)\,
 \tau_1(u)q_u^{it}
 1_{(u,\mathfrak R)=1}1_{s\mid u},\\
 &\hspace{40mm}u=\Pi l_1l_2.
 \end{split}
 \label{eq:centered-coefficient-form}
\end{equation}
Here \(\tau_1\) is one zero-extended product of a fixed finite-ray
character and moving residue-symbol factors; \(\mathfrak R\) is a
fixed squarefree extra mask; \(s\) is a fixed squarefree ideal; and
\(t\in\mathbb R\).  The condition \(s\mid u\) is omitted when
\(s=1\).  The slots retain their original disjoint underlying
supports.  They may share primes with either plain variable.

These common data factor multiplicatively on the \emph{full} product.
For any ideals \(v_1,\ldots,v_r\), even with common primes,
\[
 \chi_{\prod v_i}(h)=\prod_i\chi_{v_i}(h),\quad
 q_{\prod v_i}^{it}=\prod_iq_{v_i}^{it},\quad
 1_{(\prod v_i,\mathfrak R)=1}
   =\prod_i1_{(v_i,\mathfrak R)=1}.
\]
The first identity includes all zeros: a present prime whose total
exponent is divisible by six gives the zero-extended principal
factor, not the constant one.  The same multiplicativity holds for
\(\tau_1\).  A fixed-ray character evaluated on a full product also
factors with the same character on every variable.

We fix a support convention that will also control the slot-mesh
quantifier.  Let \(N\) be the original fixed number of slots, and
choose fixed intervals \([a_i,b_i]\) containing the support of each
\(W_i^{\rm slot}\).  Put
\(h_i=\max\{|\log a_i|,|\log b_i|\}\).  Choose a fixed number
\(H_N\) at least \(\sum_i h_i\), enlarged to include the two plain
profile windows, \(\log C_{\rm ref}\), fixed arithmetic normalizations, the relative dyadic
boxes, and the finitely many support enlargements through the \(D\)
operation stages.  The number \(H_N\) may depend on all the fixed
data and on \(N\), but not on \(Z\) or any moving label.  For every
subset \(I\) of live or frozen slots,
\begin{equation}
 \left|\log\prod_{i\in I}\frac{q_{p_i}}{P_i}\right|
 \le\sum_{i\in I}h_i\le H_N,
 \qquad \theta_N:=\frac{H_N}{\log Z}.
 \label{eq:centered-aggregate-support}
\end{equation}
Every residual full product divided by its nominal product scale lies
in \(\exp([-H_N,H_N])\), after increasing \(H_N\) once for the
fixed list of operations.  Both rectangles use that same box.  Extracted
plain prime powers use their exact norms, and a frozen slot contributes
its ratio \(q_p/P_i\) just once.  If a nonempty plain scale below one
is clipped to one, its error is at most the logarithm of that plain
window's fixed endpoint divided by \(\log Z\), once for that plain.
Thus a whole subset of slots or a whole divisor extraction contributes
one aggregate \(O(\theta_N)\) boundary error, not one copy of a
preselected tolerance for each slot or prime.

\subsection{The first Poisson transform and its target bound}\label{sec:plain-first-transform}

We now estimate either the uncentered product or the centered
difference selected above.  Put \(H=Z^m\) and \(X=Z^A\).  Choose a
fixed nonnegative smooth radial function that majorizes the row ball.
The full index product in either rectangle has norm in a fixed
multiple of \(X\).  All comparisons of exponents in this subsection
are first made on fixed dyadic norm intervals.  Their bounded
relative widths change a logarithmic length by \(O(1/\log Z)\);
the final loss discussion includes these changes.

We use the following common localization for both Poisson formulas.
Fix a nonnegative smooth dyadic partition
\(\sum_\lambda\omega_\lambda(q/T_\lambda)=1\) for \(q>0\),
with each weight supported in \(C_d^{-1}\le q/T_\lambda\le1\)
for one fixed \(C_d\).  A
sector fixes dyadic boxes for the finite list of aggregate outer norms,
not a separate box for each slot.  Let \(T_{\rm sec}\) be the
supremum in that sector of the nominal frequency scale.  Its ratio to
the scale at any one set of outer labels is at most a fixed
\(C_{\rm sec}\).  Retain exactly the whole weights whose support
meets
\[
 0<q\le T_{\rm sec}Z^{\xi/2}.
\]
Their union is contained in \(q\le C_dT_{\rm sec}Z^{\xi/2}\);
every discarded weight is supported above \(T_{\rm sec}Z^{\xi/2}\).
The selection depends only on the sector, never on a live column.
After \(N\) and the support data are fixed, take \(Z\) large enough
that
\begin{equation}
 2\theta_N+\frac{\log(C_dC_{\rm sec})}{\log Z}<\frac\xi4.
 \label{eq:centered-localization-threshold}
\end{equation}
One fixed enlargement of \(C_{\rm sec}\) covers all the finitely many
sector endpoint factors.

Here is a direct tail bound that justifies this operation on the
genuine sums.  In both applications the kernel argument is at least
\(q/(e^{2H_N}T_{\rm sec})\).  On every discarded weight it is
therefore at least \(Z^{\xi/4}\), by
Equation~\eqref{eq:centered-localization-threshold}.
Lemma~\ref{lem:kernel-seminorms} gives arbitrary decay
\((1+\text{argument})^{-B}\).  For the first formula use
\(|G(a,h)|\le q_a^{1/2}\), and for the second use the finite
definition \(|F(u,v;j)|\le q_uq_v\).  Absolute ideal counting and
the divisor bounds for the fixed number of factors bound all raw
columns and prefactors in a sector by \(C_NZ^{B_0}\) times a fixed
polynomial in the retained heights, where \(B_0\) depends only on
the bounded total lengths and chosen subsidiary power shares.  A
lattice norm dyad of scale \(T\) has \(O(1+T)\) frequencies.
Summing the radial decay over discarded dyads consequently gives
\(C_{N,B}Z^{B_1-B\xi/4}\) times that height polynomial, for a
bounded \(B_1\) independent of \(B\) and \(N\), and a convergent
geometric sum when \(B>1\).
Choose \(B\) sufficiently large to obtain any prescribed power
saving, including the polynomial number of frozen outer labels.
This is an absolute bound for the original terms, before any
off-coprime extension or Fourier absolutization.  It uses neither
the later \(s\)-radical saving nor a factorized off-coprime identity.
On the retained weights the full smooth kernel is kept; no sharp
condition comparing a row norm with the two live column norms is
inserted.  A retained range below the first nonzero lattice norm is
empty, apart from a bounded boundary dyad covered by the clipping
convention below.

At zero frequency in the first row Poisson formula, a character mean
can be nonzero only if its exponent at every prime is zero modulo
six.  In particular, no prime occurs to total multiplicity one in
the product of the two full index products.  This product is
therefore a powerful ideal.  There are \(O_\epsilon(X^{1+\epsilon})\)
such products of norm \(O(X^2)\): each powerful ideal is a square
times a cube of a squarefree ideal, and summing over the latter gives
the usual \(O(Y^{1/2+\epsilon})\) bound up to norm \(Y\).
Allocations to the fixed number of factors are divisor-bounded.
The central factor is \(X^{-1}\), and the row mean has size at most
\(H\).  Thus the zero frequency is \(O(Z^{m+\epsilon_1})\).

For the nonzero frequencies write the two full products as \(Ca,Db\),
where \(C,D\) contain their complete common prime support and
\[
 (a,b)=1,\qquad (ab,CD)=1.
\]
At each common prime, fix its exact
valuation in each plain variable and whether it is supplied by a
slot.  Dividing out these valuations adds that prime to the mask of
\emph{every} remaining factor on the side.  A slot that supplied the
prime is frozen and removed.  This description is valid even when a
slot and one or both plain variables supplied that prime.  The
allocation is made once for the coefficient, so the same ideals
\(\mathfrak b_i\) occur in the two terms of \(D_{\mathbf b}\).
An impossible allocation is a zero term.  In particular, a scalar
forced to vanish by an old moving zero or a common mask is not
replaced by its absolute upper bound before these support conditions
have been imposed.

Let \(c,d\) be the logarithmic norms of \(C,D\), and let \(p\) be the logarithmic norm of
their common radical.  Let \(\mathfrak r\) be the product of common
primes whose net exponents are nonzero modulo six; its logarithmic
norm is \(R\).  The corresponding character
\(\xi_{\mathfrak r}\) is primitive modulo \(\mathfrak r\).
In the M\"obius expansion of the complementary common row mask,
write \(\mathfrak e\) for the selected divisor and \(E\) for its
logarithmic norm.  In particular
\[
 R\le p,\qquad E\le p-R.
\]

Let \(A_C(a)\) and \(A_D(b)\) be the allocated convolution
coefficients, with the common character \(\tau\) omitted.  They
include all profiles, live slot coefficients, and fixed masks; in
the centered case each contains the entire allocated difference.
Set
\[
 \tau_C(a)=\tau(a)\chi_a(\mathfrak e\mathfrak r)
                       \xi_{\mathfrak r}(a),\qquad
 \tau_D(b)=\tau(b)\chi_b(\mathfrak e\mathfrak r)
                       \overline{\xi_{\mathfrak r}(b)}.
\]
Residue-class Poisson, with the self-dual lattice measure from
Section~\ref{sec:arithmetic}, gives for this allocation the following
nonzero-frequency expression, up to a scalar of bounded modulus in
the frozen labels:
\begin{equation}
 \begin{split}
 \frac{H}{Xq_{\mathfrak e}\sqrt{q_{\mathfrak r}}}
 \sum_{h\ne0}G_{\xi_{\mathfrak r}}(\mathfrak r,h)
 \sum_{(a,b)=1}
 &\frac{A_C(a)\overline{A_D(b)}}{\sqrt{q_aq_b}}\,
 \tau_C(a)\overline{\tau_D(b)}
 \overline{\mathcal R(a,b)}\,
 G(a,h)\overline{G(b,-h)}
 \\
 &\quad\cdot
 \widehat\Phi_1\left(
    \frac{Hq_h}{q_{\mathfrak e}q_{\mathfrak r}q_aq_b}
                    \right).
 \end{split}
 \label{eq:first-poisson-bridge}
\end{equation}
Here \(G_{\xi_{\mathfrak r}}\) is the normalized primitive Gauss
sum and has modulus at most one.  To check the normalization, the
substitution \(k=\mathfrak e k'\) and Poisson modulo
\(\mathfrak r ab\) give \(H/(q_{\mathfrak e}q_{\mathfrak r}q_aq_b)\).
The three unnormalized Gauss sums restore
\(\sqrt{q_{\mathfrak r}q_aq_b}\).  The global squared central
normalization is \(X^{-1}\).  For the phase, CRT for the pairwise coprime moduli
\(a,b,\mathfrak r\) supplies
\[
 \chi_a(b\mathfrak r)\overline{\chi_b(a\mathfrak r)}
       \xi_{\mathfrak r}(ab).
\]
The substitution \(k=\mathfrak e k'\) supplies
\(\chi_a(\mathfrak e)\overline{\chi_b(\mathfrak e)}\), up to a
scalar in the frozen labels.  Reciprocity changes
\(\chi_a(b)\overline{\chi_b(a)}\) into
\(\overline{\mathcal R(a,b)}\); the remaining factors are precisely
\(\tau_C(a)\overline{\tau_D(b)}\).  This proves
Equation~\eqref{eq:first-poisson-bridge}.  Each new character acts on
its whole residual product.  Its new moving primes belong to the
extracted support and puncture every residual factor.  Their full displayed
union is counted even if the factors at \(\mathfrak r\) cancel on units.

The new full displayed moving support has logarithmic norm at most
\(\widetilde q=q+R+E\), including any canceled factors at
\(\mathfrak r\).  Its nominal frequency scale is exactly
\[
 T_1(C,D,\mathfrak e,\mathfrak r)
 =\frac{q_{\mathfrak e}q_{\mathfrak r}X^2}
        {Hq_Cq_D}=Z^{K_0},
 \qquad K_0=2A-c-d+R+E-m.
\]
The product support convention gives
\(q_a/(X/q_C),q_b/(X/q_D)\in\exp([-H_N,H_N])\), so the
argument in Equation~\eqref{eq:first-poisson-bridge} is at least
\(q_h/(e^{2H_N}T_{\rm sec})\) for the supremum of \(T_1\)
in the outer sector.  Apply the preceding whole-dyad localization
to this genuine coprime bridge.  If \(K=\log_ZT_\lambda\) is
the upper length of a retained dyad, then
\begin{equation}
 \widetilde q=q+R+E,\qquad
 K\le K_0+\delta_{{\rm fr},1},\qquad
 0\le\delta_{{\rm fr},1}:=\frac\xi2+
       \frac{\log(C_dC_{\rm sec})}{\log Z}<\xi.
 \label{old-eq:2.3}
\end{equation}
The zero frequency already estimated above was the zero term of the
original common smooth row ball; it is not restored on a truncated
row domain.  The later ledgers retain the possible inequality
\(K_0-K\ge-\delta_{{\rm fr},1}\).

For each retained dyad and genuine common-support allocation, define
\(\mathcal C_1(a,b;h)\) to be the full bridge summand also on noncoprime
residual pairs individually disjoint from \(CD\) and allowed by the old
masks.  Use the fixed bicharacter \(\mathcal R(a,b)\), the full
zero-extended \(G(a,h),G(b,-h)\), the displayed whole-product characters,
and the same dyad weight, kernel, inverse roots, and formal product norms.
CRT identifies this definition with the genuine summand only on
\((a,b)=1\).  On each finite column shell the exact identity is
\[
 \sum_{(a,b)=1}\mathcal C_1(a,b;h)
 =\sum_{a,b}\mathcal C_1(a,b;h)\sum_{s\mid a,b}\mu(s).
\]
The full squarefree M\"obius sum annihilates the artificial noncoprime
pairs.  Insert it before estimating independent factors, and put
\(s_0=\log_Zq_s\).
Separate the fixed-ray phases and all smooth factors in the normalized
row norm and the two whole-product norms.  In these variables the
kernel is \(\widehat\Phi_1(R_{\rm sc}x/(y_1y_2))\) on fixed logarithmic
boxes.  If its aggregate scale ratio \(R_{\rm sc}\) varies over the sector,
include that single normalized outer ratio as another coordinate.
The cutoffs are chosen on the common product annulus of
Equation~\eqref{eq:centered-aggregate-support}, before fixing live slot
labels.  Thus one Fourier coefficient measure is common to the rows and
all live labels.  It supplies a row phase and only one norm power on each
whole column, together with phases in frozen outer norms.  It introduces
no separate powers on the two plain variables or the two rectangles.
The full kernel and inverse roots are kept until this separation;
taking their absolute supremum inside \(D_{\mathbf b}\) would not
preserve the coefficient.  Apply Cauchy--Schwarz in \(h\) only after this separation,
and only then use \(|G_{\xi_{\mathfrak r}}|\le1\).  The resulting positive norm
on the \(C\) side is
\[
 \mathcal N_C=
 \sum_{h\ne0}\omega_K(q_h/Z^K)
 \left|Z^{-(A-c)/2}
   \sum_{s\mid a}A_C(a)\tau_C'(a)q_a^{it}G(a,h)\right|^2,
 \qquad \omega_K\ge0,
\]
and there is an analogous \(D\) norm.  Here \(\tau_C'\) includes
one separated fixed-ray character.  All fixed masks in \(A_C\)
remain present.  Complete extraction and multiplicativity therefore leave
the coefficient in Equation~\eqref{eq:centered-coefficient-form}, now
multiplied by \(G(a,h)\).  In particular the two rectangles retain the
same allocated plain powers, character, puncture, and norm power.

The factors outside these two norms have exponent
\(m-A-R/2-E\).  The absolute number of common-support labels and
M\"obius labels has exponent \(p+s_0+\epsilon_1\).  To justify
the common-support count uniformly, first count its radical, giving
\(O(Z^{p+\epsilon_1})\).  For a fixed radical \(\mathfrak c\) of
polynomial norm and any fixed \(T\), Rankin's bound gives
\[
 \#\{v:\operatorname{rad}(v)\mid\mathfrak c,\ q_v\le Z^T\}
 \le Z^{aT}\prod_{\substack{r\mid\mathfrak c\\r\ {\rm prime}}}
             (1-q_r^{-a})^{-1}
 \ll Z^{aT+\epsilon_1}
\]
for every fixed \(a>0\).  Choose \(a\) small and use the
polynomial-size Euler-product estimate.  This bounds the choices of
the powers in \(C,D\) by an arbitrarily small power.  Their
allocations and the choice of \(\mathfrak e\) have divisor-bounded
multiplicity.  Finally there are \(O(Z^{s_0+\epsilon_1})\)
possible \(s\).

The allowance \(\epsilon_G\) below denotes the error envelope for the
current induction depth, together with its local small-power shares.
These envelopes will be chosen compatibly when the finite induction is
completed, using the same slot mesh throughout.
We will prove the following sufficient bound for the squared \(C\)
norm:
\begin{equation}
 \mathcal N_C\ll
 Z^{A-c+\widetilde q+\mathcal B_c-s_0+\epsilon_G},
 \qquad
 \mathcal B_c=\max\{0,(3c-5d-R)/6\}.
 \label{old-eq:2.5}
\end{equation}
The analogous quantity is
\(\mathcal B_d=\max\{0,(3d-5c-R)/6\}\).
The complete exponent ledger for this implication is
\[
 \begin{aligned}
 &(m-A-R/2-E)+(p+s_0)\\
 &\quad+\tfrac12\{A-c+\widetilde q+\mathcal B_c-s_0
                  +A-d+\widetilde q+\mathcal B_d-s_0\}\\
 &=M+\tfrac12\{\mathcal B_c+\mathcal B_d-(c+d-2p-R)\}.
 \end{aligned}
\]
The last brace is nonpositive:
\begin{equation}
 \mathcal B_c+\mathcal B_d\le c+d-2p-R.
 \label{old-eq:2.6}
\end{equation}
Indeed, at a common prime of multiplicities \(i\ge j\ge1\), put
\(r=1_{6\nmid i-j}\).  Only the \(i\) side can have a positive
local numerator.  Its contribution is at most
\((3i-5j-r)_+/6\), whereas the right side contributes
\(i+j-2-r\).  The latter is nonnegative.  If the former is
positive, six times their difference is
\(3i+11j-12-5r\), which is nonnegative: for \(r=0\) it is at
least \(2\), and for \(r=1\) one has \(i\ge j+1\), giving at
least \(0\).  The positive part of a sum is at most the sum of
positive parts.  Multiplication by each prime's logarithmic norm
and summation proves Equation~\eqref{old-eq:2.6}.  Thus it remains
to establish Equation~\eqref{old-eq:2.5}.

\subsection{Enlarging the Gauss-row norm}\label{sec:plain-gauss-enlargement}

This subsection describes three possible positive norms to which
the second transform will be applied.  Let \(w\) denote a length
removed from the \(C\) column, and let \(w_o\le w\) be the
additional moving-radical length created by that removal.  Put
\(a_0=A-c-w\).  When a squared extraction coefficient of size
\(Z^{-w_o}\) has been removed, Equation~\eqref{old-eq:2.5}
allows the unweighted remaining norm the exponent
\begin{equation}
 \Lambda_c=
 a_0+\widetilde q+w_o+w+\mathcal B_c-s_0+\epsilon_G.
 \label{old-eq:2.7}
\end{equation}
This is just \(A-c+\widetilde q+\mathcal B_c-s_0+\epsilon_G+w_o\).

For the initial norm, with \(w=w_o=0\) and \(a_0=A-c\), the first
transform gives the following conditional summand after fixing \(\Pi\)
and omitting its outer scalar:
\begin{equation}
 Z^{-a_0/2}\sum_{l_1,l_2}
 D_{\mathbf b}(l_1,l_2)\tau_1(l_1l_2)q_{l_1l_2}^{it}
 1_{(\Pi l_1l_2,\mathfrak R)=1}1_{s\mid\Pi l_1l_2}
 G(\Pi l_1l_2,h).
 \label{old-eq:2.18c}
\end{equation}
The slot sums are restored before this polynomial is squared.
The same statement with the second rectangle omitted applies to
uncentered products.  The local calculations below will show that the
extracted terms retain this form at their shortened length \(a_0=A-c-w\).

Define
\begin{equation}
 J=d-c+(K_0-K)-2w+w_o,\qquad J_+=\max(J,0).
 \label{old-eq:2.8}
\end{equation}

In the zero-slot proof take \(w=w_o=0\), and set
\[
 \ell=0,\qquad g=J_++\sigma.
\]
The positive norm \(\mathcal N_C\) is at most the corresponding
norm over a smooth ball of length \(K+g\).  There is no
multiplication of rows in this case.

For positive slots, start with \(w=w_o=0\).  Set
\(\ell_*=\sigma/3\), choose the slot mesh \(\eta<\sigma/6\),
and take the fixed pool
\[
 \mathcal P=\{p: Z^{\ell_*}/2<q_p\le Z^{\ell_*},\
                    p\notin\mathcal S\}.
\]
The prime ideal theorem in the fixed field, equivalently its fixed
ray-class form \cite[Theorem 1.1]{TZ}, gives
\(|\mathcal P|=Z^{\ell_*+o(1)}\).  Because the live slot lengths
are at most \(\eta\) and their relative annular supports are fixed,
the single inequality
\(Z^{\ell_*-\eta}>2\max_i b_i\) makes this pool disjoint from
every live slot window.  Its exponent gap is at least \(\sigma/6\),
independently of \(N\); only the threshold depends on the fixed
windows.  For a row \(h\ne0\), omit pool primes dividing
\(h\), \(s\), or any frozen support.  Each such integer or ideal
has polynomial norm, so \(O(\log Z)\) primes are omitted.
The remaining set \(\mathcal P_h\) has size comparable to
\(|\mathcal P|\), uniformly in the row and frozen labels.

To compare the rows \(h\) and \(hp^6\), write a full modulus as \(p^iu\) with
\((p,u)=1\).  CRT and reciprocity give the exact formula
\[
 G(p^iu,h)
 =\mathcal R(p,u)^i\chi_p(u)^{2i}G(p^i,h)G(u,h).
\]
Also \(G(u,hp^6)=G(u,h)\), since \(p\) is a unit modulo \(u\)
and \(\chi_u(p^6)=1\).
For \(p\nmid h\), Equation~\eqref{eq:gauss-local} shows that
replacing \(h\) by \(hp^6\) changes only \(i=1,6,7\).
The factors \(G(p^i,h)\) that occur are a scalar in the row and
\(p\); the displayed remaining factor is one fixed-ray character
times one residue character on the whole \(u\).  Since \(p\) is
outside every live slot window, \(p^i\) is allocated only to the two
plain variables.  The residual product is punctured at \(p\), and
the same allocation updates \(D_{\mathbf b}\) in both rectangles.
Because \(p\nmid s\), the condition \(s\mid u\) remains.

Here is the precise averaging argument.  Write the normalized
polynomial in \(\mathcal N_C\) as \(\mathcal H(h)\).  For each
eligible \(p\), local Gauss evaluation gives
\(\mathcal H(h)=\mathcal H(hp^6)\) plus the extracted terms with
\(p\)-adic column valuations \(1,6,7\).  There are only boundedly
many allocations of each of these valuations to the two plain
variables.  Jensen's inequality, first in \(p\) and then for this
fixed finite sum, gives
\[
 |\mathcal H(h)|^2\ll
 \frac1{|\mathcal P_h|}\sum_{p\in\mathcal P_h}
 \left\{|\mathcal H(hp^6)|^2+
        \sum_{i=1,6,7}\sum_{\rm allocations}
        |c_{p,i}(h)|^2|\mathcal H_{p,i}(h)|^2\right\}.
\]
The \(\mathcal H_{p,i}\) are normalized at their shortened column
scales.  The local identity and allocation just described are applied to the
whole centered coefficient, so each error retains
Equation~\eqref{eq:centered-coefficient-form}.

On the support of the original row weight, the new row \(hp^6\)
has norm \(O(Z^{K+6\ell_*})\).  A fixed output row has only
boundedly many representations as \(hp^6\) with \(p\in\mathcal P_h\):
every such \(p\) divides that row, and all such primes have norm at
least \(Z^{\ell_*}/2\), while the output norm has bounded
logarithmic length.  Summing the first term over \(h\) therefore
gives the factor \(|\mathcal P|^{-1}=Z^{-\ell_*+o(1)}\)
times a positive norm on the new rows.  For this main term set
\[
 \ell=\ell_*,\qquad g=J_++2\sigma.
\]
Since \(6\ell_*=2\sigma\), a smooth ball of length \(K+g\)
contains all the new rows.

For the extracted terms put \(P=q_p\).  When \(p\nmid h\),
Equation~\eqref{eq:gauss-local} gives the following central
coefficients.  At valuation one, the old Gauss sum has modulus one
and the new one is zero, so extraction gives squared coefficient
\(P^{-1}\).  At valuation six, the new Gauss sum is
\(P^3(1-P^{-1})\); its central factor is \(P^{-3}\), giving
\((1-P^{-1})^2\).  At valuation seven, the new Gauss sum has
modulus \(P^3\), and the central factor \(P^{-7/2}\) gives
\(P^{-1}\).  The row phase is \(\overline{\chi_p(h)}\) for
valuations one and seven, and is one for valuation six.  It
multiplies both rectangles.

Let \(\ell_p=\log_ZP\).  For sufficiently large \(Z\),
\(\ell_*/2\le\ell_p\le\ell_*\).  The removal and new
moving-radical lengths for the three terms are
\begin{equation}
 (w,w_o)=(i\ell_p,e_i\ell_p),\qquad
 i=1,6,7,\quad e_1=e_7=1,\quad e_6=0.
 \label{old-eq:2.9}
\end{equation}
The two squared factors \(P^{-1}\) give exactly \(Z^{-w_o}\).
For the valuation-six term, the exact factor
\(\chi_p(u)^{12}=1_{(u,p)=1}\) is represented by the already
common fixed puncture at \(p\), with its fixed-ray phase \(\mathcal R(p,u)^6\) retained.
At valuations one and seven the displayed local factor remains, and its
prime is counted in the moving radical.  In each
error, freeze \(p\), retain its common
column puncture, and discard its row eligibility restriction only
after taking the positive norm.  The average remains
\(|\mathcal P|^{-1}\sum_{p\in\mathcal P}\), so a bound uniform in
the frozen \(p\) introduces no prime-count factor.  For each error
set
\[
 \ell=0,\qquad g=J_++\sigma
\]
and enlarge its original rows directly to a smooth ball of length
\(K+g\).  Errors are not amplified again.  All three types of
norm satisfy
\begin{equation}
 0\le w_o\le w\le7\sigma/3.
 \label{old-eq:2.10}
\end{equation}

The row zero is added only at the final smooth ball.  Put
\(Y_{\rm col}=e^{H_N}Z^{a_0}=Z^{a_0+\theta_N}\).  A nonempty column
shell satisfies \(Y_{\rm col}\ge1\), and hence \(a_0\ge-\theta_N\).
By Equation~\eqref{eq:gauss-local}, \(G(u,0)=0\) unless \(u\) is a
sixth power, and \(|G(u,0)|\le q_u^{1/2}\le Y_{\rm col}^{1/2}\).
There are \(O(Y_{\rm col}^{1/6})\) sixth powers on this support.
If \(s\mid u\), then \(q_s^6\le q_u\le Y_{\rm col}\), so
\(s_0\le(a_0+\theta_N)/6\).  Including the assigned divisor-bounded
convolution loss, the squared contribution after central normalization is
\[
 \ll Z^{-a_0}\bigl(Y_{\rm col}^{1/6}Y_{\rm col}^{1/2}\bigr)^2
          Z^{\epsilon_1}
 =Z^{a_0/3+4\theta_N/3+\epsilon_1}
 \le Z^{a_0-s_0+2\theta_N+\epsilon_1}.
\]
The last inequality follows from \(a_0\ge-\theta_N\) and the displayed
bound for \(s_0\).  Thus Equation~\eqref{old-eq:2.7} bounds this added
row with the numerical \(2\theta_N\) correction included in the
\(C_*\xi\) stage allowance.  Here \(C_*\ge1\) denotes the common
constant for aggregate support errors in one stage, chosen independently
of the slot count \(N\).  The estimates below establish that one such
choice covers all operations in a stage.  Zero is never multiplied by a
pool prime.

\subsection{The second transform and smaller-width products}\label{sec:plain-second-transform}

Restore all live slot sums before expanding the square.  Let
\(B(u)\) denote the full allocated convolution coefficient in
one of the preceding Gauss polynomials, including its fixed mask
and \(s\)-divisibility condition.  It contains the whole difference
when centering is used.  Its moduli satisfy
\(q_u/Z^{a_0}\in\exp([-H_N,H_N])\).  Let \(\Phi_2\) be the nonnegative
smooth radial function defining the final row ball.  Apart from
the \(Z^{o(1)}\) loss in the prime density, the norm to be bounded is
\[
 Z^{-\ell}\sum_h\Phi_2(q_h/Z^{K+g})
 \left|Z^{-a_0/2}\sum_u B(u)\tau_1(u)q_u^{it}G(u,h)\right|^2.
\]
The Fourier identity in Lemma~\ref{lem:full-correlation} gives its
exact expansion
\[
 \begin{split}
 Z^{K+g-\ell-a_0}\sum_{u,v}
 \frac{B(u)\overline{B(v)}\tau_1(u)\overline{\tau_1(v)}
       q_u^{it}q_v^{-it}}{\sqrt{q_uq_v}}
 \sum_j F(u,v;j)
 \widehat\Phi_2\left(\frac{Z^{K+g}q_j}{q_uq_v}\right).
 \end{split}
\]
The kernel that occurs here is
\begin{equation}
 \widehat\Phi_2\left(\frac{Z^{K+g}q_j}{q_uq_v}\right).
 \label{old-eq:2.18b}
\end{equation}
Both the displayed inverse square roots and this kernel are retained
as functions of full \(u,v\) until their whole-product separation below.  In particular the
formula includes every live prime and every shared-prime multiplicity.

At \(j=0\), Lemma~\ref{lem:full-correlation} leaves only \(u=v\),
with \(F(u,u;0)=\varphi(u)\le q_u\).  The number of supported
moduli divisible by \(s\), on a nonempty shell, is at most
\[
 C\frac{Y_{\rm col}}{q_s}=C Z^{a_0-s_0+\theta_N},
\]
because writing \(u=sv\) gives \(Y_{\rm col}/q_s\ge1\) on that shell.
The divisor-bounded coefficients use their separate \(\epsilon_1\)
share.  Thus the diagonal exponent is
\(K+g-\ell-s_0+\theta_N\), including when \(a_0-s_0\) is slightly
negative.  Subtracting the allowance in Equation~\eqref{old-eq:2.7}
from its nominal part, without \(\epsilon_G\), gives
\begin{equation}
 \begin{split}
 &(K+g-\ell-s_0)
 -(a_0+\widetilde q+w_o+w+\mathcal B_c-s_0)\\
 &\quad=(A-M)-d-(K_0-K)-w_o-\mathcal B_c+g-\ell\\
 &\quad\le A-M+5\sigma/3+\delta_{{\rm fr},1}.
 \end{split}
 \label{old-eq:2.12}
\end{equation}
For the inequality, put \(D_0=d+K_0-K\).  The contribution
\(-D_0-w_o+J_+\) is at most \(\delta_{{\rm fr},1}\): it is at most
\(-c-2w\) when \(J\ge0\), and when \(J<0\) use
\(D_0\ge-\delta_{{\rm fr},1}\).
The largest remaining increment is
\(2\sigma-\ell_*=5\sigma/3\) in the amplified main norm;
the other norms have increment \(\sigma\).  Thus the diagonal
requires only the displayed terminal loss \(5\sigma/3\), plus
the already reserved frequency perturbation \(\delta_{{\rm fr},1}\)
and numerical support correction \(\theta_N\), when \(A\le M\),
and an additional \(\delta\) in the padded zero-slot core.  The
\(\theta_N\) correction is included in the \(C_*\xi\) stage allowance.

For \(j\ne0\), first perform the whole-dyad localization on this
genuine full \((u,v,j)\) sum, before extracting any common support.
The nominal scale is
\(T_2=Z^{2a_0-K-g}\), and the kernel argument is at least
\(q_j/(e^{2H_N}T_{\rm sec})\) for its supremum in the current
Gauss sector.  The tail estimate above applies using
\(|F(u,v;j)|\le q_uq_v\), with the literal conditions
\(s\mid u,v\) still present.  Write
\[
 \Omega_{\rm ret}(q_j)
 :=\sum_{\lambda\ {\rm retained}}\omega_\lambda(q_j/T_\lambda).
\]
This common row weight satisfies \(0\le\Omega_{\rm ret}\le1\).
The full kernel in Equation~\eqref{old-eq:2.18b} remains on every
retained term.  All subsequent signed Fourier separations are performed
one retained \(\lambda\) at a time, so the normalized row variable
stays on a fixed log box; \(\Omega_{\rm ret}\) records their sum and
common domain.  There are \(O(\log Z)\) relevant retained dyads in
the bounded polynomial ranges.  The preceding \(j=0\) term is exactly the diagonal
of the same final \(\Phi_2\) ball, including the previously added
Gauss row zero; neither zero term is restored on a different domain.

Now write \(u=D_2a,\ v=E_2b\), extracting the genuine complete
common support, so that \((a,b)=1\) and \((ab,D_2E_2)=1\).
Allocate all extracted powers to the plain variables and slots as at the
first transform.  Lemma~\ref{lem:complete-support-correlation} applies on
this genuine locus and gives
\[
 F(D_2a,E_2b;j)=F(D_2,E_2;j)\mathcal R(a,E_2)
   \overline{\mathcal R(b,D_2)}\mathcal R(a,b)
   \chi_a(j)\overline{\chi_b(-j)}.
\]
It permits arbitrary prime powers in all four moduli and leaves a row
scalar at the common primes.  Every extracted prime punctures all
remaining factors on its side, and every slot supplying that prime is
frozen.

This also explains explicitly the cases where a slot shares a
prime with a plain variable.  If \(p\) occurs only on one side, with
full multiplicity \(i\), its residual factor is
\(\chi_p(j)^i\).  Splitting \(i\) among the two plain variables and
the possible slot gives precisely the same factor by multiplicativity,
including when \(6\mid i\) and \(p\mid j\).  For example, a plain
\(p^5\) and a slot \(p\) give the zero-extended factor
\(\chi_p(j)^6\).  If \(p\) occurs on both sides, its full powers are
removed.  In the unequal case \(i>j_0\ge1\), the local correlation
is zero unless \(6\mid j_0\) and the frequency is \(p^{j_0}k\)
with \(p\nmid k\); when it is nonzero it equals
\[
 P^{j_0-1}(P-1)\chi_p(k)^{i-j_0},\qquad P=q_p.
\]
This is a row scalar, even if the excess valuation on the first side
came partly from a live slot.  For instance \(i=7,j_0=6\) with a
plain \(p^6\) and slot \(p\) leaves the scalar
\(P^5(P-1)\chi_p(k)\), freezes that slot, and punctures both
remaining plain variables at \(p\).  Equal multiplicities have the
scalar factors in Equation~\eqref{eq:correlation-local} and the
same puncture conclusion.  Complete rather than gcd-only extraction
is what makes these local factors independent of the residual
variables.

Define
\[
 c_2=\log_Zq_{D_2},\quad d_2=\log_Zq_{E_2},\quad
 b_2=(c_2+d_2)/2.
\]
Let \(p_2\) be the logarithmic norm of their common radical,
let \(G_c=(D_2,E_2)\), and put \(g_2=\log_Zq_{G_c}\).
The correlation vanishes unless \(G_c\mid j\).  At a common
prime with equal multiplicity \(i\not\equiv0\pmod6\), call the
divided frequency \(j/G_c\) a unit or nonunit according as that
prime does not or does divide it.  Let \(t_2\) be the total
radical length of these unit primes.  Let \(V_{\rm id}\) be
the product of these nonunit primes and put
\(V=\log_Zq_{V_{\rm id}}\).

Equation~\eqref{eq:correlation-local} gives the following absolute
local bounds:
\[
 \begin{array}{c|c}
 \text{common multiplicities and divided frequency}&
       \text{absolute correlation bound}\\ \hline
 i=i,\ 6\nmid i,\ \text{unit}&P^{i-1}\\
 i=i,\ 6\nmid i,\ \text{nonunit}&P^i\\
 i=i,\ 6\mid i,\ \text{either}&P^i\\
 i>j_0,\ 6\mid j_0,\ \text{unit}&P^{j_0}.
 \end{array}
\]
Every unequal case not in the last line is zero.  Fix the indicated
unit/nonunit partition and write \(j=G_cV_{\rm id}h'\).  The
partitioned scalar
\[
 1_{\rm part}(h')
 \frac{F(D_2,E_2;G_cV_{\rm id}h')}{Z^{g_2-t_2}}
\]
is defined to be zero off that partition and has modulus at most one
everywhere with this definition.  We do not assert the unit bound for
the unpartitioned correlation on other rows.  Every frozen allocation
forced to be zero by the old masks is still discarded before this
absolute bound is used.

The complete-support identity leaves the row factor
\(\chi_n(G_cV_{\rm id}h')\) on each whole residual column.
To count its fixed moving support, put \(v=G_cV_{\rm id}\) and
\(e_p=v_p(v)\bmod 6\).  Products of the canonical primary
generators of these good ideals are primary, so the exact all-input
identity is
\[
 \chi_n(v)=
   \prod_{\substack{p\mid v\\e_p\ne0}}\chi_n(p)^{e_p}
   \prod_{\substack{p\mid v\\e_p=0}}1_{(n,p)=1},
\]
with any unit of the original row retained in \(h'\).  All primes in this identity
already puncture every residual factor, because they are in the
genuine extracted support.  The active \(e_p\ne0\) primes are
contained in the unit set counted by \(t_2\) and the nonunit set
counted by \(V\).  Equal six-divisible and unequal-minimum-six-divisible
primes have \(e_p=0\) and are represented solely by the common
puncture in this fixed factor.  A nonunit prime can also have
\(e_p=0\), in which case counting it in \(V\) only enlarges the
bound.  This factors only \(\chi_n(v)\): the natural row factor
\(\chi_n(h')\), with all its zeros, and all fixed-ray reciprocity
phases remain.  It does not reclassify a cancellation with the
varying row as an extra puncture.  Thus the full new moving support
has length at most \(\widetilde q+w_o+t_2+V\).

Define the nominal row and total widths by
\begin{equation}
 \begin{aligned}
 m'&=2a_0-K-g-g_2-V,&
 q'&=\widetilde q+w_o+t_2+V,\\
 M'&=m'+q'
    =M+J-g-g_2+t_2
    \le M-\sigma.
 \end{aligned}
 \label{old-eq:2.13}
\end{equation}
Here \(g_2\ge t_2\), and the definitions of \(g\) give
\(g-J\ge\sigma\).  The identities follow by substituting
\(a_0=A-c-w\) and the definition of \(K_0\) preceding
Equation~\eqref{old-eq:2.3}.  Put
\[
 0\le\delta_{{\rm fr},2}:=\frac\xi2+
       \frac{\log(C_dC_{\rm sec})}{\log Z}<\xi.
\]
The sector constants here are those for the second transform.
The retained row weight pulls back exactly to
\(\Omega_{\rm ret}(q_{G_cV_{\rm id}}q_{h'})\).  On its support,
\[
 q_{h'}\le
 \frac{C_dT_{\rm sec}Z^{\xi/2}}{q_{G_cV_{\rm id}}}
 \le\frac{C_dC_{\rm sec}Z^{2a_0-K-g+\xi/2}}
          {q_{G_cV_{\rm id}}}
 =Z^{m'+\delta_{{\rm fr},2}}.
\]
Thus it lies in that common enclosing row ball.  On a
nonempty retained range define the declared nonnegative row length
\(m'_{\rm act}=\max\{0,m'+\delta_{{\rm fr},2}\}\) and
\(M'_{\rm act}=m'_{\rm act}+q'\).  Nonemptiness implies
\(m'+\delta_{{\rm fr},2}\ge0\), so
\(0\le M'_{\rm act}-M'\le\delta_{{\rm fr},2}\).
Regard the pulled-back weight as zero on the rest of this enclosing
ball throughout the signed calculation, and do the same for the
partitioned scalar.  This is an exact extension by zero.  They will
be replaced by their absolute bounds only after a nonnegative child
norm or an exceptional absolute product has been formed.
Complete extraction only shortens a plain variable or freezes an
entire slot, so the surviving slot length satisfies \(z'\le z\).

We record every other exponent in this second transformation.  The
condition \(s\mid u,v\) has not been dropped: it implies that every
prime of \(s\) is in the second complete common radical.  Thus after the complete extraction there is no remaining
\(s\)-divisibility condition on the residual columns.  The number of
possible radicals of length \(p_2\), for this fixed
\(s\), is \(O(Z^{p_2-s_0+\epsilon_1})\): write that radical as
\(s\mathfrak r_2\) with \((s,\mathfrak r_2)=1\), and count the
squarefree \(\mathfrak r_2\) of the remaining norm.  If \(p_2<s_0\)
outside the fixed shell boundary, there are none.  The same Rankin
argument used for \(C,D\) bounds their power and allocation
multiplicities.  Choosing the unit/nonunit partition costs at most
\(2^{\omega(\operatorname{rad}(D_2E_2))}\), another divisor-bounded
factor included in \(\epsilon_1\).  The exponents are
\[
 \begin{array}{l|r}
 \text{factor}&\text{exponent}\\ \hline
 \text{squared central normalization}&-a_0\\
 \text{row Poisson factor and normalized inverse roots}&K+g-a_0\\
 \text{prime density, when present}&-\ell\\
 \text{common-support count with fixed }s&p_2-s_0\\
 \text{common correlation}&g_2-t_2\\
 \text{conversion to normalized residual products}&a_0-b_2
 \end{array}
\]
The last line is
\(\{(a_0-c_2)+(a_0-d_2)\}/2\).
Their sum is
\[
 K+g-\ell-a_0+p_2-s_0+g_2-t_2-b_2.
\]
Subtracting this sum from the allowance
Equation~\eqref{old-eq:2.7} leaves the exponent permitted for
the inner plain products.  This is the bound to be supplied either by
smaller-width moments or by the exceptional-row estimate:
\begin{equation}
 M'+\Delta_{\rm child},\qquad
 \Delta_{\rm child}=b_2-p_2+w+\mathcal B_c+\ell
                   \ge w+\ell.
 \label{old-eq:2.14}
\end{equation}
The inequality uses \(b_2\ge p_2\), since both multiplicities
at each common prime are at least one.

We now remove the residual coprimality before forming independent
children.  For the fixed genuine \(D_2,E_2\), define
\[
 \widetilde F_{D_2,E_2}(a,b;j)
 :=F(D_2,E_2;j)\mathcal R(a,E_2)
   \overline{\mathcal R(b,D_2)}\mathcal R(a,b)
   \chi_a(j)\overline{\chi_b(-j)}
\]
on all residual pairs individually coprime to \(D_2E_2\) and allowed by
the old masks.  It equals the genuine correlation only when \((a,b)=1\).
Define the remaining
\(\mathcal C_2(a,b;j)\) on all individually allowed residual pairs
by the allocated convolution formulas and the old common masks,
including \((ab,D_2E_2)=1\).  Keep the formal full-product norms
\(q_{D_2}q_a,q_{E_2}q_b\), both inverse roots, the full smooth
kernel, and the same pulled-back row weight extended by zero on its
enclosing ball.  For each retained row the exact identity is
\[
 \begin{split}
 \sum_{(a,b)=1}\mathcal C_2(a,b;j)F(D_2a,E_2b;j)
 &=\sum_{\mathfrak t\ {\rm squarefree}}\mu(\mathfrak t)
   \sum_{\mathfrak t\mid a,b}\mathcal C_2(a,b;j)
                \widetilde F_{D_2,E_2}(a,b;j).
 \end{split}
\]
This is the full M\"obius indicator, not a truncation.  The individual
column shells and the retained row ball are finite.  Its equality
therefore follows by interchanging finite sums and using the genuine
correlation identity only when \((a,b)=1\).  Extra common primes of
\(\mathfrak t\) are not part of the genuine \(D_2,E_2\) support or
of its frequency restrictions.  Since the residual masks already
exclude that support, every nonzero \(\mathfrak t\) is disjoint from
it and from \(s\).

On each retained dyad, separate the full kernel and inverse roots in
the normalized \(h'\) norm and the two whole-product norms, as for the first transform.
The frozen factor \(G_cV_{\rm id}\) contributes only an outer phase
to the row Fourier power.  The fixed support boxes are chosen before
the current live labels, so the resulting coefficient measure is
common to them, to both rectangles, and to the row.  Then fix \(\mathfrak t\).  For each of its primes \(p\),
use the exact factor-allocation identity
\[
 1_{p\mid\prod_i n_i}
 =\sum_{\varnothing\ne J}(-1)^{|J|+1}
       \prod_{i\in J}1_{p\mid n_i},
\]
where the factors \(n_i\) are the two plain variables and the live
slots.  For a selected plain variable write \(n_i=pl_i\), with no
restriction on \(l_i\).  This extracts only a row scalar and
\(q_p^{it}\), and replaces both \(X_i,Y_i\) by
\(X_i/q_p,Y_i/q_p\).  It introduces no one-variable puncture.
A selected slot is frozen.  If two distinct divisor primes select
the same prime slot, the term is zero.  The quotients and unselected
factors may still contain \(p\), and may overlap the opposite side;
no new coprimality is imposed.  Old common masks remain common, and
the product of the two new plain scales is equal in the two rectangles.
The extracted row scalars, including zero scalars, are retained in
the signed identity.

Only now, for a fixed retained dyad and fixed Fourier parameters,
is each separated summand a product of two independent residual
convolutions, where \(\mathbf J\) records the factor allocations.
Their row characters are
\[
 \tau_1(n)\rho(n)\chi_n(G_cV_{\rm id}h'),\qquad
 \tau_1(n)\rho'(n)\chi_n(-G_cV_{\rm id}h'),
 \quad \rho,\rho'\in\Theta.
\]
They are understood with the fixed-factor support representation
above.  In particular their inducing-character ratio belongs to
\(\Theta\), including the supplementary factor
\(n\mapsto\chi_n(-1)\); this assertion takes no quotient at a zero.
Their natural row zeros are retained.  Extracting
the selected factors contributes row scalars, not new factors of the
residual character and not new support in \(q'\).  These scalars,
including zeros, remain in the signed identity.

\begin{lemma}[Common coefficient under complete extraction]
\label{lem:centered-coefficient-invariant}
For a centered input, the two transforms and the intervening Gauss-row
enlargement just constructed preserve the following coefficient data.
Each Gauss polynomial and each amplifier error retains
Equation~\eqref{eq:centered-coefficient-form}, multiplied by \(G(u,h)\),
apart from bounded frozen scalars and row scalars of modulus at most one.
For an uncentered input, omit the second rectangle throughout.
After the second transform and the full M\"obius factor allocation, each
separated child side has that coefficient form without \(G\) and without
\(s\mid u\).  Every surviving slot has the child's one whole-product
character and its original coefficient \(\nu_i\), with no Gauss coefficient.
Within a side the plain variables have the same character, puncture mask,
and norm power in both rectangles.  The two sides of one squared Gauss norm
may have different norm powers; their inducing characters differ by a
member of \(\Theta\).

If the new inducing character belongs to \(\Theta\), fixing the live slot
labels leaves the plain coefficient
\[
 \vartheta(l_1)\vartheta(l_2)
 1_{(l_1l_2,\mathfrak R_*)=1}q_{l_1l_2}^{it}D_{\mathbf b}(l_1,l_2)
\]
for one \(\vartheta\in\Theta\), one squarefree mask
\(\mathfrak R_*\) of polynomial norm, and one real \(t\).
The mask may depend on the frozen row, but is common to both variables
and both rectangles.  All coefficients are understood with their retained
frozen scalars, including zeros, and the exact normalizations recorded below.
\end{lemma}

\begin{proof}
The first-transform calculation proved the Gauss coefficient form; the
local \(p\)-power calculation proved it for each amplifier error.
The genuine complete-support identity and the full M\"obius allocation
above proved the child form, preserving one character and norm power on
each whole residual product.  Thus ``common'' concerns one separated side;
the displayed character ratio is the relation between the two sides.
When the inducing character belongs to \(\Theta\), it equals some
\(\vartheta\in\Theta\) on units.  Its redundant natural zeros and
the fixed punctures combine into one \(\mathfrak R_*\) of polynomial
norm.  For each fixed \(\Pi\), multiplicativity factors its value,
mask, and norm power on \(\Pi l_1l_2\) into a scalar in \(\Pi\)
times the three factors stated in the conclusion.  This proves the exceptional assertion.
\end{proof}

We next record the exact normalization of the two residual convolutions.  Put
\[
 \alpha_1=a_0-c_2,\qquad \alpha_2=a_0-d_2,\qquad
 \frac{\alpha_1+\alpha_2}{2}=a_0-b_2.
\]
At the entrance to the current two-transform stage, let
\(\mathcal I_{\rm in}\) be its live slot set and let \(X_1,X_2\)
be the formal scales of its first rectangle.  Their exact convention is
\(X_1X_2\prod_{i\in\mathcal I_{\rm in}}P_i=Z^A\).
For side \(j\in\{1,2\}\), let \(\mathcal F_j\) be the slots from this
entrance set frozen by the first and second genuine common-support
extractions, let \(\mathcal I_j\) be the slots still live before the
final \(\mathfrak t\)-allocation, and let \(Q_{{\rm plain},j}\) be
the product of the exact norms of the plain powers extracted at those
steps and at the amplifier.  Slots removed in an ancestor stage are not
included in \(\mathcal F_j\).  Let \(T_j\) be the common formal
pre-\(\mathfrak t\) product of the two plain scales in its rectangles;
for an uncentered input use its one formal plain product.  Before using
slot ratios, discard a frozen-slot profile-zero term, which is identically
zero by its frozen data independently of the row and live labels.
Complete extraction gives the exact identities
\[
 \begin{aligned}
 Z^{A-\alpha_j}
   &=Q_{{\rm plain},j}\prod_{i\in\mathcal F_j}q_{p_i},\\
 T_j\prod_{i\in\mathcal I_j}P_i
   &=\frac{Z^A}{Q_{{\rm plain},j}\prod_{i\in\mathcal F_j}P_i}
     =Z^{\alpha_j+e_j},\\
 e_j&=\sum_{i\in\mathcal F_j}\log_Z(q_{p_i}/P_i),
 \qquad |e_j|\le\theta_N.
 \end{aligned}
\]
The amplifier contributes only exact plain powers to this calculation.
Each earlier frozen slot occurs once, even if it shared its prime with
a plain.  The already separated inverse roots, kernel, and fixed
normalization constants remain outside \(e_j\).

For side \(j\), let \(d_{j,i}\) be the product of final
\(\mathfrak t\)-primes selected in plain \(i\), and let
\(\mathcal J_j\subseteq\mathcal I_j\) be the newly frozen slots.  Put
\[
 a_{j,i}=\log_Zq_{d_{j,i}},\qquad
 \widetilde r_j=a_{j,1}+a_{j,2}+\sum_{i\in\mathcal J_j}z_i\ge0,
 \qquad
 \omega_j=\sum_{i\in\mathcal J_j}\log_Z(q_{p_i}/P_i).
\]
The corresponding formal scale is divided by \(q_{d_{j,i}}\) in both
rectangles.  An assignment selecting one prime slot at two distinct
\(\mathfrak t\)-primes, or a newly frozen slot with zero profile value,
is identically zero from the frozen data and is discarded before these
ratio bounds are used.  All other extracted zeros remain until the
nonnegative or absolute estimate described below.

Use lower-endpoint divisor dyads \(T\le q_{\mathfrak t}<2T\),
\(T\ge1\), and put \(t_-:=\log_ZT\).  Each dyad contains
\(O(Z^{t_-})\) ideals.  The actual selected product on side \(j\)
is divisible by \(\mathfrak t\), so
\begin{equation}
 \begin{aligned}
 \widetilde r_j+\omega_j
 &=\log_Z\left(q_{d_{j,1}d_{j,2}}
                  \prod_{i\in\mathcal J_j}q_{p_i}\right)
 \ge\log_Zq_{\mathfrak t}\ge t_-,\\
 |\omega_j|&\le\theta_N,\qquad |e_j+\omega_j|\le\theta_N
 \quad(j=1,2).
 \end{aligned}
 \label{eq:centered-divisor-boundary}
\end{equation}
The last inequality uses the disjoint subsets \(\mathcal F_j\) and
\(\mathcal J_j\) of the current stage entrance slots.  It does not
include ancestor slots or fixed separation constants.

Writing \(T'_j=T_j/q_{d_{j,1}d_{j,2}}\), the post formal factor product
and the nominal raw child scale are
\begin{equation}
 T'_j\prod_{i\in\mathcal I_j\setminus\mathcal J_j}P_i
   =Z^{\alpha_j+e_j-\widetilde r_j},\qquad
 N_{{\rm post},j}=Z^{\alpha_j-\widetilde r_j}.
 \label{eq:centered-raw-normalization}
\end{equation}
Define \(P_{C,\mathfrak t,\mathbf J}(h')\) and
\(P_{D,\mathfrak t,\mathbf J}(h')\) to be their residual convolutions
multiplied by \(N_{{\rm post},j}^{-1/2}\).  Relative to the nominal
pre-normalizer \(Z^{-\alpha_j/2}\), the exact extraction coefficient
on side \(j\) is \(Z^{-\widetilde r_j/2}\), apart from the frozen slot
amplitudes, retained extracted row scalars, and already separated outer
factors.  These raw children retain the formal signed rectangles and are
not moment-lemma invocations; their formal logarithmic lengths may be
negative.  No clipping has occurred.

The factor assignments are divisor-bounded for fixed \(N\); their
total is bounded using \(C_N^{\omega(\mathfrak t)}
\ll_{N,a}q_{\mathfrak t}^a\) for any fixed \(a>0\), not by
spending a fixed exponent at each prime.  The \(O(\log Z)\) divisor
dyads in the bounded column range use their existing logarithmic share.
These discrete losses are separate from the numerical \(\theta_N\) terms.

For precision, the other common row scalar after extracting
\(Z^{g_2-t_2}\) is
\[
 v_{{\rm part},\lambda}(h')=
 \omega_\lambda(q_{G_cV_{\rm id}}q_{h'}/T_\lambda)
 1_{\rm part}(h')
 \frac{F(D_2,E_2;G_cV_{\rm id}h')}{Z^{g_2-t_2}}\zeta(h'),
 \qquad |\zeta(h')|\le1,
\]
where \(\zeta\) contains the separated row phases for this dyad and
these Fourier parameters.  It is zero off the retained partition and
enclosing ball, and \(|v_{{\rm part},\lambda}|\le1\).
The subsequent estimates are made dyad by dyad and then summed with
the already allowed \(O(\log Z)\) mass.  Split the already factorized row sum according to membership of the
inducing character in \(\Theta\).  On rows outside \(\Theta\),
Cauchy--Schwarz yields two nonnegative child norms; on rows in
\(\Theta\), take a pointwise absolute product.  Only at these steps
may the absolute values of the common row weight, partition scalar,
or extracted row scalars be bounded by one and the rows enlarged to
the common \(m'_{\rm act}\) ball.  This can admit extra exceptional
rows that violate a former unit restriction, but the character-only
count below includes them.  Both sides use the same eligibility class
and compare with Equation~\eqref{old-eq:2.14}.

Consider first rows whose child inducing character is outside
\(\Theta\).  The two sides have the same eligibility condition
because their characters differ by a member of \(\Theta\).
On such rows, first bound a centered child norm by the sum of the norms
of its two rectangles; for an uncentered child there is one rectangle.
For one fixed side \(j\) and rectangle \(\varrho\), write its pre and
post formal plain scales as \(S_{\varrho,i}\) and
\(S'_{\varrho,i}=S_{\varrho,i}/q_{d_{j,i}}\).  Choose a fixed upper
support endpoint \(B_i\) for each current plain profile type before
the current row, divisor, and live labels.  Omit a rectangle only when
\(S'_{\varrho,i}B_i<1\) for some \(i\); then that plain factor is zero
on every nonzero integral ideal.  Retain equality and every other profile,
mask, row-scalar, or arithmetic zero.  This test is independent of the
row and live labels.

For a retained rectangle put
\(x_i=\log_ZS_{\varrho,i}\) and \(y_i=x_i-a_{j,i}\).  The fixed
endpoint test and \(a_{j,i}\ge0\), with \(H_N\) enlarged to contain
the two positive parts of the upper endpoint logs, give the exact clipped identities
\begin{equation}
 \begin{aligned}
 \pi_{0,j,\varrho}&=\sum_{i=1}^2(-x_i)_+,\qquad
 \pi_{j,\varrho}=\sum_{i=1}^2(-y_i)_+,
 \qquad 0\le\pi_{0,j,\varrho}\le\pi_{j,\varrho}\le\theta_N,\\
 r_{{\rm clip},j,\varrho}
 &=\sum_{i\in\mathcal J_j}z_i+
       \sum_{i=1}^2\bigl((x_i)_+-(y_i)_+\bigr)
 =\widetilde r_j-(\pi_{j,\varrho}-\pi_{0,j,\varrho})\ge0,\\
 A_{{\rm clip},j,\varrho}
 &=\sum_{i=1}^2(y_i)_++\sum_{i\in\mathcal I_j\setminus\mathcal J_j}z_i
 =\alpha_j+e_j-\widetilde r_j+\pi_{j,\varrho}\\
 &=\alpha_j+e_j+\pi_{0,j,\varrho}-r_{{\rm clip},j,\varrho}.
 \end{aligned}
 \label{eq:centered-clipped-rectangle}
\end{equation}
In particular the clipped total decreases by \(r_{{\rm clip},j,\varrho}\)
from its own pre-clipped total \(\alpha_j+e_j+\pi_{0,j,\varrho}\).
Different rectangles can have different clipped totals.  A retained
subunit scale lies in the fixed interval \([1/B_i,1]\), so its dilation
to scale one preserves finite seminorm bounds.  The row character, common
mask, and surviving slot weights are unchanged.  The exact coefficient
from the pre-normalizer \(Z^{-\alpha_j/2}\) to this standard clipped
product is
\[
 Z^{(A_{{\rm clip},j,\varrho}-\alpha_j)/2}
 =Z^{(-r_{{\rm clip},j,\varrho}+e_j+\pi_{0,j,\varrho})/2}.
\]
For a fixed pair of retained rectangles, suppress their rectangle indices.
Equation~\eqref{eq:centered-divisor-boundary} then bounds the paired
divisor count and these coefficients by
\begin{equation}
 \begin{split}
 t_- -\frac{r_{{\rm clip},1}+r_{{\rm clip},2}}2
     +\frac{e_1+\pi_{0,1}+e_2+\pi_{0,2}}2
 &\le\frac{(e_1+\omega_1)+\pi_1+(e_2+\omega_2)+\pi_2}{2}\\
 &\le2\theta_N.
 \end{split}
 \label{eq:centered-clipped-shell}
\end{equation}
There are at most two rectangles per side, hence at most four such pairs;
their triangle factor is fixed.  No common clipped reduction is used for
the signed difference.

On the \(c_2\) side, before the last coprimality extraction, the
nominal total length is \(A'=\alpha_1=A-c-w-c_2\).  For the positive-slot parameters,
\begin{equation}
 \begin{split}
 (A'-M')-(A-M)
 &=g+w-d-c_2-(K_0-K)-w_o+g_2-t_2\\
 &\le6(w+\ell)+\delta_{{\rm fr},1}.
 \end{split}
 \label{old-eq:3.14}
\end{equation}
To prove the inequality, first use \(g_2-t_2\le c_2\).
In the amplified main norm, \(w=w_o=0\) and
\[
 g=J_++2\sigma
 \le d+(K_0-K)+6\ell+\delta_{{\rm fr},1}.
\]
For an error,
\(g\le d+(K_0-K)+\sigma+\delta_{{\rm fr},1}\), so the excess is at most
\(w-w_o+\sigma+\delta_{{\rm fr},1}\).  These inequalities use
\(d+K_0-K\ge-\delta_{{\rm fr},1}\) and the
\(1\)-Lipschitz property of the positive part.  The three choices in
Equation~\eqref{old-eq:2.9}, together with
\(\ell_p\ge\sigma/6\), give
\(w-w_o+\sigma\le6w\).  This proves
Equation~\eqref{old-eq:3.14}.

For each retained clipped rectangle, if no slot survives, apply the
completed unrestricted zero-slot assertion at the declared width
\(M'_{\rm act}\).  If slots survive, first apply the algebraic mask deletion in
Equations~\eqref{old-eq:2.1a}--\eqref{old-eq:2.1b}.
It expresses this standard clipped product as natural products and only
decreases its nonnegative affine expression.  For each actual product, after both frequency
enclosures, the \(\mathfrak t\)-allocation, and mask deletion, let
\(A_{\rm act},z_{\rm act}\) be its declared total and slot lengths
and put
\[
 F_{\rm act}
 =\bigl(A_{\rm act}-M'_{\rm act}+(6\kappa-1)z_{\rm act}\bigr)_+.
\]
The parent satisfies Equation~\eqref{old-eq:3.9}.  Before this mask
deletion, let \(z'\le z\) be the surviving slot length of the clipped
rectangle.  Equations~\eqref{old-eq:3.14} and
\eqref{eq:centered-clipped-rectangle}, together with \(M'_{\rm act}\ge M'\), give
\[
 \begin{split}
 A_{{\rm clip},j,\varrho}-M'_{\rm act}+(6\kappa-1)z'
 &\le \alpha_j-M'+(6\kappa-1)z
       +e_j+\pi_{j,\varrho}-\widetilde r_j-(M'_{\rm act}-M')\\
 &\le6(w+\ell)+\delta_{{\rm fr},1}+e_j+\pi_{j,\varrho}.
 \end{split}
\]
The same calculation holds on the other side with \(c_2,d_2\) exchanged.
Taking the positive part and then deleting the fixed mask therefore gives
\[
 F_{\rm act}\le6(w+\ell)+\delta_{{\rm fr},1}
                 +(e_j+\pi_{j,\varrho})_+
 \le6(w+\ell)+\delta_{{\rm fr},1}+2\theta_N.
\]
Here \(z_{\rm act}\le z'\le z\).  All the displayed ledgers use the fixed list
of aggregate lengths
\[
 A,z,c,d,p,R,E,s_0,K,w,w_o,g,\ell,c_2,d_2,p_2,g_2,t_2,V.
\]
Their affine coefficients and positive-part Lipschitz constants are
numerical and independent of \(N\).  The displayed frequency, raw
normalization, and rectangle-clipping bounds are therefore included in
\(C_*\xi\), for a fixed \(C_*\ge1\) independent of \(N\), once the
aggregate threshold has been imposed.  The existence of this fixed error bound uses the single
\(\theta_N=H_N/\log Z\) before comparing lengths, not \(N\) separate
copies of \(\xi\).  We enlarge \(C_*\) below to cover the fixed number
of operations in a stage.

If \(F_{\rm act}>0\), remove whole slots from this product until the
remaining product satisfies Equation~\eqref{old-eq:3.9} or
until no slot remains.  Removal here means applying the pointwise
bound Equation~\eqref{old-eq:3.5} to the entire selected prime
polynomial; no prime label is frozen.  A removed slot of length
\(d\) decreases \(A_{\rm act}+(6\kappa-1)z_{\rm act}\) by \(6\kappa d\)
and contributes \(\kappa d\) to the squared \(Z\)-exponent, with the
fixed seminorm and height factor retained separately.  Since each
slot has length at most \(\eta\), order the positive live lengths
and take the first prefix reaching \(F_{\rm act}/(6\kappa)\).
Its preceding prefix is smaller than that threshold and its final
slot has length at most \(\eta\).  If no prefix reaches the threshold,
remove all slots, whose total is smaller.  Thus exactly one slot can
cause an overshoot, and the total removed length \(d_z\) satisfies
\begin{equation}
 \begin{split}
 d_z&\le\min\left\{z_{\rm act},
                     \frac{F_{\rm act}}{6\kappa}+\eta\right\},\\
 \kappa d_z&\le\frac{F_{\rm act}}6+\kappa\eta
 \le\Delta_{\rm child}+\frac{\delta_{{\rm fr},1}}6
                       +\frac{\theta_N}{3}+\eta.
 \end{split}
 \label{old-eq:3.16}
\end{equation}
If the slots disappear before the affine boundary is reached, the
remaining plain lengths may be arbitrary; this is exactly why the
completed smaller-width zero-slot assertion includes all lengths.
Otherwise apply the positive-slot induction to the remaining
product.  Its slots still have their original coefficient class,
and its inducing rows remain outside \(\Theta\).  Cauchy--Schwarz
averages the errors of the two separately clipped children, so their
possibly different rectangles and slot sets do not double \(\eta\).
Combining Equation~\eqref{old-eq:3.16} with the paired coefficient
bound in Equation~\eqref{eq:centered-clipped-shell} and
\(M'_{\rm act}-M'\le\delta_{{\rm fr},2}\), the strict-edge scale
cost beyond the child envelope is at most
\begin{equation}
 \delta_{{\rm fr},2}+\frac{\delta_{{\rm fr},1}}6
        +\eta+\frac{7\theta_N}{3}.
 \label{eq:centered-child-edge-cost}
\end{equation}
If no slot survives, the same bound holds without needing a greedy cost.
The numerical \(\theta_N\) terms and the two frequency corrections
give the single \(O(\xi+\eta)\) edge loss in the \(\epsilon_G\)
reserve inherited from Equation~\eqref{old-eq:2.7} when using
Equation~\eqref{old-eq:2.14}; the existing local small-power and fixed
analytic factors remain separate.  The pointwise estimate is requested
once for the entire removed product; its internal small-power shares
may depend on \(N\).  This one greedy operation occurs only after
all boundary defects of the actual child have been included in
\(F_{\rm act}\).  The argument for the \(d_2\)
side is the same with \(c_2,d_2\) exchanged.

We have now bounded all child rows whose inducing characters lie outside
\(\Theta\) using only smaller widths.  For the remaining rows, the next
count and volume bound handle the uncentered range; the centered range
uses the subtraction retained in the coefficient lemma.

\subsection{Rows with inducing characters in \texorpdfstring{\(\Theta\)}{Theta}}

For this subsection, call a child row \emph{exceptional} when its
inducing character belongs to \(\Theta\).  By the coefficient
lemma, either both separated sides are exceptional or neither is.
Every prime in the existing full displayed moving support is a common
column zero, as is every extra common puncture.  This is true initially,
remains true for \(\mathfrak e,\mathfrak r\) in the first transform,
and remains true for an amplifier prime.  For a nonzero genuine
second allocation, the factors \(\tau_1(D_2),\tau_1(E_2)\) and the
old common masks therefore force its complete radical to be disjoint
from all those supports.  This conclusion is made before replacing
any frozen scalar by an upper bound.  The artificial residual
extension keeps \(D_2,E_2\) and those masks fixed and cannot revive
an impossible allocation.

We make explicit the finite-ray reduction for row factors at the fixed
primes.  As in Equation~\eqref{eq:row-fixed-ray-reduction}, write a nonzero row as
\(h'=\varepsilon h_{\mathcal S}h_{\rm good}\), with
\(\varepsilon\) a unit and the two other factors supported on
\(\mathcal S\) and its complement.  For a primary element \(n\)
prime to \(\mathcal S\), reciprocity gives, with every zero retained,
\[
 \chi_n(h')=\chi_n(\varepsilon h_{\mathcal S})
   \mathcal R(n,h_{\rm good})
   \prod_{p\mid h_{\rm good}}\chi_p(n)^{v_p(h')}.
\]
By Lemma~\ref{lem:fixed-numerator-ray}, the first factor belongs to
a finite family of ray characters supported on \(\mathcal S\) after
the unit and the \(\mathcal S\)-valuations modulo six are fixed.
The second belongs to the fixed reciprocity family after fixing the
good ray sector.  We do not absorb any moving good prime into this
fixed family.  The family supplied by the first factor need not be
contained in \(\Theta\); there are simply finitely many such choices.
Every remaining displayed good-prime factor has its actual local
sextic exponent, with nontrivial exponents ramified at that prime.

Let \(v_1\) be the radical length of the primes counted in \(V\)
whose equal multiplicity is one, and put \(f=2v_1\).  At such a
prime the fixed factor \(G_cV_{\rm id}\) in the row has valuation
two.  There is no existing moving character at that prime, and
every character of \(\Theta\) is unramified there.  Sextic
reciprocity therefore shows that exceptional induction requires
\[
 v_p(h')+2\equiv0\pmod6,\qquad\text{that is,}\qquad
 v_p(h')\equiv4\pmod6.
\]
More generally, at every prime outside \(\mathcal S\), exceptional
induction prescribes a single residue class modulo six for
\(v_p(h')\), determined by the frozen moving character and
\(G_cV_{\rm id}\).  Outside their supports that residue is zero.
For each of the finitely many choices at \(\mathcal S\) and of
unit and fixed-ray data, the ideal of \(h'\) consequently has a
unique form
\[
 (h')=\mathfrak h_0\mathfrak v^6,
\]
where \(\mathfrak h_0\) is fixed and sixth-power-free.  The
displayed valuation-four conditions give
\(q_{\mathfrak h_0}\ge Z^{4v_1}=Z^{2f}\).
Ideal counting up to \(q_{h'}\ll Z^{m'_{\rm act}}\) now gives the
stronger bound \(O(Z^{(m'_{\rm act}-2f)/6+\epsilon_1})\) on every
nonempty range, with the usual bounded-scale convention.  We use
only the weaker bound
\begin{equation}
 \#\{h':q_{h'}\ll Z^{m'_{\rm act}},\ \text{exceptional}\}
 \ll Z^{(m'_{\rm act}-f)/6+\epsilon_1}
 \le Z^{(m'-f)/6+\delta_{{\rm fr},2}/6+\epsilon_1}.
 \label{eq:exceptional-row-count}
\end{equation}
If the exponent would describe a scale below one, the forced ideal
\(\mathfrak h_0\) makes the range empty except at the same
bounded-scale boundary.  This argument includes the principal
character and every other member of \(\Theta\).  Additional
conditions at old moving primes can only reduce the count.
It also covers rows admitted when the partition scalar was bounded
after positivity.  At a unit prime, such an added row can be
exceptional even though it was absent from the original partition;
no extra forcing saving from the unit set \(t_2\) is used.

On exceptional rows take the absolute product for each already
allocated pair of children and multiply by
Equation~\eqref{eq:exceptional-row-count}.  Let
\(c_{\mathfrak t,\mathbf J}(h')\) denote its central extraction
coefficient and bounded extracted scalars.  For an allocation with raw
reductions \(\widetilde r_1,\widetilde r_2\),
\(|c_{\mathfrak t,\mathbf J}(h')|\ll_N
 Z^{-(\widetilde r_1+\widetilde r_2)/2}\) after taking absolute values.
Using absolute volume rather than cancellation, the unnormalized plain
difference on side \(j\) is \(O(T'_j)\) at every positive formal scale,
including subunit scales, and the unnormalized live slots contribute
\(O_N(\prod_{i\in\mathcal I_j\setminus\mathcal J_j}P_i)\).
Equation~\eqref{eq:centered-raw-normalization} therefore gives the raw
normalized volume
\[
 |P_{j,\mathfrak t,\mathbf J}(h')|
 \ll_N N_{{\rm post},j}^{-1/2}T'_j
             \prod_{i\in\mathcal I_j\setminus\mathcal J_j}P_i
 =Z^{(\alpha_j-\widetilde r_j)/2+e_j}
\]
up to the fixed seminorm and height factors, where \(j\) denotes its
\(C\) or \(D\) side.  Thus the exponent for an allocated pair and one
lower-endpoint divisor dyad, including its count, is
\[
 a_0-b_2+e_1+e_2+t_- -\widetilde r_1-\widetilde r_2
 \le a_0-b_2+e_1+(e_2+\omega_2)
 \le a_0-b_2+2\theta_N.
\]
Here Equation~\eqref{eq:centered-divisor-boundary} gives
\(t_- -\widetilde r_2\le\omega_2\), and \(\widetilde r_1\ge0\).
Consequently
\[
 \sum_{\mathfrak t,\mathbf J}
 |c_{\mathfrak t,\mathbf J}(h')|
 |P_{C,\mathfrak t,\mathbf J}(h')
       P_{D,\mathfrak t,\mathbf J}(h')|
 \ll Z^{a_0-b_2+2\theta_N+\epsilon_1}.
\]
The displayed sum is over a fixed lower-endpoint divisor dyad; its
allocations have already been included in
\(\epsilon_1\).  It does not assert an independent product before
M\"obius allocation.  Subtracting the inner allowance from the
reference volume and the nominal exceptional count gives the exact
algebraic identity
\begin{equation}
 \frac{m'-f}{6}+a_0-b_2-(M'+\Delta_{\rm child})
 =A-\frac56M-F_1-F_2,
 \label{old-eq:2.15}
\end{equation}
where
\begin{equation}
 \begin{aligned}
 F_1&=c/6+5\{d+(K_0-K)\}/6+w/3+\widetilde q/6
       +w_o+\mathcal B_c-5g/6+\ell,\\
 F_2&=2b_2-\tfrac56g_2-p_2+t_2+V/6+f/6.
 \end{aligned}
 \label{old-eq:2.16}
\end{equation}
The actual excess is bounded by the nominal expression in
Equation~\eqref{old-eq:2.15} plus
\(\delta_{{\rm fr},2}/6+2\theta_N+\epsilon_1\).
For an explicit check, the left side of
Equation~\eqref{old-eq:2.15} first equals
\[
 a_0-2b_2+p_2-w-\mathcal B_c-\ell
       -\tfrac56m'-q'-f/6.
\]
Substitution of Equation~\eqref{old-eq:2.13} and
\(K=2A-c-d+R+E-m-(K_0-K)\) gives
Equations~\eqref{old-eq:2.15}--\eqref{old-eq:2.16}.

The following lower bounds are the reason complete common supports
do not consume the available exponent:
\begin{equation}
 F_1\ge\tfrac23(c+w)-3\sigma-\tfrac56\delta_{{\rm fr},1},\qquad
 F_2\ge\tfrac23b_2.
 \label{old-eq:2.17}
\end{equation}
To prove the first, put \(D_0=d+K_0-K\), initially set
\(g=J_+\) and \(\ell=0\), and recall
\(J=D_0-c-2w+w_o\).  When \(J\ge0\), direct simplification
gives
\[
 F_1=c+2w+\widetilde q/6+w_o/6+\mathcal B_c
       \ge c+2w.
\]
When \(J<0\), use \(\widetilde q\ge R\) and
\[
 \mathcal B_c+\widetilde q/6
 \ge (3c-5D_0)_+/6-\tfrac56\delta_{{\rm fr},1}.
\]
Indeed, the left side is at least
\(\{(3c-5d-R)_++R\}/6\), which is at least
\((3c-5d)_+/6\), while
\(D_0\ge d-\delta_{{\rm fr},1}\).  The positive part is
\(1\)-Lipschitz, giving the displayed correction.
It follows that
\[
 F_1\ge
 \frac{c+5D_0+(3c-5D_0)_+}{6}+\frac w3
       -\frac56\delta_{{\rm fr},1}
 \ge\frac23(c+w)-\frac w3-\frac56\delta_{{\rm fr},1}.
\]
The actual choices have \(g\le J_++2\sigma\) and
\(\ell\ge0\).  Equation~\eqref{old-eq:2.10} therefore bounds
the additional loss from \(2(c+w)/3\) by
\[
 w/3+5\sigma/3+\tfrac56\delta_{{\rm fr},1}
 \le22\sigma/9+\tfrac56\delta_{{\rm fr},1}
 <3\sigma+\tfrac56\delta_{{\rm fr},1}.
\]

For the second inequality in Equation~\eqref{old-eq:2.17}, compute
prime by prime from the definition of \(F_2\).  In units of the
prime's logarithmic norm, the contributions are
\[
 \begin{array}{c|c|c}
 \text{common case}&F_2&b_2\\ \hline
 i=i,\ 6\nmid i,\ \text{unit}&7i/6&i\\
 i=i,\ 6\nmid i,\ \text{nonunit}
       &(7i-5)/6+\tfrac13 1_{i=1}&i\\
 i=i,\ 6\mid i,\ \text{either}&7i/6-1&i\\
 i>j_0,\ 6\mid j_0,\ \text{unit}
       &i+j_0/6-1&(i+j_0)/2 .
 \end{array}
\]
The extra \(1/3\) in the second line is \(f/6\).  The first
line exceeds \(2b_2/3\).  In the second line equality holds at
\(i=1\), and for \(i\ge2\) the difference is
\((3i-5)/6\ge0\).  In the third line the difference is
\((i-2)/2\ge0\), because \(i\ge6\).  In the last line it is
\((4i-j_0-6)/6\ge0\), because \(j_0\ge6\) and
\(i\ge j_0+1\).  Summing proves the claim.

For the uncentered range \(A\le5M/6\),
Equations~\eqref{old-eq:2.15}--\eqref{old-eq:2.17}
bound the nominal exceptional excess by
\(3\sigma+5\delta_{{\rm fr},1}/6\).  The actual-count and aggregate
support corrections recorded above are further \(O(\xi)\) terms
in the fixed stage allowance.  Together with the
diagonal estimate and the smaller-width estimates, this proves the
uncentered stage at the current width with the reserved terminal
loss.  The comparisons constructed earlier may consequently use
that stage.  It remains to estimate the centered exceptional terms
when \(A>5M/6\).

For one separated side of a fixed exceptional row, fix the live slot
labels.  Lemma~\ref{lem:centered-coefficient-invariant} then gives the same
character, puncture, and norm power in the two plain variables and in both
rectangle terms.  The next lemma shows that the coefficient of \(X^{1+it}\)
in each plain sum is independent of \(X\), so the two product main terms
agree at equal products of scales.

\begin{lemma}[Masked rectangle cancellation]
\label{lem:centered-lattice-cancellation}
Let \(\vartheta\) range over a fixed finite set of finite-order
ray characters with conductor primes in \(\mathcal S\).  Let
\(\mathfrak R_*\) be squarefree with \(q_{\mathfrak R_*}\le Z^{B_*}\)
for a fixed \(B_*\), and let \(W_1,W_2\) be smooth profiles on
fixed annuli.  For \(X>0\) and \(t\in\mathbb R\), define
\[
 \mathcal L_i(X,t):=
 \sum_l\vartheta(l)1_{(l,\mathfrak R_*)=1}
           q_l^{it}W_i(q_l/X).
\]
Then
\begin{align}
 \mathcal L_i(X,t)
 &=c_{\vartheta,\mathfrak R_*}X^{1+it}I_i(t)
   +O\left(Z^{\epsilon_1}(1+|t|)^J\right),
 \label{old-eq:2.18d}\\
 |\mathcal L_i(X,t)|&\ll X.
 \label{old-eq:2.18e}
\end{align}
Here \(J\) is fixed, the constants use finitely many seminorms,
\(c_{\vartheta,\mathfrak R_*}\) is independent of \(X,t\), and
\(I_i(t)\) is a fixed measure constant times
\(\int_0^\infty W_i(y)y^{it}\,dy\).
Both estimates are uniform for \(X>0\), for the stated masks, and
for the finite character set.

Let \(D_{\mathbf b}\) be as in
Equation~\eqref{old-eq:2.18a}, and put
\[
 U_i=X_i/q_{\mathfrak b_i},\qquad
 V_i=Y_i/q_{\mathfrak b_i},\qquad
 T=U_1U_2=V_1V_2.
\]
If \(U_i,V_i\ge Z^r\) for \(i=1,2\) and some \(r\ge0\),
then
\begin{equation}
 \begin{split}
 T^{-1/2}\left|
 \sum_{l_1,l_2}
 \vartheta(l_1)\vartheta(l_2)
 1_{(l_1l_2,\mathfrak R_*)=1}
 q_{l_1l_2}^{it}D_{\mathbf b}(l_1,l_2)
 \right|
 \ll Z^{\epsilon_1}(1+|t|)^{2J}T^{1/2}Z^{-r}.
 \end{split}
 \label{old-eq:2.18f}
\end{equation}
Without a nonnegative lower length, the same expression is
\(O(T^{1/2})\).
\end{lemma}

\begin{proof}
First omit the mask.  The primary generators representing ideals
outside \(\mathcal S\), with the fixed character weight
\(\vartheta\), are a finite weighted collection of residue
classes in a fixed lattice.  Apply Poisson on that lattice to
\[
 f_{X,t}(z)=q_z^{it}W_i(q_z/X)
           =X^{it}f_t(z/\sqrt X),\qquad
 f_t(z)=q_z^{it}W_i(q_z).
\]
The Fourier transform is
\(X^{1+it}\widehat f_t(\sqrt X\,\xi)\).
Because \(f_t\) is supported on a fixed annulus, integration by
parts for any fixed \(J_0>2\) gives
\[
 |\widehat f_t(\xi)|
 \ll_{J_0} (1+|t|)^{J_0}(1+|\xi|)^{-J_0},
\]
using finitely many seminorms of \(W_i\).  For \(X\ge1\), the
sum over nonzero points of the fixed dual lattice is therefore
\[
 \ll (1+|t|)^{J_0} X
       \sum_{\xi\ne0}(1+\sqrt X|\xi|)^{-J_0}
 \ll (1+|t|)^{J_0}
\]
after increasing \(J_0\) if necessary.  For \(0<X<1\), both the
lattice sum and its zero-frequency main term are \(O(1)\) by
absolute counting on the fixed annulus.  The zero frequency is
\(c_\vartheta X^{1+it}I_i(t)\), where \(c_\vartheta\) is the
fixed weighted mean of the residue classes.  This proves the
unmasked version of Equation~\eqref{old-eq:2.18d}.

Now insert the mask by inclusion--exclusion.  First remove from
\(\mathfrak R_*\) its primes in \(\mathcal S\), since every
summation ideal already avoids them.  Thus all divisors used below
are outside \(\mathcal S\).  With \(\mathcal L_i^0\) denoting
the unmasked sum, multiplicativity gives
\[
 \mathcal L_i(X,t)
 =\sum_{d\mid\mathfrak R_*}
       \mu(d)\vartheta(d)q_d^{it}\mathcal L_i^0(X/q_d,t).
\]
The main coefficient is
\[
 c_{\vartheta,\mathfrak R_*}
 =c_\vartheta\sum_{d\mid\mathfrak R_*}
             \frac{\mu(d)\vartheta(d)}{q_d}.
\]
The \(q_d^{it}\) from extraction has canceled the
\(q_d^{-it}\) in the main term at \(X/q_d\).  This proves that
the coefficient is independent of both \(X\) and \(t\).
The errors are multiplied by at most the number of divisors
\(\#\{d:d\mid\mathfrak R_*\}\ll_{\epsilon_1,B_*}Z^{\epsilon_1}\).
The coefficient itself is also \(Z^{\epsilon_1}\)-bounded by
the polynomial-size Euler-product estimate.  This proves
Equation~\eqref{old-eq:2.18d}.

For Equation~\eqref{old-eq:2.18e}, take absolute values in the
original sum.  It is empty below a fixed positive scale; whenever
it is nonempty, ideal counting on the fixed annulus gives \(O(X)\)
points.  This is uniform in \(t\) and in the mask.

The masked sum in Equation~\eqref{old-eq:2.18f} is exactly
\[
 \mathcal L_1(U_1,t)\mathcal L_2(U_2,t)
 -\mathcal L_1(V_1,t)\mathcal L_2(V_2,t).
\]
The two product main terms in
Equation~\eqref{old-eq:2.18d} are both
\(c_{\vartheta,\mathfrak R_*}^2T^{1+it}I_1(t)I_2(t)\).
They cancel.  Each cross term with one error is bounded by
\(Z^{\epsilon_1}(1+|t|)^{2J}\) times one of
\(U_1,U_2,V_1,V_2\), and the product of errors has the same
bound after reducing the subsidiary power loss.  If all four
scales are at least \(Z^r\), each is at most \(TZ^{-r}\),
and \(1\le TZ^{-r}\) because \(T\ge Z^{2r}\).
The difference is therefore
\(O(Z^{\epsilon_1}(1+|t|)^{2J}TZ^{-r})\).
Division by \(T^{1/2}\) proves
Equation~\eqref{old-eq:2.18f}.  When no nonnegative lower length
is available, Equation~\eqref{old-eq:2.18e} bounds each product
by \(O(T)\), giving the last assertion.
\end{proof}

Apply this lemma to each already allocated exceptional child.  The
hypotheses on its coefficient and common mask follow from
Lemma~\ref{lem:centered-coefficient-invariant}; in particular the
main terms cancel for the same row, mask, and whole-product norm
power before any absolute value.  Initially all four plain lengths
are at least \(L\).  Before the selected \(\mathfrak t\)-factors
are removed, the \(c_2\) side has lower plain length at least
\(L-c-w-c_2\) and nominal total length \(\alpha_1=a_0-c_2\):
no one plain loses more than the total \(c+w+c_2\) extracted by
the two genuine common supports and the amplifier.  The other side
has the analogous bounds with \(d_2\).  Put
\[
 r=(L-c-w-\min(c_2,d_2))_+,
\]
and call the side with this reference saving the first side, relabeling
its associated raw data together.  If \(\widetilde r_1<r\), each
selected plain length satisfies \(a_{1,i}\le\widetilde r_1\), so all
four formal post plain scales have logarithmic length at least
\(r-a_{1,i}\ge r-\widetilde r_1>0\).  Apply
Equation~\eqref{old-eq:2.18f} on these exact equal-product formal scales.
If \(\widetilde r_1\ge r\), including equality, or if \(r=0\), use
the all-scale absolute-volume fallback in the same lemma.  No formal
centered scale is clipped in this branch, and an identically zero
rectangle remains in the formal difference.  The resulting saving is
\((r-\widetilde r_1)_+\).  With absolute volume on the other side,
this gives the aggregate bound on each lower-endpoint divisor dyad,
\begin{equation}
 \begin{split}
 &\sum_{\mathfrak t,\mathbf J}
 |c_{\mathfrak t,\mathbf J}(h')|
 |P_{C,\mathfrak t,\mathbf J}(h')
       P_{D,\mathfrak t,\mathbf J}(h')|\\
 &\hspace{15mm}\ll
 Z^{a_0-b_2-r+2\theta_N+\epsilon_1}.
 \end{split}
 \label{old-eq:2.18}
\end{equation}
Here and below a fixed polynomial in the separated heights is
understood.  To check the bound, the raw coefficients and volumes give
the reference exponent \(a_0-b_2+e_1+e_2\) and reduction
\(-\widetilde r_1-\widetilde r_2\), while the divisor count gives
\(t_-\).  Equation~\eqref{eq:centered-divisor-boundary} gives the
one-line inequality
\begin{equation}
 \begin{split}
 t_- -\widetilde r_1-\widetilde r_2-(r-\widetilde r_1)_+
 &=(t_- -\widetilde r_2)-\max(r,\widetilde r_1)\\
 &\le\omega_2-r.
 \end{split}
 \label{old-eq:2.18h}
\end{equation}
Adding \(e_1+e_2\) leaves at most
\(e_1+(e_2+\omega_2)-r\le2\theta_N-r\).
The divisor-bounded allocations contribute only \(\epsilon_1\),
proving Equation~\eqref{old-eq:2.18}.
This applies to the artificial terms even when quotients retain
\(\mathfrak t\)-primes, because the coefficient lemma preserves the
common mask, equal product scales, and one norm power within each
rectangle difference.

The live slots are summed only after the finite transform has removed
their Gauss coefficients.  Their unnormalized absolute sums contribute
\(O_N(\prod_{i\ {\rm live}}P_i)\).  Combining this with the nominal
raw normalizer gives the full \(e_j\) volume factor in
Equation~\eqref{eq:centered-raw-normalization}, as included above; it
is not replaced by a factor-product normalizer.  The Fourier measure was fixed before
their labels, so conditioning on those labels changes neither the
common mask nor the centered saving.  A zero conditional slot scalar
is discarded only in the present absolute bound.

Set \(v=c+w+\min(c_2,d_2)\).  By
Equation~\eqref{old-eq:2.17}, \(F_1+F_2\) is at least
\(2v/3-3\sigma-5\delta_{{\rm fr},1}/6\), since
\(b_2\ge\min(c_2,d_2)\).
The centered saving in Equation~\eqref{old-eq:2.18} now bounds
the exceptional deficit, apart from \(3\sigma\) and the recorded
frequency and aggregate support perturbations, by
\begin{equation}
 A-\tfrac56M-\tfrac23v-(L-v)_+
 \le (A-M)_+.
 \label{old-eq:2.19}
\end{equation}
Indeed, for \(0\le v\le L\) the left side is
\(A-5M/6-L+v/3\), which increases with \(v\); for
\(v\ge L\) it is \(A-5M/6-2v/3\), which decreases.
Its maximum is attained at \(v=L=M/4\) and equals \(A-M\).
For positive slots, Equation~\eqref{old-eq:3.9} implies
\(A\le M\).  For the zero-slot core,
Equation~\eqref{old-eq:2.1h} gives \(A-M\le\delta\).
Thus the centered exceptional terms require only the terminal
loss \(3\sigma\), or \(\delta+3\sigma\) in the padded core.
The additional \(\delta_{{\rm fr},1}\), \(\delta_{{\rm fr},2}\),
and \(\theta_N\) terms are included in the same \(C_*\xi\)
stage allowance.
Together with Equation~\eqref{old-eq:2.12}, this completes the
centered stage at the current width.

\subsection{Completion of the finite induction}

We finish by verifying the quantifier order and the finite induction.
Every nonterminal call is to a child with nominal width \(M'\) in
Equation~\eqref{old-eq:2.13}, at least \(\sigma\) below its parent.
The declared row width is \(M'_{\rm act}\), and all nonexceptional
moment-input lengths are taken after the specified support and clipping
operations.  Exceptional raw scales remain formal and are not clipped.
The two frequency enclosures, the aggregate support errors, and these
clippings alter the width comparison by at most \(C_*\xi\).  Choose
this below \(\sigma/2\).  Every nonempty child then has nonnegative
width and leaves its band of length \(\sigma/4\).  An empty
nonzero-frequency range is discarded; a nonempty formal range just
below scale one has already been included by its enclosing length and
clipping error.

At each smaller width, Equations
\eqref{old-eq:2.1a}--\eqref{old-eq:2.1b} delete the fixed extra masks
before the natural positive-slot estimate.  The unrestricted zero-slot
assertion follows once from the padded core by reflecting at most its
two plains.  A centered comparison calls only the earlier uncentered
stage in its band, using its strict margin.  Each Gauss norm has at
most one amplification, whose errors go directly to the second
transform.  Equation~\eqref{old-eq:3.16} is invoked once for each
actual natural child, after all its boundary defects have been
included in \(F_{\rm act}\).  Thus there is no same-band cycle.
The integer \(D\) defined above bounds the strict calls by \(D-2\),
because each drops the width by at least \(\sigma/2\) from an initial
width at most \(M_{\max}\).

Here is why \(C_*\) and the mesh can be chosen independently of the
fixed slot count.  Equation~\eqref{eq:centered-aggregate-support}
bounds the entire logarithmic error of any slot subset by
\(\theta_N=H_N/\log Z\).  Each extracted plain power is measured
exactly, and each of the two possible plain clippings in one
nonexceptional rectangle, after triangle inequality, has one fixed
endpoint error.  The exceptional calculation uses raw reductions and
the disjoint-subset bounds without clipping.  The first and second frequency enclosures use only
their one aggregate outer scale and one common row dyad.  All the
local ledgers are affine or positive parts of affine expressions in
the fixed list of aggregate lengths displayed above.  Their numerical
Lipschitz constants do not depend on \(N\); the local \(F_2\)
inequalities and the radical counts use exact prime norms.  In
particular the change in the positive-slot affine expression is at most
\begin{equation}
 |\Delta M|+|\Delta A|+5|\Delta z|,
 \label{old-eq:2.1j}
\end{equation}
because \(0\le6\kappa-1\le5\).  The total change of \(z\) is
aggregated before this inequality is used.  These observations give
one numerical \(C_*\), enlarged for the fixed number of operations
per stage, that covers all frequency, normalization, and boundary
errors by \(C_*\xi\).  The single greedy prefix contributes at
most \(\eta\), not one \(\eta\) per removed slot.

The analytic separations introduce no derivative-order multiple of
\(\xi\).  On a fixed full-product logarithmic box a radial kernel is
\(\widehat\Phi_i(R_{\rm sc}\exp(L(\mathbf x)))\), where \(L\) is a fixed
linear form in the row and whole-column log norms.  Every derivative
is a fixed combination of Euler derivatives of \(\widehat\Phi_i\).
Lemma~\ref{lem:kernel-seminorms} bounds these uniformly for all
\(R_{\rm sc}>0\), with any prescribed radial decay and derivative orders.
One does not use the weaker bound \(R_{\rm sc}^J\) for
\(R_{\rm sc}\le Z^\xi\).
After its displayed central power has been extracted, an inverse root
is \((q_u/Z^{a_0})^{-1/2}\) on the fixed box; its derivatives cost
only fixed constants depending on \(H_N\) and their order.
Lemma~\ref{lem:smooth-calculus} uses the full weighted Fourier
integral, not a supremum over heights below \(Z^\xi\).  A required
height degree \(J\), which may depend on \(N\), raises an input
seminorm order and a fixed constant, not a \(Z^{J\xi}\) exponent.
Reflection similarly acts on at most two plains with fixed gamma
shape; larger derivative orders raise finite seminorm and height
orders, while the upper-annulus subshare \(\xi/2\) remains fixed.
The fixed polynomial height factor in Equation~\eqref{old-eq:3.5}
is integrated by these weighted Fourier measures at its required finite
order, without changing the previously chosen exponent of \(Z\).

The discrete label counts are also used only once.  After extracting
the displayed exponent for a family of common supports, divisors, or
allocations, divide its absolute weighted counting measure by that
total mass before applying a separated tuple seminorm.  This leaves a
measure of mass at most one; its already extracted exponent is not
counted again in the smooth norm.  Each Fourier measure is common to
the live labels by the whole-product construction.

\paragraph{Choices before specifying the slot count.}
Choose the parameters in the following order.  First, from the bounded
real ranges and \(\epsilon\), choose \(\rho,\sigma,\delta>0\), with
\(\delta\) small compared to \(\rho,\sigma\), so that
\[
 T_{\rm term}:=\rho+\delta+\rho/6+3\sigma+5\sigma/3<\epsilon/4.
\]
This bounds a terminal width-floor, diagonal, or exceptional loss;
the displayed frequency and numerical support corrections belong instead to the per-stage
\(\xi\) allowance.  With \(D\) and \(C_*\ge1\) as above, choose,
still before specifying \(N\) or its fixed profiles,
\begin{equation}
 \begin{aligned}
 \xi&<\min\left\{\frac\delta2,\frac\rho{30},
                  \frac{\sigma}{4C_*},
                  \frac{\epsilon}{16C_*D}\right\},\\
 \eta&<\min\left\{\frac\sigma6,
                   \frac{\epsilon}{16C_*D}\right\},\qquad
 \epsilon_0<\frac{\epsilon}{16C_*D}.
 \end{aligned}
 \label{old-eq:2.1i}
\end{equation}
These choices make the strict drop at least \(\sigma/2\), preserve
the comparison margins and padded core, and are uniform for
\(\kappa\in[3/4,1]\).

\paragraph{Choices for a fixed slot system.}
Now fix any \(Z\)-independent \(N\), arithmetic data, and profile
windows.  Form \(H_N\) and the finite full support boxes through
depth \(D\).  Their fixed factors \(\exp(O(H_N))\) belong to the
constants.  After imposing an eventual threshold
\(\log Z\ge C H_N/\xi\), the threshold in
Equation~\eqref{eq:centered-reflection-length}, and
Equation~\eqref{eq:centered-localization-threshold}, all occurring
column, radical, and mask norms have bounded logarithmic ranges
independent of moving labels.  In those ranges choose every
arbitrarily small power estimate with local shares whose sum in a
stage is at most \(\epsilon_0\).  These shares may depend on \(N\).
For example, the two divisor masses and the at most \(N\) frozen-slot
masses in mask deletion may each use a share
\(\epsilon_0/(10(N+2))\).  In the prime estimate, the principal
squared exponent is exactly \(\kappa z\), and the contour displacement
cost is \(2e z\), with bounded total \(z\), not \(N\eta\).  Choose
\(e\) for its assigned share; the fixed character expansions and
powers of \(\log Z\) are absorbed after an \(N\)-dependent threshold.
The same reasoning applies as individual \(z_i\) approach zero,
since \(P_i\ge1\).  For divisor allocations use the global bounds
\(\tau_{N+2}(v)^C\ll_{N,C,a}q_v^a\) and
\(C_N^{\omega(v)}\ll_{N,a}q_v^a\) with a sufficiently small
\(a>0\), not a fixed loss at every prime.  Rankin's fixed-radical
bound is treated with the same local shares.  The one normalized pool
average has only its single density loss in a stage.

The induction is simultaneous for every surviving subset of these
original \(N\) slots, with the same mesh.  To make this precise, for
\(0\le d\le D-2\), let
\[
 \mathcal E_d
 :=T_{\rm term}+(d+1)C_*(\eta+\xi+\epsilon_0)
\]
be the permitted error when at most \(d\) strict calls remain.
A terminal group, including its bounded reflection or comparison
preprocessing, has error at most \(\mathcal E_0\).  A strict edge
and its fixed number of same-band operations use the child envelope
\(\mathcal E_{d-1}\) plus at most
\(C_*(\eta+\xi+\epsilon_0)\), giving \(\mathcal E_d\).
The Gauss allowance \(\epsilon_G\) in
Equation~\eqref{old-eq:2.5} denotes this appropriate envelope and
the current local shares; it is not a request to reapply the moment
lemma with an \(N\)-dependent loss and a new mesh.  Triangle
inequalities take the maximum error up to the already counted mass,
and Cauchy--Schwarz averages the errors of the two children.
Consequently one terminal loss occurs along a branch, not at every
ancestor.  The largest path error is at most
\(T_{\rm term}+C_*D(\eta+\xi+\epsilon_0)<\epsilon\) by
Equation~\eqref{old-eq:2.1i}.

Finally choose the required finite kernel, reflection, prime, and
terminal seminorm and polynomial-height orders backwards through
these \(D\) stages.  The full weighted Fourier estimates and uniform
Euler-kernel estimates above make every such choice finite.  Choose
one final lower threshold for the aggregate support inequalities,
pool separation and density, the raw frequency tails, and all fixed
logarithmic losses.  The orders, constants, and threshold may grow
with \(N\) and the fixed data, but \(\eta,\xi,D,C_*\) in the
exponent comparison do not.

Each new extra mask is supported on frozen columns or a frozen
amplifier prime, and its logarithmic norm increases by a bounded
total length at a stage.  It is fixed within every child row sum.
The only row-dependent zeros there are the natural zeros of the row;
the declared moving zeros remain in the fixed twist and remain counted.
Over depth \(D\) the masks remain of polynomial norm.
On an exceptional row, the redundant part of the full moving union
and the row radical, together with these extra masks, forms the
common polynomial-size \(\mathfrak R_*\); the lattice cancellation
is uniform for it.  The fixed-numerator ray lemma controls the
finite choices at \(\mathcal S\), without absorbing a moving good
prime into a fixed modulus.

When this estimate and the inverse moment are applied together, all internal
orders in both arguments, including the reflection-kernel orders,
are fixed first.  The later external Fourier-tail order in
Lemma~\ref{lem:smooth-calculus} lies outside this internal
propagation: it raises only the external input seminorm and does
not change a previously fixed internal height order.  This proves
the asserted finite-order uniformity.  The finite induction proves
the natural assertion, and the initial mask deletion proves
Lemma~\ref{lem:plain} in its full stated form.

For the later physical application, \(q=0\) and the row \(u\)
is sixth-power-free.  If a prime outside \(\mathcal S\) has
valuation \(1,\ldots,5\) in \(u\), its local character has order
\(6/\gcd(6,v_p(u))>1\), so the inducing character cannot belong
to \(\Theta\).  The exceptional physical rows are therefore
supported on \(\mathcal S\); sixth-power-freeness makes this a
finite set, up to the finite unit group.  With abstract moving
twists, additional exceptional rows can arise by cancellation.
They were included in Equation~\eqref{eq:exceptional-row-count}
and the fixed-character volume argument.
\section{Prime amplitudes and refined row counts}\label{sec:refined-row-counts}

We use the current Part~II arithmetic data and the physical slots of
Equation~\eqref{eq:compensated-probe-definition}.  Instantiate the shared
zero detector of Section~\ref{sec:detector} with these same fixed data.
The main and error factors below belong to the retained dynamic decomposition
in Equation~\eqref{eq:dynamic-compensated-decomposition}; no individual local
quotient is asserted outside that region.  Every retained external coordinate obeys
the single height allocation in Equation~\eqref{eq:stage-height-allocation}.

\subsection{Prime amplitudes}\label{sec:amplitudes}

For a main physical slot of scale \(P_i=Z^{\ell_i}\), write its
central factor as
\[
 Q_i(u;z)=P_i^{-1/2}\sum_{p\in1_T}
   \overline{\chi_p(u)}W_i(q_p/P_i)(q_p/P_i)^{z-1}.
\]
The main part \(\mathcal Q_i\) in Proposition~\ref{prop:probe-errors} is exactly
\[
 \mathcal Q_i(u;z)=-\sum_{p\in1_T}\overline{\chi_p(u)}q_p^{z-1}W_i(q_p/P_i)
 =-P_i^{z-1/2}Q_i(u;z).
\]
As in Section~\ref{sec:compensation}, each sum over \(p\in1_T\) retains
the same allowed set \(\mathcal P_i(Z)\), including its exclusion of \(S\).
The original zero extension is retained, so a prime dividing \(u\)
does not contribute to this main slot.  The real part of \(z\) is
in a fixed bounded range and its imaginary part is one of the
external frequencies restricted by the common height allowance.

\begin{lemma}[Prime bound in a bin]\label{lem:bin-prime-bound}
Under the hypotheses of Lemma~\ref{lem:detector-dyads}, with the
prime-annulus Mellin frequency included in its cumulative allowance,
for every \(\epsilon_1>0\)
\[
 |Q_i(u;z)|\ll_{\mathcal A,e,\epsilon_1}
 U^{\epsilon_1}P_i^{a-1/2+O(e)}.
\]
The implied \(O(e)\) is uniform for \(51/100\le a\le1\) and the
fixed real ranges.  It holds for every main slot in the row.
\end{lemma}

\begin{proof}
Expand \(1_{p\in1_T}=|T|^{-1}\sum_{\theta\in\widehat T}\theta(p)\).
Every resulting prime character belongs to \(\mathcal X_u\).
Lemma~\ref{lem:logarithmic-control}, applied to the buffered disks,
bounds its logarithmic derivative on \(\Re s=a+8e\) by
\(O_{\mathcal A,e}(\log U)\); the logarithmic derivative of the
deleted product has the same bound.  Mellin inversion of the
corresponding smooth von Mangoldt annulus, shifted only in the
retained central range, gives
\(U^{\epsilon_1}P_i^{a-1/2+O(e)}\) after normalization.
The pure twist \((q_p/P_i)^{i\Im z}\) is translated into the
logarithmic-derivative argument before the added Mellin frequency
is truncated.  The joins have bounded real length.  On them the
logarithmic derivative is \(O_{\mathcal A,e}(\log U)\) inside its
allocated buffer, while on the starting line \(\Re s=2\) it is
absolutely bounded.  Apply the pointwise estimate in
Equation~\eqref{eq:pointwise-mellin-tail} to the untwisted transform
for the joins, and Equation~\eqref{eq:late-fourier-tail} for the
absolute-line tails.  Choosing the external order after their fixed
polynomial scale and height bounds gives the asserted estimate.

To pass from von Mangoldt coefficients to primes, divide the
annular weight by \(\log(P_i y)\).  On the fixed annulus,
\[
 (y\partial_y)^j\frac1{\log(P_i y)}
 =\frac{(-1)^j j!}{\{\log(P_i y)\}^{j+1}},
\]
so this preserves every fixed smooth seminorm for sufficiently
large \(P_i\), without introducing a power of \(P_i\) depending on
the derivative order.  Bounded \(P_i\) are handled by absolute
counting.  Prime powers
contribute \(P_i^{o(1)}\) after central normalization, since their
number in a norm annulus is \(O(P_i^{1/2+o(1)})\).
The polynomial-size punctures and their zero extensions are already
included in the logarithmic derivative.  This proves the bound.
\end{proof}

Here is a precise amplitude subdivision.  Choose a fixed bin width
\(\vartheta>0\).  First reduce \(e\) and \(\epsilon_1\), after the
slot lengths have been fixed, so that the preceding bound is at
most \(P_i^{\delta/2+\vartheta}\) for all sufficiently large \(Z\).
This is possible because
\(\log U/\log P_i=d/\ell_i\) stays in a fixed bounded range for
each fixed mesh.  For a main slot with \(Q_i\ne0\), put
\[
 g_i=\min\left\{\frac{\delta}{2},
   \max\left(0,\vartheta
     \left\lfloor\frac{\log|Q_i|}{\vartheta\log P_i}\right\rfloor
   \right)\right\}.
\]
Put \(g_i=0\) if \(Q_i=0\), and also put \(g_i=0\) for an error
slot from Equation~\eqref{old-eq:5.13e}.  Then, for a main slot,
\[
 |Q_i|\le P_i^{g_i+\vartheta},\qquad
 g_i>0\ \Longrightarrow\ |Q_i|\ge P_i^{g_i}.
\]
No lower bound is asserted for a slot with \(g_i=0\).
The finitely many possible vectors \((g_i)\), with the possible
endpoint value \(\delta/2\), partition the rows into amplitude
bins.  Define their length-weighted mean by
\begin{equation}
 q=\frac{\sum_i\ell_i g_i}{\ell},\qquad
 0\le q\le\delta/2,\qquad \ell=\sum_i\ell_i.
 \label{old-eq:6.6}
\end{equation}
This \(q\) records prime amplitude; it is not the conductor
exponent denoted \(q\) in the auxiliary fourth moment.

\subsection{Selected prime slots and the row counts}\label{sec:selection}

Use \(\Delta\) and \(\kappa\) from Equation~\eqref{eq:part-II-bootstrap}:
\[
 0<\Delta\le\frac1{24},\qquad
 \kappa=2\beta_*-1=\frac34+2\Delta\in(3/4,5/6],\qquad
 \alpha:=\frac56.
\]
The parameter \(\kappa\) remains dynamic; \(\alpha\) is the fixed slope
from sixth-power amplification.  Since \(\kappa<1\) and
\(\beta_*=(1+\kappa)/2\), the positive-slot hypothesis of
Lemma~\ref{lem:plain} is satisfied by equality.  Equation~\eqref{eq:part-II-bin-ceiling}
gives \(\delta\le\kappa\le\alpha\) for the current Part~II bins, including
the possible endpoint \(\delta=\alpha\).

Fix a dynamic and amplitude bin and fix the external physical
Mellin parameters.  Write \(z_{\rm phys}\) for the fixed third Mellin
parameter, so the physical slots are \(Q_i(u;z_{\rm phys})\).
All \(g_i\), and hence \(q\), are now fixed
within its row sum.  At base \(U\), the length of slot \(i\) is
\(w_i=\ell_i/d\), and the total available length is \(\ell/d\).
For a requested length \(0\le z\le\ell/d\), order the positive
\(g_i\) decreasingly and fill \(z\) fractionally in that order.
The gained exponent is at least \(qz\): if positive slots suffice
to fill \(z\), their initial weighted average is at least the
average \(q\) of all slots; otherwise retaining them all gives
gain \(q\ell/d\ge qz\).  Removing the one possibly fractional
slot loses at most \((\max_i w_i)\delta/2\).  Thus a fixed
subcollection of whole positive slots of length at most \(z\)
satisfies
\[
 \left|\prod_{i\ {\rm selected}}Q_i\right|^2
 \ge U^{2qz-\delta\max_iw_i}.
\]
When a strict moment inequality is needed, we first replace \(z\)
by \(z-\nu_0\) for a fixed small \(\nu_0>0\), or select no slot if
\(z\le\nu_0\).  The resulting loss in this display is at most
\(2q\nu_0+\delta\max_iw_i\), except in the stated zero-capacity
neighborhood, where an unweighted moment will be used.  Both losses
can be made smaller than any prescribed positive power.

The selected factors have exactly the coefficient class required
by the moments.  Conjugate both witness factors, including their
profiles, if necessary and write their row character as
\(\psi(n)=\nu(n)\overline{\chi_n(u)}\), \(\nu\in\Theta\).
Relative to this row, a physical prime has coefficient
\[
 \overline{\nu(p)}1_{p\in1_T}
 =\frac1{|T|}\sum_{\theta\in\widehat T}
       (\overline{\nu}\theta)(p).
\]
This is a fixed finite combination of members of \(\Theta\);
it is independent of the moving row in the fixed row sum.
There is no requirement that \(\eta(p)=1\).
The factor \((q_p/P_i)^{z_{\rm phys}-1}\) is part of its smooth profile.
Underlying prime supports remain disjoint before masks.
The marked moment permits this negative common row orientation.
To apply Lemma~\ref{lem:plain}, whose row character has positive
orientation, conjugate the whole product of both plain witness
factors and all selected prime factors.  Its absolute square is
unchanged, and its common row character becomes
\(\overline{\psi(n)}=\overline{\nu(n)}\chi_n(u)\).
Both witness profiles and every selected slot profile are conjugated,
and each selected slot coefficient becomes
\(\nu(p)1_{p\in1_T}\), again a fixed finite combination of members
of \(\Theta\).  Conjugation retains all zero extensions and underlying
prime supports.  It also preserves whether the inducing character
belongs to \(\Theta\), since \(\Theta\) is a group.
The witness and physical heights need not be equal.  Apply
Lemma~\ref{lem:smooth-calculus} to the rowwise witness parameters
after fixing the physical parameters and selected indices.
Derivatives in those parameters insert only logarithmic profile
weights.  The cost is a fixed power of \(1+T_1\), uniform over
moving rows and outer labels.  Every retained large row induces
outside \(\Theta\), because it ramifies at a prime outside \(S\).
Thus the \(z>0\) family condition of Lemma~\ref{lem:plain} holds.

For an actual plain length \(m\le1/2\), its largest zero-loss
capacity from Lemma~\ref{lem:plain}, at effective row width one, is
\begin{equation}
 z_P(m)=\frac{1-2m}{6\kappa}
       =\frac{1-2m}{9/2+12\Delta}.
 \label{eq:plain-zero-loss-capacity}
\end{equation}
Indeed the moment is applied to two copies of the plain witness,
so its condition is \(2m+6\kappa z\le1\).
For an inverse witness of length \(r<1\), put
\(z_M(r)=(1-r)/2\).  We use this capacity only where the second
strict inequality in Lemma~\ref{lem:marked} also has a fixed margin.

\begin{proposition}[Row counts from the witnesses]\label{prop:detector-counts}
Let \(\mathcal B\) be a fixed dynamic and amplitude bin of retained
sixth-power-free physical rows \(q_u\asymp U=Z^d\), with
\(a>51/100\), \(0<\delta=2a-1\le\alpha\), and mean amplitude
\(q\in[0,\delta/2]\).  Put \(x=q/\delta\), so \(0\le x\le1/2\),
and define
\[
 D_x=3-\frac{17x}{9},\qquad
 P_x=\left(2-\frac{8x}{9}\right)(1-x).
\]
Assume the total available prime length exceeds \(7/37\) by a
fixed positive amount.  For every \(t\in[1,3/2]\) and every
\(\epsilon>0\), the capacity decrements and slot mesh can be chosen
using only \(\epsilon,\Delta\) and the bounded real ranges so that
\[
 \#\mathcal B\ll_{\mathcal A,\epsilon}
 U^{\max\{R_{\rm short}(t),L(t)\}+\Delta/4+\epsilon}
 (1+T_1)^{A_{\mathcal A}},
\]
where \(A_{\mathcal A}<\infty\) is uniform over moving rows and
\[
 R_{\rm short}(t)=1-\delta+\frac{\delta P_x}{D_x}(3/2-t),
 \qquad
 L(t)=1-\delta+(\alpha-\delta)(t-1).
\]
The estimate is valid for the rowwise witnesses of
Proposition~\ref{prop:detector-witness}, with its height condition.
If no prime slots are selected, then \(t=1\) instead gives
\[
 \#\mathcal B\ll_{\mathcal A,\epsilon}
 U^{1-2\delta/3+\epsilon}(1+T_1)^{A_{\mathcal A}},
\]
without a prime-supply hypothesis.  At zero inverse capacity use
Lemma~\ref{lem:inverse-amplification}; at zero plain capacity use
the zero-slot case of Lemma~\ref{lem:plain}.
\end{proposition}

\begin{proof}
Subdivide by the witness presentation and dyadic pair.  The number
of choices is \(O_{\mathcal A}((\log U)^2)\); the remaining
rowwise smooth parameters are handled by the preceding Sobolev
argument.  Work in one subdivision, denoting it again by
\(\mathcal B\), and let \(r,m\) be the actual lengths
from Proposition~\ref{prop:detector-witness}.  Whenever the selected
slots satisfy the corresponding moment hypotheses and have total
requested capacity \(z\), their spike and the individual witness
spikes imply, after reducing preliminary losses,
\[
 \begin{array}{ll}
 \#\mathcal B
 \ll U^{1-\delta r-2qz+\epsilon}(1+T_1)^{A_{\mathcal A}},
 &\text{from Lemma~\ref{lem:marked}},\\[2mm]
 \#\mathcal B
 \ll U^{1-2\delta m-2qz+\epsilon}(1+T_1)^{A_{\mathcal A}},
 &\text{from Lemma~\ref{lem:plain}}.
 \end{array}
\]
In the second line use two copies of \(S_m\), so the denominator
is \(|S_m|^4\), not \(|S_m|^2\).  The effective width is one
because the twist \(\nu\) is fixed within the row sum and has no
moving conductor radical.  Each formula includes the arbitrarily
small capacity, rounding, moment, and witness losses.

\paragraph{Inverse witnesses without selected primes.}
Whenever no inverse slot is selected, use instead the sixth-power
amplification of Lemma~\ref{lem:inverse-amplification}.  It applies
to these physical rows because their ideal valuations are at most
five and their original zero extensions are unchanged.  After the
present subdivision, \(U,t,D,D_*\) are common, and the inverse base
profile is
\[
 W_{\rm base}(y)=W_1(y)V_{\le}(Dy/D_*).
\]
Its fixed logarithmic seminorms are uniformly bounded: a derivative
of the second factor is supported where its scaled argument lies in
a fixed compact interval.  The remaining rowwise parameters are
\(\sigma\) in a fixed compact interval and the pure twist
\(-(\gamma-\nu)\), of absolute value at most \((3I+1)T_1\).
The rowwise assertion of Lemma~\ref{lem:inverse-amplification}
therefore applies with that height range.  Replacing its factor
\((1+(3I+1)T_1)^A\) by
\((3I+2)^A(1+T_1)^A\) changes only a fixed constant.

For clarity, this use is uniform even when \(r\) approaches one
with \(U\).  If \(0\le r\le R_0\), Equation~\eqref{eq:amplified-note}
with preliminary loss \(\epsilon_0\) has exponent
\[
 \max\left\{1,\frac{1+5(1+c)r}{6}\right\}
       +(1+r)\epsilon_0
 \le e(r)+\frac{5cR_0}{6}+(1+R_0)\epsilon_0,
 \qquad e(r)=\max\left\{1,\frac{1+5r}{6}\right\}.
\]
For a requested loss \(\epsilon_m>0\), take, for example,
\(c=3\epsilon_m/(10\max\{R_0,1\})\) and
\(\epsilon_0=\epsilon_m/(4(1+R_0))\).  These are fixed before
\(U\), and the two extra terms are at most \(\epsilon_m/2\).
Thus division by the inverse witness spike gives
\begin{equation}\label{eq:no-slot-inverse-count}
 \begin{aligned}
 \#\mathcal B&\ll
 U^{e(r)-\delta r+\epsilon}(1+T_1)^{A_{\mathcal A}},\\
 e(r)-\delta r&=
 \begin{cases}
  1-\delta r,&0\le r\le1,\\
  1-\alpha+(\alpha-\delta)r,&r\ge1.
 \end{cases}
 \end{aligned}
\end{equation}
In particular this is not an application of the strict marked
moment with the shrinking margin \(1-r\).

\paragraph{Cases requiring no selected primes.}
We first dispose of the cases in which a witness already gives the
required count without selecting primes.
For \(r\ge1\), Equation~\eqref{eq:no-slot-inverse-count} uses effective
moment exponent \((1+5r)/6=1-\alpha+\alpha r\).
Division by the inverse spike gives
\(1-\alpha+(\alpha-\delta)r\).
Because \(\delta\le\alpha\) and \(r\le t+O(\epsilon)\), this is at
most
\begin{equation}
 L(t)=1-\alpha+(\alpha-\delta)t
     =1-\delta+(\alpha-\delta)(t-1)
 \label{eq:long-count}
\end{equation}
up to \(O(\epsilon)\).  This includes \(r=1\), where the two
branches in Equation~\eqref{eq:no-slot-inverse-count} agree.
If \(m\ge1/2\), the zero-slot case of Lemma~\ref{lem:plain}
gives count exponent \(1-2\delta m\le1-\delta\).
Since \(R_{\rm short}(t)\ge1-\delta\), this also satisfies the
stated bound.  We may therefore assume for the remaining argument that
\[
 r<1,\qquad m<1/2.
\]
Both capacities \(z_M(r)\) and \(z_P(m)\) are then positive;
when either is too small for the fixed decrement, we will use its
unweighted estimate below.

\paragraph{Comparing the positive capacities.}
For the inverse capacity \(z_M(r)\), the resulting ideal exponent
is \(A_I(r)\) below.  To compare the plain exponent first replace
\(z_P(m)\) by its value at \(\Delta=0\), namely
\(2(1-2m)/9\).  Since the actual \(m\) is at least
\(t-r-O(\epsilon)\), the resulting ideal comparison exponent is
\begin{equation}
 \begin{aligned}
 A_I(r)&=1-\delta\{x+(1-x)r\},\\
 S_t(r)&=1-\delta\left\{\frac{4x}{9}
          +\left(2-\frac{8x}{9}\right)(t-r)\right\}.
 \end{aligned}
 \label{eq:selected-counts}
\end{equation}
Both the actual plain exponent
\(1-2\delta m-2q(1-2m)/(9/2+12\Delta)\) and its baseline
version decrease with \(m\): their derivatives are respectively
\[
 -2\delta+\frac{4q}{9/2+12\Delta}<0,\qquad
 -2\delta+\frac{8q}{9}<0.
\]
The \(O(\epsilon)\) replacement of \(m\) is therefore legitimate.

The two affine expressions in Equation~\eqref{eq:selected-counts}
cross at
\[
 r_*(t)=
 \frac{(2-8x/9)t-5x/9}{D_x}.
\]
Here \(D_x\ge37/18>0\).  Directly,
\[
 r_*(3/2)=1,\qquad
 r_*(1)=\frac{2-13x/9}{3-17x/9}\ge\frac{23}{37},
\]
and
\[
 t-r_*(t)=\frac{(1-x)t+5x/9}{D_x}.
\]
The last expression is increasing in \(t\), equals \(1/2\) at
\(t=3/2\), and at \(t=1\) is
\((1-4x/9)/(3-17x/9)\ge1/3\).
Thus, throughout the stated ranges,
\[
 r_*(t)\ge\frac{23}{37},\qquad
 \frac13\le t-r_*(t)\le\frac12,\qquad r_*(t)\le1.
\]

Use the plain count when \(r\le r_*(t)\), and the inverse count
when \(r_*(t)\le r<1\), except within the fixed small
zero-capacity neighborhoods.  On the inverse side \(r\ge23/37\).
After decreasing \(z_M(r)\) by \(\nu_0\),
\[
 1-r-2z\ge2\nu_0>0,\qquad
 3-2r-8z=4(1-r-2z)+(2r-1)>0.
\]
The second inequality has a margin at least \(9/37\) before its
positive first term.  These are precisely the two strict width
conditions of Lemma~\ref{lem:marked} at row width one.
The inverse capacity is at most
\((1-23/37)/2=7/37\).  On the plain side,
\(m\ge1/3-O(\epsilon)\), so its capacity is at most
\(2/27+O(\epsilon)\).  At \(d=h\) the available length is
\[
 \frac{\ell}{h}=\frac8{39},\qquad
 \frac8{39}-\frac7{37}=\frac{23}{1443}>0.
\]
The assumed positive supply margin and a sufficiently fine mesh
therefore permit every selection just made.

On the plain side, replacing the baseline capacity by the actual
capacity in Equation~\eqref{eq:plain-zero-loss-capacity} raises
the count exponent by
\begin{equation}
 \begin{split}
 2q(1-2m)
 \left(\frac29-\frac1{9/2+12\Delta}\right)
 &=\frac{24q(1-2m)\Delta}{(9/2)(9/2+12\Delta)}\\
 &\le\frac{\Delta}{4}+O(\epsilon).
 \end{split}
 \label{eq:capacity-comparison}
\end{equation}
For the inequality use \(m\ge1/3-O(\epsilon)\), \(2q\le1\),
and the lower bound four for each denominator.  The constants in
the \(O(\epsilon)\) term are uniform.

The weighted average
\begin{equation}
 \begin{aligned}
 R_{\rm short}(t)
 &=\frac{(2-8x/9)A_I(r)+(1-x)S_t(r)}{D_x}\\
 &=1-\delta+\frac{\delta P_x}{D_x}(3/2-t)
 \end{aligned}
 \label{eq:short-crossing-count}
\end{equation}
is independent of \(r\): the \(r\)-coefficients cancel, and the
weights sum to \(D_x\).  It is the common value at \(r_*(t)\).
Since \(S_t\) is increasing in \(r\) and \(A_I\) is decreasing,
the chosen short count is at most \(R_{\rm short}(t)+\Delta/4+O(\epsilon)\).

\paragraph{Small capacities and the conclusion.}
If \(r<1\) but \(z_M(r)\le\nu_0\), the same no-slot estimate,
using \(e(r)=1\), gives
\[
 1-\delta r\le1-\delta+2\delta\nu_0.
\]
If \(z_P(m)\le\nu_0\), then
\(1-2m\le6\kappa\nu_0\le6\nu_0\), and its count is at most
\(1-\delta+6\delta\nu_0\).  Since \(R_{\rm short}(t)\ge1-\delta\),
choosing \(\nu_0\) within the prescribed \(\epsilon\) preserves
the short bound.  Finally, \(r=1/2-O(\epsilon)\) and \(t\ge1\)
force \(m\ge1/2-O(\epsilon)\), so the same unweighted plain
estimate applies.  No negative capacity is requested.
Together with the cases requiring no selected primes, this proves the stated selected
count after adding the finitely many subdivisions.

Finally, use no prime slots and take \(t=1\).  For the inverse
side use Equation~\eqref{eq:no-slot-inverse-count}, including
\(e(r)=1\) for every \(r\le1\); for the plain side use the
zero-slot case of Lemma~\ref{lem:plain}.  The same short calculation
with selection gain zero, that is, with \(x=0\) in the two comparison
lines, gives \(R_{\rm short}(1)=1-2\delta/3\); the long bound is
\(L(1)=1-\delta\).  Neither input has a prime-supply hypothesis,
proving the final assertion.
\end{proof}

\begin{remark}[Unselected inverse witnesses beyond the Part II bin ceiling]
The derivation of Equation~\eqref{eq:no-slot-inverse-count} uses only the inverse spike from
Proposition~\ref{prop:detector-witness} and
Lemma~\ref{lem:inverse-amplification}.  It can therefore be repeated for any
instance of the shared detector satisfying those hypotheses, independently
of the current Part~II bin ceiling.  Consequently, for each fixed
\(t\in[1,3/2]\), the bound in Equation~\eqref{eq:no-slot-inverse-count}
holds on each fixed presentation/dyadic subdivision of a fixed dynamic
bin of retained rows with actual witnesses from that proposition whenever
\[
 a>51/100,\qquad 1/50<\delta=2a-1\le1,
\]
and \(r\) lies in the prescribed bounded nonnegative range.  It uses
the common inverse profile \(W_{\rm base}\), rowwise parameter bounds,
and witness loss and height hypotheses used in the preceding proof,
but requires neither
\(\delta\le\alpha\) nor a prime-supply hypothesis.  This conclusion
does not include all rows in the floor bin \(a=51/100\), which need
not have an actual witness.
\end{remark}

The formulas in Proposition~\ref{prop:detector-counts} describe the
only row exponents needed below.  The cardinality factor
\((1+T_1)^{A_{\mathcal A}}\) is kept explicit here.  We verify the
additional target-dependent height ceiling required by
Lemma~\ref{lem:late-height-closure}.  Let \(\epsilon_{\rm ht}>0\) be the
minimum of the finitely many detector height allowances already chosen
with the real losses.  For the fixed target, after the internal profile
orders are fixed, let \(A_{{\rm ht},\eta}\ge0\) dominate their finitely many
height orders.  Set
\[
 \tau_{0,\eta}:=
 \frac{d_{\min}\epsilon_{\rm ht}}{20(A_{{\rm ht},\eta}+1)}>0.
\]
For \(0<\tau\le\tau_{0,\eta}\), \(T_1=Z^\tau\), and sufficiently large
\(Z\), the inequality \(U\ge Z^{d_{\min}}\) gives
\[
 (1+T_1)^{A_{{\rm ht},\eta}}
 \le 2^{A_{{\rm ht},\eta}}Z^{d_{\min}\epsilon_{\rm ht}/20}
 \le U^{\epsilon_{\rm ht}/10}.
\]
Thus this ceiling enforces all the preceding detector height hypotheses.
It is fixed before the external tail order and \(Z\); increasing that
order changes only external test seminorms, not these internal height
orders.  Lemma~\ref{lem:late-height-closure} then makes the final height
choice without changing the positive real margins or the slot mesh.

\section{The seven-eighths bound}\label{sec:conclusion}

We complete the contradiction assumed in Part~II.  Recall from
Equations~\eqref{eq:part-II-bootstrap} and~\eqref{eq:part-II-bin-ceiling}
that
\[
 0<\Delta=\beta_*-\frac78\le\frac1{24},\qquad
 \kappa=\frac34+2\Delta\le\frac56,\qquad
 \delta=2a-1\le\kappa
\]
for every retained bin.  The compensated low estimate is
Proposition~\ref{prop:probe-low}. For the same physical probe, we first
normalize the principal term of its high expansion, then bound the remaining
rows and verify the target-independent margins required by
Proposition~\ref{lem:continuation-criterion}.

The geometry and Mellin exponent are
\[
 h=\frac{13}{16},\qquad \ell=\frac16,\qquad
 l_x=\frac{17}{48},\qquad l_y=\frac{23}{48},\qquad
 h=1-l_x+\ell,
\]
\[
 C(s)=C_{\mathrm{II}}(s)=s-\frac{11}{16},
 \qquad C(7/8)=\frac3{16}.
\]
These are the values in Equations~\eqref{old-eq:5.1}
and~\eqref{eq:part-II-mellin-target}.  Write \(s\) for the local variable
called \(x\) in Section~\ref{sec:local-compensation}; the amplitude ratio
\(x=q/\delta\) below is a different real number.

Use Definition~\ref{def:stage-high-data} with
\[
 \mathscr I_\eta(Z)=I_{\eta,\mathrm{modified}}(Z),\qquad
 \mathfrak H_{\eta,u,Z}(s,w,z)
     \ \text{as in Equation~\eqref{eq:holomorphic-selected-tuple}}.
\]
In the shared analytic lemmas take \(\sigma_0=7/8\).
Section~\ref{sec:local-compensation} verifies every part of these data:
Equation~\eqref{eq:compensated-poisson-identity} is the absolutely
convergent high identity for the independently defined finite expression
in Equation~\eqref{eq:compensated-probe-definition}, and the full
correction is holomorphic in the two required Euler regions and satisfies
Equation~\eqref{eq:stage-all-height-majorant}.  The excluded set, physical
row masks, and calibration are unchanged.  In particular there is no
additional Euler factor outside this full correction.  Its scalar
denominator is \(L_F^S(s,\eta\overline{\chi_\bullet(u)})\), and its
Gaussian is \(\Phi(s+z-1)\).

In the estimates below, \(0<e<10^{-3}\) is an admissible detector width.
Its final choice, together with the remaining real losses, is made in
the concluding order of choices.

\subsection{The principal normalizer}\label{sec:principal-signal}

We verify the two correction hypotheses of
Lemma~\ref{lem:principal-residue-interface}.  First,
Equation~\eqref{eq:principal-local-tuple-bound} gives
\[
 |\mathfrak H_{\eta,1,Z}(s,w,z)|\ll Z^{\ell\Re z}
\]
uniformly in all imaginary parts on every fixed real box in the second
Euler region.  In particular this holds on the entire rectangle in
Equation~\eqref{eq:principal-correction-bound}, with height degree zero.
This is a bound for the full tuple correction.

For the residue value, define
\[
 S_i(Z)=\sum_{p\in\mathcal P_i(Z)}W_i(q_p/P_i)q_p^{-5/6},
 \qquad P_i=Z^{\ell_i}.
\]
Lemma~\ref{lem:ray-prime-normalizer}, including its assertion about deletion
of a fixed finite set, gives
\[
 S_i(Z)\sim\frac{P_i^{1/6}}{|T|\log P_i}
       \int_0^\infty W_i(y)y^{-5/6}\,dy.
\]
Each leading constant is positive.  Since the slot count is fixed,
all \(S_i(Z)\) are positive for every sufficiently large \(Z\), with
a common lower threshold allowed to depend on the fixed data.  Set
\begin{equation}\label{eq:principal-scalar}
 A_T(Z)=(-1)^K Z^{-\ell/6}\prod_{i=1}^K S_i(Z).
\end{equation}
Using \(\log P_i=\ell_i\log Z\) and \(\sum_i\ell_i=\ell\), we obtain
\[
 A_T(Z)\sim
 \frac{(-1)^K}{|T|^K(\log Z)^K}
 \prod_{i=1}^K\left\{\frac1{\ell_i}
       \int_0^\infty W_i(y)y^{-5/6}\,dy\right\}.
\]
Consequently \(A_T(Z)\ne0\) on that range and
\[
 |A_T(Z)|\asymp_{T,K,(\ell_i,W_i)}(\log Z)^{-K},
 \qquad |A_T(Z)|^{-1}\ll_\epsilon Z^\epsilon
\]
for every \(\epsilon>0\).  The lower threshold may depend on the target's
finite excluded set; no uniform prime asymptotic in a moving ray conductor
is being used.

On the principal second Euler region, Section~\ref{sec:local-compensation}
defines \(\mathcal B_p=G_p/H_p\) with \(H_p\ne0\), and
Equation~\eqref{eq:principal-local-approximation} gives
\(\mathcal B_p=-1+O(q_p^{-7/8})\), uniformly in all imaginary parts and
target unit phases.  Apply the principal factorization in
Equation~\eqref{eq:principal-slot-factorization} at \(w=1,z=1/6\), where
it remains valid even if the original quotient by \(P_p\) was undefined.
It gives
\[
 \mathfrak H_{\eta,1,Z}(s,1,1/6)
 =H_\eta(s)\prod_{i=1}^K
       \left(\sum_{p\in\mathcal P_i(Z)}
           W_i(q_p/P_i)q_p^{-5/6}\mathcal B_p\right),
\]
where \(H_\eta\) is the function in Equation~\eqref{eq:shared-principal-data}
for this same excluded set \(S\).  By nonnegativity of \(W_i\), its \(i\)th
slot equals
\[
 -S_i(Z)\{1+\rho_i(s)\},\qquad
 |\rho_i(s)|\ll P_i^{-7/8}
 \quad(\Re s=\beta_*+e,\ \Im s\in\mathbb R).
\]
Indeed the sum of the absolute local errors is at most a fixed multiple of
\(P_i^{-7/8}S_i(Z)\).  Choose any pretarget number
\[
 0<\kappa_P<\frac78\min_i\ell_i.
\]
Since \(K\) is fixed, \(\mathcal R_{\eta,Z}(s):=\prod_i(1+\rho_i(s))-1\)
satisfies \(|\mathcal R_{\eta,Z}(s)|\ll Z^{-\kappa_P}\) on that entire line.
Thus the exact residue correction is
\[
 \mathfrak H_{\eta,1,Z}(s,1,1/6)
 =H_\eta(s)Z^{\ell/6}A_T(Z)\{1+\mathcal R_{\eta,Z}(s)\},
\]
which is Equation~\eqref{eq:principal-correction-residue} with
\(A_\eta=A_T\) and \(\mu=\kappa_P\).  This factorization is asserted
only at the principal residue; it is not the correction used for general
contour moves.

Let \(c_S>0\) be the scalar in Equation~\eqref{eq:shared-principal-data},
whose positivity is part of Lemma~\ref{lem:principal-residue-interface}.
For sufficiently large \(Z\), define the normalized physical probe
\[
 J_{\mathrm{II},\eta}(Z)
   =\frac{I_{\eta,\mathrm{modified}}(Z)}{c_S A_T(Z)}.
\]
This is distinct from \(J_{\mathrm I,\eta}\), and both its numerator and
normalizer are independent of the analysis height \(T_1\).
Proposition~\ref{prop:probe-low} and the subpower bound for \(A_T^{-1}\)
give, for every \(\epsilon>0\),
\begin{equation}\label{eq:part-II-normalized-low}
 |J_{\mathrm{II},\eta}(Z)|
 \ll_{\eta,\epsilon} Z^{C(7/8)+\epsilon}.
\end{equation}
The same scalar is used for the high comparison.

Choose the fixed cutoff as required for
Equation~\eqref{eq:shared-principal-nonvanishing}.  The associated
\(H_\eta\) is holomorphic on \(\Re s>7/8\), satisfies the contraction
there, and uses this same \(S\).  Let \(\mathscr P_{\mathrm{II},\eta}\)
denote the \(u=1\) term of Equation~\eqref{eq:compensated-poisson-identity},
and set
\[
 f_{\mathrm{II},\eta}(Z)=\frac1{2\pi i}\int_{(2)}
 Z^{C(s)}e^{(s-5/6)^2}\frac{H_\eta(s)}{L_F^S(s,\eta)}\,ds.
\]
All hypotheses of Lemma~\ref{lem:principal-residue-interface} have now
been verified with \(\sigma_0=7/8\).  Its remainder estimate is
\[
 \begin{split}
 \left|\frac{\mathscr P_{\mathrm{II},\eta}(Z)}{c_S A_T(Z)}
       -f_{\mathrm{II},\eta}(Z)\right|
 \ll{}&
 Z^{C(\beta_*)+(1+h)e-m_w+\epsilon}
 +Z^{C(\beta_*)+e-m_z+\epsilon}\\
 &+Z^{C(\beta_*)+e-\kappa_P+\epsilon},
 \end{split}
\]
where the two geometric margins are
\begin{equation}\label{old-eq:5.14}
 m_w:=\frac{l_y}{20}=\frac{23}{960},\qquad
 m_z:=\frac h{600}=\frac{13}{9600}.
\end{equation}
Lemma~\ref{lem:principal-residue-interface} includes the residue paths,
the scalar Jacobian, and
the rightward shift of the main integral.  In particular the last term
is the error estimated on its original global line; it is not part of
\(f_{\mathrm{II},\eta}\).  Both this signal and the physical probe are
independent of \(T_1\).

\subsection{The compensated high exponent}

The low estimate and the principal-row comparison now concern the functions
$J_{\mathrm{II},\eta}$ and $f_{\mathrm{II},\eta}$ required by the continuation
criterion. It remains to prove, for a common $\sigma>0$,
\[
 \left|\frac{I_{\eta,\mathrm{modified}}(Z)
                -\mathscr P_{\mathrm{II},\eta}(Z)}{c_S A_T(Z)}\right|
 \ll_\eta Z^{C(\beta_*)-\sigma}.
\]
We estimate these nonprincipal contributions first relative to the raw
scale $Z^{C(7/8)}$. Division by $c_S A_T(Z)$ costs an arbitrarily small
power, reserved once in the final choice of margins.


\begin{lemma}[A high exponent for one row bin]\label{lem:high-bin}
Let \(0<d_{\min}<d_{\max}<\infty\), let \(U=Z^d\) with
\(d_{\min}\le d\le d_{\max}\), and let
\(\mathcal B\) be the finite set of retained nonprincipal physical rows
\(q_u\asymp U\) in one fixed dynamic bin \((i,a)\).  Put
\(\delta=2a-1\).  Assume the bin and height hypotheses of
Lemmas~\ref{lem:buffered-bins}, \ref{lem:detector-dyads}, and
\ref{lem:bin-prime-bound}, with
\(0<e<10^{-3}\), \(T_1=Z^\tau>2\), \(0<\tau\le d_{\min}/100\), and the cumulative
allocation in Equation~\eqref{eq:stage-height-allocation}.
The bounded real ranges, the positive losses \(e,\vartheta,\epsilon\), and the
slot system are fixed before the target.

For each retained tuple of external parameters and each subset of error
slots, partition \(\mathcal B\) pointwise into the amplitude sets of
Section~\ref{sec:amplitudes}, and into the witness subdivisions of
Section~\ref{sec:selection} when a witness count is used.  Suppose each
such set \(\mathcal C\) has main-slot mean \(q\in[0,\delta/2]\), as in
Equation~\eqref{old-eq:6.6}, and
\[
 \#\mathcal C\ll_{\mathcal A,\epsilon}
 U^{R+\epsilon}(1+T_1)^{A_{\mathcal A}},
\]
uniformly in the retained tuple and moving labels.  The occurring \(R,q\)
range over a fixed bounded set and may depend on the pointwise set, \(d,a\),
and \(\beta_*\), but not otherwise on the target.  For each occurring pair
define
\begin{equation}\label{eq:common-high-exponent}
 \begin{split}
 E(d)
 &=a-\frac78+h\left(\frac{17}{50}-\frac16\right)
       -a l_y-(1-a)\ell-(\delta/2-q)\ell
       +d\left(R+\frac\delta2-\frac{17}{50}\right)\\
 &=C_0+\frac23\delta+\frac q6-h(1-R)
       +(d-h)\left(R+\frac\delta2-\frac{17}{50}\right),
       \qquad C_0=-\frac1{48}.
 \end{split}
\end{equation}
If \(\mathcal B\ne\varnothing\), let \(E_{\max,Z}(d)\) be the supremum
of these values over all pointwise sets at all retained tuples.

On the central contours
\[
 \Re s=a+16e,\qquad \Re w=1-a-6e,\qquad \Re z=\frac{17}{50},
\]
the contribution of the original fixed dynamic-bin sum is bounded by
\[
 Z^{C(7/8)+E_{\max,Z}(d)+O(e+\vartheta+\epsilon)}
 (1+T_1)^{A'_{\mathcal A}},
\]
where \(\vartheta\) is the amplitude-bin width and
\(A'_{\mathcal A}<\infty\) is uniform in moving labels.  An empty bin
contributes zero.  The constant in the \(O(e+\vartheta+\epsilon)\) term
depends only on the bounded real ranges and the fixed slot system, not on
the target.  No separate integral of an individual pointwise set is
asserted.  After fixing \(\tau>0\), all discarded external pieces and joins
are \(O_{\mathcal A,N}(Z^B T_1^{-N})\), for a fixed \(B\) and every fixed
\(N\).
\end{lemma}

\begin{proof}
We verify the central hypothesis of
Lemma~\ref{lem:bin-contour-accounting}, which supplies the contour
move and its tails for the full correction just specified.  Its bin
ceiling holds by Equation~\eqref{eq:part-II-bin-ceiling}.  The physical
\(\Im z\) coordinate is included in its height allocation because it
enters the prime profiles.  The other witness, dyadic, and prime-annulus
coordinates are the fixed finite list used in the estimates of
Section~\ref{sec:selection}.  Their complete pointwise bounds include the
discarded auxiliary integrals on global or absolute lines.  Thus the
single cumulative allocation in Equation~\eqref{eq:stage-height-allocation}
applies, without renewing an allowance at a later estimate.

Only on the retained contours, Proposition~\ref{prop:probe-errors} gives
the decomposition in Equation~\eqref{eq:dynamic-compensated-decomposition}.
For \(I\subseteq\{1,\ldots,K\}\), its summand is
\[
 \mathfrak H_{\eta,u,Z}^{(I)}
 =\mathcal H_{\eta,u}(s,w,z)
       \prod_{i\in I}\mathcal D_i(u)
       \prod_{i\notin I}\mathcal Q_i(u;z).
\]
The required reflected primitive numerator bound follows from the
buffered estimate for the numerator presentation and its conjugate,
together with Lemma~\ref{lem:deleted-euler-factors}, as verified before
Proposition~\ref{prop:probe-errors}.  The retained \(w\) heights lie
in its allowed set.  Every retained numerator is nonprincipal.

Fix one amplitude set at one retained tuple.  For a main slot,
\(\mathcal Q_i=-P_i^{z-1/2}Q_i\) and
\(\lvert Q_i\rvert\le P_i^{g_i+\vartheta}\).  Assign \(g_i=0\) to
an error slot.  Equation~\eqref{old-eq:5.13e} bounds all error slots
in \(I\) jointly with the numerator; its proof uses the actual
conductor deficit in Equation~\eqref{old-eq:5.13d} simultaneously
for the distinct strict ramified labels.  It is not a separate
numerator allowance for each slot.  Since
\(\mathcal H_{\eta,u}\ll_\epsilon U^\epsilon\), multiplication of
these estimates gives, after choosing the preliminary powers,
\[
 \begin{split}
 \left|L^S(w,\chi_\bullet(u))
       \mathfrak H_{\eta,u,Z}^{(I)}(s,w,z)\right|
 \ll{}&U^{\delta/2+O(e)+\epsilon}(1+T_1)^{A_1}\\
 &\cdot
 Z^{\ell(17/50-1/2)+q\ell+O(e+\vartheta+\epsilon)}.
 \end{split}
\]
Here \(A_1<\infty\) is a fixed height order for the target and slot
system, and \(\sum_i\ell_i g_i=q\ell\), including the assigned zeros
for the errors.  No lower bound on an error slot has been used.

Multiplying by the assumed cardinality proves
Equation~\eqref{eq:stage-central-bound} with \(g=q\ell\), after taking
\(\varepsilon_c\) to be a fixed sufficiently large multiple of
\(e+\vartheta+\epsilon\).  This multiple depends only on the fixed slot
system and the bounded real ranges.  There are \(2^K\) error subsets.
The number of amplitude vectors is fixed, and the witness dyadic choices
contribute only a fixed power of \(1+\log Z\).  Thus the required
pointwise multiplicity bound holds.  These sets are formed only after
Lemma~\ref{lem:bin-contour-accounting} has moved the original dynamic-bin
sum using
\(\mathfrak H\).  They are not used for continuation and are not
integrated separately.

Apply Lemma~\ref{lem:bin-contour-accounting} with \(\sigma_0=7/8\)
and \(g=q\ell\).  Its exponent in
Equation~\eqref{eq:stage-high-exponent} is
\[
 a-\frac78+h\left(\frac{17}{50}-\frac16\right)-a l_y-\frac{\ell}{2}
       +q\ell+d\left(R+\frac\delta2-\frac{17}{50}\right).
\]
Because
\(-(1-a)\ell-(\delta/2-q)\ell=-\ell/2+q\ell\),
this is the first line of Equation~\eqref{eq:common-high-exponent}.
Substituting the geometry and \(a=(1+\delta)/2\) gives its second line.
In particular \(q\ell=q/6\) is a base-\(Z\) exponent, not \(dq/6\).
The explicit errors in Equation~\eqref{eq:stage-central-contribution}
are \(O(e+\vartheta+\epsilon)\) on the bounded \(d\)-range after the
remaining preliminary powers are chosen.  Its supremum is exactly the
one in the statement.  Lemma~\ref{lem:bin-contour-accounting} also gives
the stated external tails
with a scale degree fixed before \(N\).
\end{proof}

The floor bin \(a=51/100\) has \(\delta_0=1/50\) and requires no
zero witness.  The ideal count \(R=1\), together with
\(q\le\delta_0/2\), gives
\begin{equation}\label{eq:floor-bound}
 E(h)\le C_0+\frac34\delta_0=-\frac7{1200}<0.
\end{equation}
Its frequency slope \(R+\delta_0/2-17/50\) is positive, so this
bounds every \(d\le h\).  A small extension above \(h\) will be
controlled below.

\subsection{Small and large row norms}\label{sec:tails}

Set \(d_{\min}=1/100\).  Lemma~\ref{lem:compensated-absolute-local-bounds}
bounds the positive sum of the full selected tuples, with every \(G_p\)
retained before taking absolute values.  On the small-row lines
\[
 \Re s=\beta_*+e,\qquad \Re w=1/2,\qquad \Re z=17/50
\]
Equation~\eqref{eq:absolute-local-tuple-bound} gives
\[
 |\mathfrak H_{\eta,u,Z}(s,w,z)|
 \ll_\epsilon U^\epsilon\prod_iP_i^{17/50}
 =U^\epsilon Z^{17\ell/50},
\]
uniformly in all imaginary parts.  This is
Equation~\eqref{eq:small-correction-bound}, with height degree zero.
It remains valid at zeros of individual local factors.  The data for
Lemma~\ref{lem:outer-row-tails} were already verified above, so it
applies to every physical row \(u\ne1\) in these dyads.  Its
classification includes nontrivial unit numerators and permits a principal
denominator; no detector witness or selected moment is used here.

The \(d\)-independent part of the relative exponent is
\begin{equation}\label{eq:small-frequency-margin}
 h\left(\frac{17}{50}-\frac16\right)-\frac{l_y}{2}
 =-\frac{79}{800}.
\end{equation}
The row exponent in Equation~\eqref{eq:small-row-bound} is
\(63/50<2\).  Thus the small dyadic sum in
Equation~\eqref{eq:small-row-sum}, measured relative to \(C(\beta_*)\),
has exponent at most
\[
 -\frac{79}{800}+2d_{\min}+O(e+\epsilon)
 =-\frac{63}{800}+O(e+\epsilon).
\]
It is negative after the adjustable real losses are made sufficiently
small.  Write \(m_{\rm small}:=63/800\) for this fixed error-free saving.

For the large rows, fix \(z_\infty>2\).  On
\((\Re s,\Re w,\Re z)=(2,2,z_\infty)\), the same
Equation~\eqref{eq:absolute-local-tuple-bound} gives
\[
 |\mathfrak H_{\eta,u,Z}(s,w,z)|
 \ll_\epsilon U^\epsilon Z^{\ell z_\infty}
\]
for every physical row and all imaginary parts.  This verifies
Equation~\eqref{eq:large-correction-bound}.  Lemma~\ref{lem:outer-row-tails}
therefore gives, with \(B_0=l_x/2+1+l_y=53/32\),
\[
 |\mathscr R_\eta(U;Z)|
 \ll Z^{B_0+hz_\infty+\epsilon}U^{1+\epsilon-z_\infty},
\]
and, for every fixed \(\zeta>0\) after choosing
\(0<\epsilon<z_\infty-1\),
\[
 \sum_{U>Z^{h+\zeta}}|\mathscr R_\eta(U;Z)|
 \ll
 Z^{B_0+(h+\zeta)(1+\epsilon)-\zeta z_\infty+\epsilon}.
\]
Here \(\mathscr R_\eta\) is the row contribution of
Lemma~\ref{lem:outer-row-tails} for the
present physical expression.  A fixed sufficiently large \(z_\infty\)
makes the last exponent as negative as required.  It is chosen after
\(\zeta\) but before the target.  These invocations use the quotient-free
tuple bounds in Section~\ref{sec:local-compensation}, not the central
main/error factorization.

\subsection{The endpoint inequality}\label{sec:adaptive-endpoint}

We next bound \(E(h)\), first without adjustable losses or height
factors.  They will be restored with a quantified order of choices.
Recall
\[
 \kappa=\frac34+2\Delta,\qquad \alpha=\frac56.
\]
At the live upper endpoint \(\delta=\alpha\), take \(t=3/2\) in
Proposition~\ref{prop:detector-witness}.  Then
\(r\ge1-O(\epsilon)\).  Apply the no-slot amplified estimate
in Equation~\eqref{eq:no-slot-inverse-count} on both sides of
\(r=1\).  For \(r\ge1\), its exponent is
\[
 1-\alpha+(\alpha-\delta)r=1-\delta.
\]
For \(r<1\) in the stated \(O(\epsilon)\) neighborhood, the same
estimate uses \(e(r)=1\), so its exponent is
\(1-\delta r\le1-\delta+O(\epsilon)\).  The fixed amplification
margin is independent of this neighborhood.
Thus no selected primes are needed and the ideal row exponent is
\(R=1-\delta\).  Using \(q\le\delta/2\) in
Equation~\eqref{eq:common-high-exponent} gives
\begin{equation}
 E(h)\le C_0+\left(\frac34-h\right)\delta+O(\epsilon)
       =-\frac1{48}-\frac{\delta}{16}+O(\epsilon)<0.
 \label{eq:large-delta-endpoint}
\end{equation}

Now assume \(1/50<\delta<\alpha\).  Put \(x=q/\delta\in[0,1/2]\),
and use \(D_x,P_x,R_{\rm short},L\) from
Proposition~\ref{prop:detector-counts}.  The parameter \(\kappa\)
remains the dynamic value \(3/4+2\Delta\), not the upper bound
\(5/6\).  Thus the actual plain capacity used in that Proposition is
\[
 z_P(m)=\frac{1-2m}{6\kappa}
       =\frac{1-2m}{9/2+12\Delta},
\]
as in Equation~\eqref{eq:plain-zero-loss-capacity}.  The comparison in
Equation~\eqref{eq:capacity-comparison} is for this exact capacity and
raises the row exponent by at most \(\Delta/4\), apart from requested
small losses.  Define
\begin{equation}
 \mathcal J=(\alpha-\delta)D_x+\delta P_x,\qquad
 t=1+\frac{\delta P_x}{2\mathcal J},\qquad
 R_*=L(t).
 \label{eq:target-cutoff}
\end{equation}
The denominator is positive.  Indeed
\[
 \frac{37}{18}\le D_x\le3,\qquad
 \frac79\le P_x\le2,\qquad
 D_x-P_x=1+x-\frac{8x^2}{9}>0,
\]
and consequently
\[
 \frac{35}{54}\le\mathcal J\le\frac52.
\]
Because \(0<\delta<\alpha\), one has
\(\mathcal J>\delta P_x>0\), so \(1<t<3/2\).
Furthermore,
\[
 D_x\{R_{\rm short}(t)-L(t)\}
 =\delta P_x(3/2-t)-(\alpha-\delta)D_x(t-1)
 =\frac{\delta P_x}{2}-\mathcal J(t-1)=0.
\]
Thus \(R_*=R_{\rm short}(t)=L(t)\) and this \(t\) balances the two counts.
The formulas also have the stated closed-endpoint values: at
\(\delta=\alpha\), one has \(\mathcal J=\alpha P_x\), \(t=3/2\),
and \(R_*=R_{\rm short}(3/2)=L(3/2)=1-\delta\).  The corresponding crossing
has \(r_*(3/2)=1\), so its inverse capacity is zero.  The preceding
equality argument uses Equation~\eqref{eq:no-slot-inverse-count} at
that boundary, not a marked estimate with a vanishing margin.

\begin{lemma}[Compensated endpoint certificate]\label{lem:balanced-endpoint}
For \(0\le\delta\le5/6\) and \(0\le x\le1/2\), define
\(\mathcal J,t,R_*\) by Equation~\eqref{eq:target-cutoff} where
\(\mathcal J>0\), and let \(E_*\) be \(E(h)\) with
\(q=x\delta\) and \(R=R_*\).  Then
\[
 -E_*\ge\frac{49}{440640}>\frac1{10000}.
\]
In particular the bound is uniform on the range
\(1/50<\delta\le5/6\), including the closed endpoint.
\end{lemma}

\begin{proof}
Set \(y=1/2-x\in[0,1/2]\) and \(v=51+41y\).  To verify the
calculation, write
\[
 p_y=7+18y+8y^2,\qquad
 j_y=185+170y+(-138+12y+96y^2)\delta.
\]
Then
\[
 D_x=\frac{37+34y}{18},\qquad
 P_x=\frac{p_y}{9},\qquad
 \mathcal J=\frac{j_y}{108}.
\]
From \(R_*=1-\delta+(\alpha-\delta)\delta P_x/(2\mathcal J)\)
and Equation~\eqref{eq:common-high-exponent},
\[
 -E_*=
 \frac{1+(3+8y)\delta}{48}
 -\frac{13}{32}\,
   \frac{(5/6-\delta)\delta P_x}{\mathcal J}.
\]
Multiplying out gives the explicit quadratic
\[
 \begin{split}
 10368\mathcal J(-E_*)
 &=2j_y[1+(3+8y)\delta]
     -468(5/6-\delta)\delta p_y\\
 &=10(37+34y)
   -8(237+377y+26y^2)\delta\\
 &\hspace{12mm}
   +48(51+131y+94y^2+32y^3)\delta^2.
 \end{split}
\]
Multiplication by \(v\) and completion of the square yields
\begin{equation}
 \begin{aligned}
 10368v\mathcal J(-E_*)
 ={}&(3+5y)\bigl\{(4v\delta-79)^2+49\bigr\}\\
 &+4y\bigl\{
 4v\delta[(1+3y)(15+32y)\delta+9-13y]
       +265+3485y\bigr\}.
 \end{aligned}
 \label{eq:balanced-endpoint-certificate}
\end{equation}
This identity can also be checked by coefficients: expansion of
its right side as a polynomial in \(\delta\) gives respectively
\[
 \begin{split}
 &10v(37+34y),\\
 &-8v(237+377y+26y^2),\\
 &48v(51+131y+94y^2+32y^3),
 \end{split}
\]
which are the constant, linear, and quadratic coefficients in
the preceding display multiplied by \(v\).

Every term on the right of
Equation~\eqref{eq:balanced-endpoint-certificate} is nonnegative:
\(y,\delta\ge0\) and \(9-13y\ge5/2\).
The first line is at least \(49(3+5y)\), while
\(v=51+41y\le17(3+5y)\).  Hence
\[
 -E_*\ge\frac{49}{176256\mathcal J}
       \ge\frac{49}{440640}>\frac1{10000},
\]
using \(\mathcal J\le5/2\).
\end{proof}

The lower bounds for \(D_x\) and \(\mathcal J\) bound all derivatives
of the crossing and cutoff on the compact parameter ranges.
The saturation losses are uniform because \(\delta\ge1/50\),
and the inverse selection has the explicit supply and width margins
proved in Proposition~\ref{prop:detector-counts}.  At a zero
capacity its unweighted bound differs from \(1-\delta\) by at
most a constant times the chosen capacity decrement.  Since
\(R_*\ge1-\delta\), these cases obey the same comparison.
Thus all detector, bin, and rounding errors have a total
\(O(\epsilon)\) with a target-independent constant once their
individual requested losses are at most \(\epsilon\).
Let \(E_{\rm actual}(h)\) denote the scale exponent relative to
\(C(7/8)\) after the observed row bound
\(R\le R_*+\Delta/4+\epsilon_{\rm row}\) is inserted.
For now the explicit height factors remain outside this exponent.
The coefficient of \(R\) in Equation~\eqref{eq:common-high-exponent}
is \(d\), so at \(d=h\) the capacity replacement contributes
\(h\Delta/4\).  Denote by \(\epsilon_{\rm total}\) the sum of
\(h\epsilon_{\rm row}\) and all other adjustable real losses; the
preceding uniformity makes it arbitrarily small with a
target-independent coefficient.  The certificate gives the
comparison actually needed:
\begin{equation}
 \begin{split}
 E_{\rm actual}(h)-\Delta
 &\le-\frac{49}{440640}-\left(1-\frac h4\right)\Delta
       +\epsilon_{\rm total}\\
 &=-\frac{49}{440640}-\frac{51}{64}\Delta
       +\epsilon_{\rm total}.
 \end{split}
 \label{eq:adaptive-endpoint}
\end{equation}
The subtraction of \(\Delta=C(\beta_*)-C(7/8)\) is essential:
the certificate is not a claim that \(E_{\rm actual}(h)<0\) for
every \(\Delta\).  When the height factors are converted to a
reserved exponent allowance below, that allowance is added to
\(\epsilon_{\rm total}\).

\subsection{Frequency ranges}

For \(1/2\le d\le h\), the available length is
\[
 \frac{\ell}{d}\ge\frac8{39}>\frac15>\frac7{37}.
\]
Passing from base \(Z\) to base \(U\) changes a slot length from
\(\ell_i\) to \(\ell_i/d\le2\ell_i\), so one mesh chosen with
this factor of two satisfies the moment requirement throughout the
interval.  If
\[
 0<\zeta<5\ell-h=\frac1{48},
\]
then for \(h<d\le h+\zeta\) the supply remains greater than
\(1/5\), whose margin above \(7/37\) is \(2/185\).

For \(1/50<\delta<\alpha\), use the ideal row upper exponent
\(R=R_*+\Delta/4\).  For \(\delta=\alpha\), use
\(R=1-\delta\).  Both are at least \(1-\delta\), so
\[
 R+\delta/2-\frac{17}{50}
 \ge\frac{33}{50}-\frac{\delta}{2}\ge\frac4{25}>0.
\]
The floor has a positive slope as well.  Thus \(d=h\) bounds
all these exponents for \(d\le h\).  The cost of extending to
\(h+\zeta\) is at most \(2\zeta\): indeed
\[
 R_*\le1-\delta+\frac{\alpha-\delta}{2}\le\frac{17}{12},
 \qquad
 R_*+\Delta/4\le\frac{139}{96},
\]
and \(139/96+1/2-17/50<2\).  The slopes at \(\delta=\alpha\)
and floor cases are smaller than two.

For \(d_{\min}\le d\le1/2\) and \(\delta\le\alpha\), choose \(t=1\)
and use no selected primes.  Proposition~\ref{prop:detector-counts}
gives the ideal exponent \(1-2\delta/3\) outside the floor.
Use the slightly larger common exponent
\[
 R=\frac{76}{75}-\frac23\delta.
\]
At \(\delta=1/50\) it equals one, so it also covers the floor.
Its slope is \(101/150-\delta/6>0\) on this range.
Using \(q\le\delta/2\) in
Equation~\eqref{eq:common-high-exponent} yields
\begin{equation}
 E(d)\le E(1/2)+O(\epsilon)
 \le-\frac{529}{2400}+\frac{25}{96}\delta+O(\epsilon)
 \le-\frac{49}{14400}+O(\epsilon)<0.
 \label{eq:adaptive-intermediate}
\end{equation}
The middle expression follows by substituting
\(d=1/2\), \(R=76/75-2\delta/3\), and \(q=\delta/2\);
the last inequality uses \(\delta\le5/6\).
At \(\delta=\alpha\), the count used in
Equation~\eqref{eq:large-delta-endpoint} and its positive slope
already control every \(d\le h\).
Together with Section~\ref{sec:tails}, these cases cover every
physical row norm.  Only \(d\ge1/2\) uses selected physical prime factors
in a row moment; every physical slot remains in the high correction in all
ranges.

To compare with \(C(\beta_*)\), subtract
\(C(\beta_*)-C(7/8)=\Delta\) from each ideal exponent.
In the adaptive range this is precisely the comparison in
Equation~\eqref{eq:adaptive-endpoint}; its only positive capacity
allowance is \(h\Delta/4\).  The other endpoint ranges have
nonpositive ideal exponents before subtracting \(\Delta\).
Thus all of them retain the target-independent main high margin
\[
 m_{\rm hi}:=\left(1-\frac h4\right)\Delta
             =\frac{51}{64}\Delta.
\]
The low exponent has margin \(m_{\rm lo}:=\Delta\).
The principal contour and approximation margins were established in
Section~\ref{sec:principal-signal}; we now choose all losses together.

\subsection{Order of choices and conclusion}

We finish by separating the target-independent real choices from the
target-dependent height and external test orders.  The numerical choices
remain part of this application; the final height selection is supplied by
Lemma~\ref{lem:late-height-closure}.

\begin{proposition}[Order of choices]\label{prop:parameter-order}
Under the contradiction in Equation~\eqref{eq:part-II-bootstrap}, there
are numbers
\[
 0<\omega<\Delta,\qquad \sigma>0
\]
and a fixed geometry, moment losses, capacity decrements, slot system,
amplitude width, bin width \(e>0\), and extension \(0<\zeta<1/48\),
all chosen before the target character, with the following property.
For every primitive finite-order target \(\eta\), one can then choose
its fixed arithmetic data, a positive height exponent \(\tau_\eta\),
a finite external tail order \(N_\eta\), and a lower threshold
\(Z_{0,\eta}\) such that, for \(Z\ge Z_{0,\eta}\),
\[
 |J_{\mathrm{II},\eta}(Z)|\ll_\eta Z^{C(7/8)+\omega},\qquad
 |J_{\mathrm{II},\eta}(Z)-f_{\mathrm{II},\eta}(Z)|
 \ll_\eta Z^{C(\beta_*)-\sigma}.
\]
The exponents \(\omega,\sigma\) do not depend on \(\eta\).
The cutoff \(T_1=Z^{\tau_\eta}\) occurs only in the estimates,
not in either function.
\end{proposition}

\begin{proof}
The ideal margins are
\[
 m_{\rm lo}=\Delta,\qquad m_{\rm hi}=\frac{51}{64}\Delta,\qquad
 m_{\rm high}:=\min\{m_{\rm hi},m_{\rm small}\}>0.
\]
The principal contour margins are \(m_w,m_z\) in
Equation~\eqref{old-eq:5.14}.  Reserve one half of each of
\(m_{\rm lo},m_{\rm high},m_w,m_z\).  On the compact real parameter
ranges, the coefficients multiplying all requested detector, moment,
capacity, and rounding losses are bounded independently of the target.
Indeed, \(\delta\ge1/50\), \(D_x\ge37/18\), and
\(\mathcal J\ge35/54\); the dyadic lengths are bounded, and
Proposition~\ref{prop:detector-counts} gives fixed inverse width and
supply margins.  The equality \(\delta=\alpha\) uses the separate
no-slot bound above.  Choose the moment losses and capacity decrement
so that their total cost in the high exponent is less than
\(m_{\rm high}/8\).

Let \(\eta_{\rm mesh}>0\) be the mesh supplied by
Lemma~\ref{lem:plain} for these losses and bounded real ranges.  Its
uniform assertion on \([3/4,1]\) applies in particular to the compact
closure \(\kappa\in[3/4,5/6]\) used here, and the mesh is independent
of the number of slots.  Only the eventual seminorm and height orders
may depend on a fixed slot count.  Let \(b_{\rm round}>0\) be small
enough that a squared-spike rounding loss below \(b_{\rm round}\), after
the bounded changes of exponent base, costs less than \(m_{\rm high}/8\).
Choose a fixed even integer \(K\) with
\[
 \frac{2\ell}{K}<
 \min\left\{\eta_{\rm mesh},b_{\rm round},\frac1{185}\right\},
 \qquad \ell_i=\frac{\ell}{K}\quad(1\le i\le K).
\]
These positive lengths sum to \(\ell\) and are independent of the
target and \(Z\).  Put \(P_i=P=Z^{\ell/K}\).  For \(1\le i\le K\), set
\[
 I_i=\left(1+\frac{2i-1}{2K+1},\,1+\frac{2i}{2K+1}\right)\subset(1,2)
\]
and choose a nonnegative, nonzero \(W_i\in C_c^\infty(I_i)\).
The intervals have positive gaps.  Hence their underlying prime windows
are disjoint for every \(Z\), before the ray, excluded-set, or row masks
are imposed.  Equation~\eqref{eq:compensated-probe-definition} keeps each
window at its original scale in every summand.

For each selected row range, \(U=Z^d\) has \(d\ge1/2\), so
\[
 w_i=\frac{\ell_i}{d}\le\frac{2\ell}{K}<\eta_{\rm mesh},
 \qquad
 \delta\max_iw_i\le\frac{2\ell}{K}<b_{\rm round}.
\]
The second inequality uses \(\delta\le5/6<1\).  The total available
length remains \(\ell/d\).  For \(d\le h\) it is at least \(8/39\),
exceeding \(7/37\) by \(23/1443\).  For \(h<d\le h+\zeta\) with
\(\zeta<1/48\), it exceeds \(1/5\), whose excess over \(7/37\) is
\(2/185\).  Thus every selected inverse capacity and the smaller
plain capacity are supplied.  The extra bound \(2\ell/K<1/185\)
is below half the latter gap.  A zero-capacity neighborhood, including
the equality endpoint, uses the no-slot moment.  Actual annular ratios
change nominal lengths by \(O_K(1/\log U)\), absorbed by the fixed
strict margins at a sufficiently large threshold.

The internal centered amplifier also stays separated from these windows.
Write \(\sigma_{\rm width}\) for the width decrement denoted \(\sigma\)
in the proof of Lemma~\ref{lem:plain}.  That proof uses the pool exponent
\(\ell_*=\sigma_{\rm width}/3\) and a mesh below
\(\sigma_{\rm width}/6\).  At base \(U\), every live physical slot has
norm at most \(2U^{\eta_{\rm mesh}}\), whereas the pool begins at
\(U^{\ell_*}/2\).  Their fixed exponent gap makes these sets disjoint
for all sufficiently large \(U\), uniformly in the selected \(d\)-range.
The marked inverse moment has no mesh condition.

The normalizer in Equation~\eqref{eq:principal-scalar} is nonzero for
this actual choice.  Put
\(c_i=\int_0^\infty W_i(y)y^{-5/6}\,dy>0\).  The asymptotic already
proved in Section~\ref{sec:principal-signal} specializes to
\[
 S_i(Z)\sim\frac{P^{1/6}c_i}{|T|\log P}>0,\qquad
 A_T(Z)\sim
 \left(\frac{K}{|T|\ell}\right)^K
       \left(\prod_i c_i\right)(\log Z)^{-K}>0.
\]
The sign uses even \(K\).  Any later fixed excluded set affects only
the common lower threshold, since its primes eventually leave all
these windows.  Choose
\[
 \kappa_P=\frac7{16}\frac{\ell}{K}=\frac7{96K},
 \qquad 0<\kappa_P<\frac78\min_i\ell_i=\frac7{48K},
 \qquad m_P:=\kappa_P.
\]
Thus the principal approximation has a positive margin chosen before
the target.  The number of subsets, the seminorms of the narrow windows,
and their finite height orders may depend on this fixed \(K\).

Choose the amplitude width, the remaining power losses, and \(e\) so
that their total high cost is less than \(m_{\rm high}/8\), and so that
\[
 (1+h)e+\epsilon_{\rm pr}<m_w/4,\qquad
 e+\epsilon_{\rm pr}<m_z/4,\qquad
 e+\epsilon_{\rm pr}<m_P/4.
\]
Here \(\epsilon_{\rm pr}>0\) includes the chosen small powers in the
principal remainder estimate.  Also require \(e<10^{-3}\) and the bounds
\(e_0\) in Lemma~\ref{lem:detector-dyads}.  The slot lengths have already
been fixed, so the small powers needed in the amplitude subdivision can
be chosen in terms of their positive minimum.  All these choices depend
only on \(\Delta\) and the pretarget slot system.

Choose
\[
 0<\zeta<\min\{1/48,m_{\rm high}/16\}.
\]
The frequency extension then costs at most
\(2\zeta<m_{\rm high}/8\).  Now choose the absolute line \(z_\infty>2\)
in Section~\ref{sec:tails} sufficiently far right that its large-row
sum has a saving larger than \(m_{\rm high}\) relative to \(C(\beta_*)\).
Its exponent depends only on
these real choices, so \(z_\infty\) also precedes the target.
Choose the pretarget cutoff needed for
Equation~\eqref{eq:shared-principal-nonvanishing}.  Any additional fixed
excluded primes required by Proposition~\ref{prop:probe-errors} after
the target is fixed may be added to \(S\): the same positive product-tail
majorant preserves the shared contraction, and neither \(T\) nor the
slot windows change.

For clarity, the nonprincipal real comparison is uniform over all
pointwise pairs in Lemma~\ref{lem:high-bin}.  Let
\(\epsilon_{\rm real}(d)\ge0\) collect the allocated real, mesh, and
moment losses for the relevant range.  The endpoint inequality,
the intermediate inequality, and the positive slopes give
\begin{equation}\label{eq:saving-criterion}
 E(d)+\epsilon_{\rm real}(d)
 \le(\beta_*-7/8)-\varepsilon_{\rm hi},
 \qquad \varepsilon_{\rm hi}>0,
\end{equation}
for every occurring \((R,q)\) and every moderate \(d\), with a fixed
fraction of \(m_{\rm high}\) still reserved.  Therefore the same
inequality holds for \(E_{\max,Z}(d)\), uniformly in \(Z\).
The small and large bounds have their stated separate savings.
Apply the subpower normalizer estimate once to the nonprincipal and
outer-row contributions, and absorb the remaining finite dyadic
multiplicities using a power below \(m_{\rm high}/8\).
Lemma~\ref{lem:principal-residue-interface} already includes its own normalizer allowance
\(\epsilon_{\rm pr}\).  The four high real allocations above cost less
than \(m_{\rm high}/2\); this last allowance leaves more than
\(3m_{\rm high}/8\).  The principal inequalities leave more than
\(3/4\) of each of \(m_w,m_z,m_P\).  Thus one may fix, before the target,
\[
 m=\frac14\min\{m_{\rm high},m_w,m_z,m_P\}>0
\]
as a common remaining high saving before retained height factors.
For the low estimate, apply Equation~\eqref{eq:part-II-normalized-low}
with the pretarget loss \(\Delta/2\), and set \(\omega=\Delta/2\).
Let \(\epsilon_{\rm ht}>0\) be the minimum of the finitely many detector
height allowances already chosen with these real losses.

Now fix a primitive target \(\eta\).  Choose its admissible fixed
arithmetic data and the final excluded set \(S\), including its conductor,
the common cutoff, and the further fixed exclusions just described.
The physical expression, \(H_\eta\), \(c_S\), and the signal all use this
same data.  The group \(T\) and the slot system remain independent of
the target.  Fix all internal moment, seminorm, and Sobolev orders
supplied by the preceding results for this datum.  Their uniformity over
moving moduli, common masks, and frozen labels gives a finite
\(A_\eta\) dominating the product of all retained high factors, including
the dyadic, prime, numerator-reflection, and row-count height factors.
It is independent of any later external test order.  The all-height
correction bounds and the direct global or absolute bounds in the shared
analytic lemmas likewise give a finite scale degree \(B_\eta\) for all
discarded physical pieces, independent of the later order \(N\).
Normalizing those pieces and summing their finitely many types only
enlarges this fixed \(B_\eta\).

We use the height ceiling verified at the end of
Section~\ref{sec:selection}.  Let \(A_{{\rm ht},\eta}\le A_\eta\)
dominate the finitely many detector height orders, enlarging \(A_\eta\)
if necessary, and put
\[
 \tau_{0,\eta}:=
       \frac{d_{\min}\epsilon_{\rm ht}}{20(1+A_{{\rm ht},\eta})}>0.
\]
This number is fixed after the retained orders for the target, but before
\(N\) and \(Z\).  If
\(0<\tau\le\min\{d_{\min}/100,\tau_{0,\eta}\}\) and \(T_1=Z^\tau\),
then, for \(U\ge Z^{d_{\min}}\) and sufficiently large \(Z\),
\[
 T_1\le U^{1/100},\qquad
 (1+T_1)^{A_{{\rm ht},\eta}}
 \le2^{A_{{\rm ht},\eta}}Z^{d_{\min}\epsilon_{\rm ht}/20}
 \le U^{\epsilon_{\rm ht}/10}.
\]
By the definition of \(\epsilon_{\rm ht}\), this proves every literal
condition \((1+T_1)^{A_{\mathcal A}}\le U^{\epsilon/10}\) in
Lemma~\ref{lem:detector-dyads} throughout the moderate range.  The first
bound also
makes the detector's crude error
\(U^{-189/100+o(1)}T_1^2\le U^{-187/100+o(1)}\) tend to zero.
Every buffered moderate range has \(d\le h+\zeta<1\); small and
absolute large rows use no buffered cutoff.

The fixed finite list of external coordinates, including the physical
\(\Im z\) coordinate, uses the single allocation in
Equation~\eqref{eq:stage-height-allocation}.  Its matrix is fixed by the
slot system and the finite retained estimates, and is independent of the
external derivative order.  For each such \(\tau\), the dyadic, witness,
and prime estimates may choose their auxiliary external orders after
\(\tau\), as their statements permit, so that their complete pointwise
bounds hold.  Choose a strict power gap when absorbing their discarded
parts; the resulting external seminorm constants are then absorbed by the
\(\tau\)-dependent lower threshold.  For any requested remaining order
\(N\), these auxiliary
orders may be increased further; this changes only external seminorm
constants and the lower threshold, not \(A_\eta\), \(B_\eta\), or
\(\tau_{0,\eta}\).  No internal moment is reapplied to an externally
differentiated profile.

Combining Lemma~\ref{lem:high-bin}, all the row ranges, and the principal
remainder estimate therefore gives, for every fixed \(N\) and sufficiently
large \(Z\) in this ceiling range,
\[
 |J_{\mathrm{II},\eta}(Z)-f_{\mathrm{II},\eta}(Z)|
 \ll_{\eta,N}
 Z^{C(\beta_*)-m}(1+T_1)^{A_\eta}
       +Z^{B_\eta}T_1^{-N}.
\]
The lower threshold is allowed to depend on \(\tau,N,\eta\).
This is Equation~\eqref{eq:late-height-hypothesis}.
Its low hypothesis is Equation~\eqref{eq:part-II-normalized-low}
with \(\omega=\Delta/2\), and the two functions do not contain \(T_1\).
Apply Lemma~\ref{lem:late-height-closure} with
\(\sigma_0=7/8\), \(C=C_{\mathrm{II}}\), this \(m\), and the verified
ceiling \(\tau_{0,\eta}\).  It chooses \(\tau_\eta\), then a common
dominating external order \(N_\eta\), and finally the threshold.
It gives the asserted high estimate with \(\sigma=m/2>0\).
Both \(\omega\) and \(\sigma\) were fixed before the target.
\end{proof}

Proposition~\ref{prop:parameter-order} and
Equation~\eqref{eq:shared-principal-nonvanishing} verify the hypotheses of
Proposition~\ref{lem:continuation-criterion} with
\(\sigma_0=7/8\), \(J_\eta=J_{\mathrm{II},\eta}\), and
\(f_\eta=f_{\mathrm{II},\eta}\).  This contradicts \(\beta_*>7/8\);
hence \(\beta_*\le7/8\).  Proposition~\ref{prop:hecke-dirichlet-transfer}
at \(\sigma_0=7/8\) extends the primitive Hecke conclusion to all
finite-order Hecke and Dirichlet \(L\)-functions, with the principal pole
allowed, and proves Theorem~\ref{thm:main}.
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\end{document}
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\title{The Quasi-Riemann Hypothesis}
\author{OpenAI}
\date{October 5, 2026}

\begin{document}
\maketitle
\begin{abstract}
We establish the quasi-Riemann hypothesis by proving that every Dirichlet $L$-function, including Riemann's zeta function, has no zeros in the half-plane $\Rea s>11/12$. More generally, we prove the same zero-free half-plane for every finite-order Hecke $L$-function over $K=\Q(\sqrt{-3})$. In particular, this rules out the existence of Landau--Siegel zeros.
\end{abstract}

\tableofcontents

\section{Introduction}\label{sec:intro}

For a primitive Dirichlet character $\chi$ of conductor $q$, the associated \emph{Dirichlet $L$-function} is defined for $\Rea s>1$ by the formula
\[
 L(s,\chi)=\sum_{n\ge1}\frac{\chi(n)}{n^s}, \qquad \Rea s>1,
\]
and admits a meromorphic continuation to $s \in \C$ \cite[\S4.6]{IK04}. When $\chi$ is the trivial character, $L(s, \chi)$ recovers the Riemann zeta function $\zeta(s)$ \cite{Rie59}.

The zeros of Dirichlet $L$-functions are of significant interest, such as for their role in governing the distribution of primes in arithmetic progressions. The \emph{Generalized Riemann Hypothesis} predicts that all zeros of $L(s, \chi)$ in the critical strip $0 < \Rea s < 1$ satisfy $\Rea s = 1/2$.
For $\zeta(s)$, the weaker assertion that there exists
$\varepsilon>0$ such that $\zeta(s)$ has no zeros in
$\Rea s>1-\varepsilon$ is called a \emph{quasi-Riemann Hypothesis}
(e.g., in \cite[\S1]{BG17}, \cite[p.~274]{MS02}, and \cite[\S2]{BR18}).
For Dirichlet $L$-functions, we consider the analogous assertion
with a single $\varepsilon>0$ valid for every primitive character
$\chi$, independently of both conductor and height.



Let $K=\Q(\sqrt{-3})$. The main result of this paper is the following.

\begin{theorem}\label{thm:main}
Every finite-order Hecke $L$-function over $K$ has no zeros in the half-plane $\Rea s>11/12$. Thus every Dirichlet $L$-function, including $\zeta(s)$, has no zeros for $\Rea s>11/12$.
\end{theorem}

Section~\ref{sec:reduction} proves the theorem assuming
Proposition~\ref{thm:ms}, whose proof is completed in
Section~\ref{sec:completion}.

Theorem~\ref{thm:main} is a natural intermediate step toward the stronger zero-free region $\Rea s>7/8$ established in \cite[Theorem~1.1]{OpenAI26}. For simplicity, we have isolated Theorem~\ref{thm:main} and its proof here.
By a standard explicit-formula argument (see \cite[Chapters~19--20]{Dav00}), Theorem~\ref{thm:main} gives the following quantitative form of the prime number theorem in arithmetic progressions.

\begin{corollary}\label{cor:primes-ap}
Let $\pi(x;q,a)$ count the primes $p\le x$ with $p\equiv a\pmod q$.
Write $\varphi$ for Euler's totient function.
For $x\ge2$,
\[
 \sup_{\substack{1\le q\le x\\(a,q)=1}}
 \biggl|\pi(x;q,a)-\frac1{\varphi(q)}
                  \int_2^x\frac{\dd t}{\log t}\biggr|
 \ll x^{11/12}\log x,
\]
where the implied constant is absolute and effective.
\end{corollary}

Theorem~\ref{thm:main} has a number of additional arithmetic consequences.
By the zero-free-region method originating with Rodosski\u{\i} \cite{Rod56},
in the form recorded in \cite[Theorem~13.12]{MV07}, the least quadratic nonresidue modulo
an odd prime $p$ is bounded by a fixed power of $\log p$, which in particular
proves Vinogradov's conjecture \cite{Vin85} that this nonresidue is
$\ll_{\varepsilon}p^{\varepsilon}$ for every $\varepsilon>0$.
See also Bhargava, Ivanyos, Mittal, and Saxena
\cite[Theorem~6.7]{BIMS17} for this consequence of a fixed zero-free half-plane.
From a computational number theory perspective, this bound allows square
roots modulo $p$ to be extracted deterministically in time polynomial in
$\log p$ using the Tonelli--Shanks algorithm
(see \cite[\S2.9]{Galbraith12}).
Theorem~\ref{thm:main} also yields a deterministic polynomial-time
implementation of Miller's primality test \cite{Mil76}; note that
polynomial-time primality testing was previously known unconditionally
via the Agrawal--Kayal--Saxena algorithm \cite{AKS04}.
For negative fundamental discriminants $D$, Littlewood's short Euler-product
argument \cite{Lit28}, applied to the fixed zero-free half-plane in
Theorem~\ref{thm:main}, together with Dirichlet's class-number formula
(see \cite[Chapter~6]{Dav00}), yields the effective class-number bound
$h(D)\gg\sqrt{|D|}/\log\log|D|$, with an absolute computable implied constant.
The class-group computations of Elsenhans, Kl\"uners, and Nicolae
\cite[Theorem~2]{EKN20}, building on Weinberger \cite{Wei73}, give a complete
list of imaginary quadratic fields with class-group exponent dividing two
when there are no Landau--Siegel zeros. Together with Grube's
characterization of idoneal numbers and his reduction to fundamental
discriminants \cite{Gru74} (see \cite[Theorem~6 and \S2.3]{Kani11}),
this shows that Theorem~\ref{thm:main} confirms the completeness of
Euler's list of 65 idoneal numbers.

\subsection{Prior work}

There is a long line of work establishing zero-free regions for Dirichlet $L$-functions. Hadamard and de la Vall\'ee Poussin proved the Prime Number Theorem
in 1896 by establishing that $\zeta(s)$ has no zeros on the line
$\Rea s=1$ \cite{Had96,VP96}. De la Vall\'ee Poussin subsequently obtained a
quantitative zero-free region to the left of this line \cite{VP99}.
For nonprincipal primitive Dirichlet characters, the classical zero-free-region theorem of Gr\"onwall and Titchmarsh
\cite{Gro13,Tit30}, in the modern form given in \cite[Theorem~5.26]{IK04}
(see also \cite[Chapter~14]{Dav00}), gives an absolute, effective constant $c_0>0$ such that
\begin{equation}\label{eq:intro-classical-region}
 L(\sigma+\mathrm i t,\chi)\ne0
 \quad\text{if}\quad
 \sigma>1-\frac{c_0}{\log(q(|t|+3))},
\end{equation}
with at most one exception for each primitive character.
Any exception is a simple real zero, can occur only if
$\chi$ is quadratic, and is called a \emph{Landau--Siegel zero}. Theorem~\ref{thm:main} rules out the existence of such a zero.

There are two parameters in \eqref{eq:intro-classical-region}:
the conductor $q$ and the height $|t|$. Even when $q$ is fixed,
its width tends to zero with the height. Vinogradov and Korobov
developed methods \cite{Vin58,Kor58} to prove that
\begin{equation}\label{eq:intro-vk-region}
 \zeta(\sigma+\mathrm i t)\ne0
 \quad\text{if}\quad
 \sigma>1-\frac{c_1}
 { (\log|t|)^{2/3}(\log\log|t|)^{1/3}},\qquad |t|\ge3,
\end{equation}
for an absolute $c_1>0$ (Ford \cite{Ford02} gives an explicit value of $c_1$). The assertion
of Theorem~\ref{thm:main} is a half-plane of fixed width, independent
of both conductor and height.

As we will see in the outline, the proof of Theorem~\ref{thm:main} draws on
modern developments in character large sieves and metaplectic theta series.
In particular, the connection between cubic Gauss sums and cubic theta
coefficients \cite{Kub69,Pat77} links the argument to work on Patterson's
conjecture by Heath-Brown and Patterson \cite{HBP79}, Heath-Brown
\cite{HB00}, and Dunn and Radziwi\l\l{} \cite{DR}.
The recursive large-sieve arguments also build on Heath-Brown's proof of
the quadratic large sieve inequality \cite{HB95}.

\subsection{Organization}

Section~\ref{sec:outline} gives a high-level outline of the argument.
Section~\ref{sec:reduction} deduces the zero-free region from a
mean-square estimate for twisted M\"obius sums.
Section~\ref{sec:initialization} reduces this estimate to a dual mean
square with cubic Gauss-sum coefficients.
Section~\ref{sec:descent} states the completed mean-square and transfer
estimates, combines them into a recursive inequality, and proves the
required mean-square bound.
Sections~\ref{sec:reflection} and~\ref{sec:transfer-proof} prove the
completed estimate and the bounds for the remaining cube-divisor sums, respectively.
Appendix~\ref{sec:arithmetic} supplies the arithmetic identities and
the detailed theta calculation. Appendix~\ref{app:weights} records the smooth
separation lemma used throughout.

\subsection{Notation}

Write $\OO=\OO_K=\mathbb Z[\omega]$, where $\omega=e^{2\pi\mathrm i/3}$.
For $a\in K$, write $\NK(a)=a\overline a=|a|^2$ for its norm.
For a nonzero integral ideal $\mathfrak a\subseteq\OO$, write
$\NK(\mathfrak a)=\#(\OO/\mathfrak a)$.
We use $\ind_{\mathcal C}$ for the indicator of a condition $\mathcal C$.
For a prime ideal represented by $p$ and a nonzero element or ideal $a$,
write $v_p(a)$ for its exponent in the prime factorization of $a$.


We use standard asymptotic notation, writing $f=O(g)$ or $f\ll g$ when $|f|\le Cg$, where $g\ge0$ and $C>0$ is an absolute constant unless otherwise specified. Subscripts indicate possible dependence of the implied constant: for example, $f\ll_{\nu,W,\varepsilon}g$ means $|f|\le C_{\nu,W,\varepsilon}g$, where the constant may depend on $\nu,W,\varepsilon$ but is uniform in all other varying parameters. For nonnegative $f,g$, we write $f\asymp g$ when $f\ll g$ and $g\ll f$.
 A dyadic norm range is an interval $R\le\NK(a)<2R$; a sum over dyadic scales uses $R=2^j$.

We use $D\ge2$ as an ambient size parameter; in the main argument, it is the original norm scale introduced 
in the outline below. Auxiliary scales may vary within ranges bounded by fixed powers of $D$. We write
\[
A\precbd B \quad\text{if}\quad A\ll_\varepsilon D^\varepsilon B
\qquad\text{for every }\varepsilon>0,
\]
uniformly in the varying parameters over their stated ranges. Implied constants may also depend on the fixed data specified in each statement.

\section{Outline of the argument}\label{sec:outline}

In this section, we sketch the proof of Theorem~\ref{thm:main}, suppressing
various coprimality conditions, local factors, and details of smoothing.
The precise statements appear in Sections~\ref{sec:reduction}--\ref{sec:transfer-proof}.

\subsection{Step 1: Reduction to power-saving estimates for twisted M\"obius sums}


Fix a finite-order Hecke character $\nu$ of $K$. Let $L_K(s, \nu)$ be the corresponding Hecke $L$-function. We extend $\nu$ by zero to ideals not coprime to its conductor. We will deduce Theorem~\ref{thm:main} from a power-saving estimate for $\nu$-twisted M\"obius sums.

Let $\mu$ denote the ideal M\"obius function on $K$. All original ideal sums
run over nonzero integral ideals.
As in Section~\ref{sec:reduction}, the ideals in these sums are understood
to be prime to $2$, $3$, and the conductor of $\nu$. For a norm scale $D>0$, consider
\[
 A_1(D)=\sum_{\mathfrak n}
   \mu(\mathfrak n)\nu(\mathfrak n)
   W(\NK(\mathfrak n)/D),
\]
where $W\in C_c^\infty((0,\infty);\mathbb C)$ is a smooth cutoff function. Thus $A_1(D)$ is a smoothed $\nu$-twisted M\"obius sum over ideals of norm comparable to $D$, with roughly $D$ terms. We seek a power saving over
this size: for a fixed $\delta>0$ and every $\varepsilon>0$,
\begin{equation}\label{eq:power-savings}
A_1(D)\ll_{\nu,W,\varepsilon}D^{1-\delta+\varepsilon} \text{ for every }W\in C_c^\infty((0,\infty);\mathbb C).
\end{equation}
The implication from \eqref{eq:power-savings} to a zero-free half-plane
is the smoothed Hecke version of the classical relation between M\"obius
sums and zero-free regions. In the zeta-function case, the equivalence
between the Riemann Hypothesis and the bound
$\sum_{n\le x}\mu(n)\ll_\varepsilon x^{1/2+\varepsilon}$ is due to
Littlewood \cite{Lit12}; see also \cite[\S1]{MM09}.
Section~\ref{sec:reduction} gives the complete argument needed here.

\subsection{Step 2: Embedding the sum in a family}

In order to estimate $A_1(D)$, we embed it into a family of such sums,
parametrized by $u\in\OO_K$, by introducing a sextic twist.
We call an element of $\OO$ \emph{primary} if it is congruent to $1$
modulo $3$. The ring $\OO=\mathbb Z[\omega]$ is Euclidean, so it has unique
factorization and every ideal is principal; for an ideal coprime to $3$,
multiplying any generator by a unique unit gives $n\equiv1\pmod3$,
since the six units represent the six invertible residue classes modulo $3$.
For a primary element $n$, we write $\mu(n)=\mu((n))$ and $\nu(n)=\nu((n))$;
divisor sums count each ideal divisor once, using its primary generator.
For an ideal $\mathfrak n$ prime to $6$, use its unique primary generator
$n$ and write
\[
 \chi_{\mathfrak n}(u)=\chi_n(u)=(u/n)_6.
\]
Here $(u/n)_6$ is the sextic residue symbol, extended by zero
when $(u,n)\neq 1$.\footnote{For a primary prime $p\nmid6$ and $p\nmid u$,
$(u/p)_6$ is the unique sixth root of unity congruent to
$u^{(\NK(p)-1)/6}$ modulo $p$; set $(u/p)_6=0$ when $p\mid u$.
Extend multiplicatively in the denominator to define
$\chi_n(u)=(u/n)_6$ for primary $n$ coprime to $6$.
For the unit ideal, $\chi_1(u)=1$ for every $u$, including $u=0$.
}
We also write $(u/n)_2$ and $(u/n)_3$ for the quadratic and cubic residue symbols, defined by the same convention with $6$ replaced by $2$ and $3$, respectively. When $(n,6)=1$, they equal $\chi_n(u)^3$ and $\chi_n(u)^2$. Restrictions that keep these symbols defined are understood when omitted in the outline.
The family is
\begin{equation}\label{eq:intro-family}
 A_u(D)=\sum_{\mathfrak n}
   \mu(\mathfrak n)\nu(\mathfrak n)\chi_{\mathfrak n}(u)
   W(\NK(\mathfrak n)/D).
\end{equation}

Fix $0<\vartheta\le1/10$. The crucial estimate, Proposition~\ref{thm:ms}, is that
\begin{equation}\label{eq:intro-ms}
 \sum_{0<\NK(u)\le H}|A_u(D)|^2
       \ll_{\nu,W,\vartheta,\varepsilon}D^{1+\varepsilon}H,
 \qquad H=D^{1+\vartheta}.
\end{equation}
In a mean square, we call the variable in the outer sum the \emph{row}
and the variable in the inner sum the \emph{column}. In
\eqref{eq:intro-ms} these are $u$ and $\mathfrak n$, respectively.
Up to the factor of $D^\varepsilon$, \eqref{eq:intro-ms} can be thought of as saying that the family $\{A_u(D)\}$ exhibits ``square-root cancellation'' on average (in the $L^2$ sense) over $u$.



The mean-square estimate \eqref{eq:intro-ms} gives the desired power saving for $A_1(D)$ because $A_{p^6}(D)\approx A_1(D)$ for many primes $p$. For a primary prime $p$ with $Y/2<\NK(p)\le Y$, the identity $\chi_n(p^6)=\ind_{p\nmid n}$ leaves only $O_W(D/Y)$ differing terms and hence gives
\begin{equation}\label{eq:A_u-vs-A_1}
 A_{p^6}(D)=A_1(D)+O_W(D/Y).
\end{equation}
By Landau's prime ideal theorem \cite{Lan03}
(see \cite[Theorem~5.33]{IK04}), there are $\asymp Y/\log Y$ choices of such $p$. Taking $Y=H^{1/6}$, considering the contribution of such terms to \eqref{eq:intro-ms} and using \eqref{eq:A_u-vs-A_1} gives
\begin{equation}\label{eq:intro-extraction}
|A_1(D)|^2
\ll D^{1+\varepsilon}H^{5/6}+D^2H^{-1/3}.
\end{equation}
With $H=D^{1+\vartheta}$, this gives
$A_1(D)\ll D^{11/12+5\vartheta/12+\varepsilon}$ for every fixed
$0<\vartheta\le1/10$. Choosing $\vartheta$
sufficiently small for each requested exponent loss yields
$A_1(D)\ll_{\nu,W,\varepsilon}D^{11/12+\varepsilon}$.
Therefore it remains to establish \eqref{eq:intro-ms}.

\subsection{Step 3: Poisson summation}

We turn to the task of proving the mean-square estimate \eqref{eq:intro-ms}. Now that we have introduced the row variable $u$, we can apply Poisson summation in this variable. This introduces sextic Gauss sums, arising from the Fourier transforms of the sextic characters. Upon application of the Gauss--Jacobi identities, these sextic Gauss sums absorb the factor of $\mu$ and turn into cubic Gauss sums. We interpret the resulting cubic Gauss sums as coefficients of Kubota's cubic theta function. In the next step, we use the automorphy of this theta function. For now, we explain the appearance of these Gauss sums in more detail.

Define the additive character
\[
 e(z)=\exp(4\pi\mathrm i\,\operatorname{Im}z/\sqrt3)\quad(z\in\mathbb C).
\]
For a squarefree Eisenstein integer $n\equiv1\pmod3$ prime to $6$ and an integer $j$, define the normalized Gauss sums
\begin{equation}\label{eq:intro-gauss-sum}
 \gamma_j(n)=\frac{1}{\sqrt{\NK(n)}}
       \sum_{x\bmod n}\chi_n(x)^j e(x/n).
\end{equation}
For $j=-1$, we interpret $\chi_n^{-1}$ as the conjugate character $\overline{\chi_n}$.

Consider the classical identity
\[
 \left(\frac{-1}{m}\right)=\left(\frac{1}{\sqrt{m}}\sum_{x\bmod m}\left(\frac{x}{m}\right)e^{2\pi ix/m}\right)^2,
\]
where $m$ is an odd squarefree positive integer and the symbols are Jacobi symbols. We would like a similar decomposition for $\mu$ instead of $\left(\frac{-1}{m}\right)$. Following the work of Hasse \cite[pp.~443--445]{Has50} and Heath-Brown \cite[(2)]{HB00}, we derive the following identity in Appendix~\ref{app:gauss-identities}, valid for squarefree primary $n$ away from a fixed set of excluded primes:
\[
 \mu(n)\gamma_{-1}(n)=\chi_n(-1)G(n)^{-1}\overline{\alpha(n)}\gamma_2(n),
 \qquad \alpha(n)=\frac{n}{|n|},\quad G(n)=\overline{\chi_n(4)}\gamma_3(n).
\]
Here $G(n)$ is a fixed ray-class factor. The factor $\gamma_{-1}(n)$ comes from Poisson summation; as promised, it combines with $\mu(n)$ to produce the cubic coefficient $\overline{\alpha(n)}\gamma_2(n)$ (up to ray-class factors). Further details are given in Section~\ref{sec:initialization}.

We call the sums obtained after applying Poisson summation \emph{dual sums}.\footnote{For this terminology, see \cite[p.~440]{PY20},
\cite[\S2.3]{KLMS20}, and \cite[arXiv abstract]{FLN10}.} Rearranging these dual sums reduces the problem of estimating
$\sum_u |A_u(D)|^2$ to estimating the following family of column sums (for clarity, we have suppressed auxiliary twists and coprimality conditions):
\begin{equation}\label{eq:intro-gauss-family}
 B_h(X)=\sum_{\substack{n\in\OO\\n\equiv1\ (3)\\n\ {\rm squarefree}}}
       \overline{\alpha(n)}\gamma_2(n)\xi(n)\chi_n(h)
       U(\NK(n)/X).
\end{equation}
Here $h$ is the new row variable, Fourier dual to $u$, while $U$ is a smooth weight restricting the column variable $n$ to norm comparable to $X$. We choose $\xi$ to be either $\nu\eta$ or $\overline{\nu}\eta$, 
where $\eta$ ranges over a finite set of ray class characters of fixed modulus.\footnote{The ray class group modulo an ideal $\mathfrak m$ is the group of fractional ideals prime to $\mathfrak m$, modulo principal ideals generated by elements congruent to $1$ modulo $\mathfrak m$. A ray class character is a character of this finite group. For an element prime to $\mathfrak m$, its ray class means the class of its principal ideal.}  For each choice of $\xi$, this defines a family of sums $B_h(X)$.

 After treating the diagonal and separating the smooth weights, we roughly get that
\begin{equation}\label{eq:intro-poisson-comparison}
 \sum_{0<\NK(u)\le H}|A_u(D)|^2
 \precbd DH+\frac HD
       \sum_{0<\NK(h)\ll\mathcal H}|B_h(D)|^2.
\end{equation}
Here $\mathcal H\asymp D^2/H$ is the dual norm scale. This schematic
comparison suppresses the common factors arising when the square is
expanded. Thus the desired estimate is
\begin{equation}\label{eq:intro-dual-ms}
 \sum_{0<\NK(h)\ll\mathcal H}|B_h(D)|^2\precbd D^2.
\end{equation}

\subsection{Step 4: \texorpdfstring{Relation}{Relation} to cubic Gauss sums and Kubota's cubic theta function}

To estimate \eqref{eq:intro-dual-ms}, we first identify the cubic
Gauss coefficients with Fourier coefficients of a theta function.
Its transformation law applies to a sum with extra cube factors,
which we will remove in Step~5.

Let $\theta(z,v)$ be Kubota's cubic theta function on hyperbolic three-space ($z\in\mathbb C$, $v \in \R_{>0}$), obtained as a residue of a cubic metaplectic Eisenstein series \cite{Kub69,Pat77}. We use the normalization of \cite[\S5.1, (5.6)--(5.8)]{DR},
recalled below in \eqref{eq:cubic-theta-definition}.

Put $\lambda=1+2\omega$. For primary elements $n,b\in\OO=\mathbb Z[\omega]$, 
with $n$ squarefree and $(nb,6)=1$, let $c_\theta(nb^3)$ denote the Fourier coefficient of $\overline\theta$ indexed by $\lambda^{-3}nb^3$ in its expansion in $z$. Patterson's formula \cite[Theorem~8.1]{Pat77}, recorded in \cite[(5.7)]{DR}, gives
\begin{equation}\label{eq:intro-theta-coefficients}
 c_\theta(nb^3)
     =3^{5/2}|b|\,\overline{\chi_n(\lambda)^2}\gamma_2(n).
\end{equation}
Taking $b=1$ in \eqref{eq:intro-theta-coefficients} and substituting into \eqref{eq:intro-gauss-family} yields
\begin{equation}\label{eq:intro-theta-family}
 B_h(X)=3^{-5/2}\sum_{\substack{n\in\OO\\n\equiv1\ (3)\\n\ {\rm squarefree}}}
 c_\theta(n)\overline{\alpha(n)}\chi_n(\lambda)^2
 \xi(n)\chi_n(h)U(\NK(n)/X).
\end{equation}
Thus \eqref{eq:intro-theta-family} expresses $B_h(X)$ as a smoothed, twisted sum of the squarefree-index coefficients of $\overline\theta$. This connection was used by Heath-Brown and Patterson to study Kummer sums \cite{HBP79}.

To use the summation formula for theta coefficients, we embed the sum \eqref{eq:intro-theta-family} in a completed sum, by including terms indexed by $nb^3$. This use of cube completion follows Dunn and Radziwi\l\l\ \cite[Lemma~5.4 and Proposition~5.3]{DR}, with the underlying theta coefficients given by Patterson \cite[Theorem~8.1]{Pat77}. For fixed $h$, $\xi$, and $U$, define
\begin{equation}\label{eq:intro-completed-theta-sum}
T_h(X):=\frac{1}{3^{5/2}\sqrt X}
 \sum_{\substack{n,b\in\OO\\n,b\equiv1\ (3)\\n\ {\rm squarefree}}}
 c_\theta(nb^3)\overline{\alpha(nb^3)}\chi_{nb^3}(\lambda)^2
 \xi(nb^3)\chi_{nb^3}(h)U(\NK(nb^3)/X).
\end{equation}
The $b=1$ terms are exactly $X^{-1/2}B_h(X)$; the other
terms supply the cube indices in the theta expansion. The first goal is to bound the mean square of $T_h(X)$ over $h$.

The automorphy of $\theta$ gives a summation formula that transforms the completed column sum $T_h(X)$, with the row $h$ held fixed, into dual sums of theta coefficients with new smooth weights and character twists.
A cusp is represented by a boundary point in $K\cup\{\infty\}$. A \emph{cusp expansion} is the Fourier expansion in the horizontal variable after a change of coordinates taking infinity to that point. We call its Fourier coefficients \emph{cusp coefficients}.
Our starting point is a variant (established in Appendix~\ref{app:fixed-ray}) of the theta transformation of Dunn and Radziwi\l\l\ \cite[\S5]{DR}, extending work of Patterson \cite{Pat77} and Yoshimoto \cite{Yos87}. The key point is how the character twists change. Suppose for
illustration that $h$ is squarefree and primary, with $(h,6)=1$. Then, by Proposition~\ref{lem:reflection}, the transformed expression is a finite linear combination of sums of the form
\begin{equation}\label{eq:intro-dual-theta-sum}
 \sum_{0\ne m\in\OO}
 \frac{d_\theta(m)\alpha(m)}{\sqrt{\NK(m)}}\,
 \chi_h(m)^3
 V_*^\sharp\!\Bigl(\frac{\NK(m)X}{\NK(h)^2}\Bigr).
\end{equation}
The dual column index is $m$, while $h$ remains the row index.
Here $d_\theta(m)$ denotes the coefficient at the Fourier index
$\lambda^{-4}m$ in a cusp expansion of $\overline\theta$, and
$V_*^\sharp$ is the transform in \eqref{eq:theta-weight}, applied to $V_*(y)=y^{1/2}U(y)$.
We have suppressed
a finite sum over these cusp expansions, fixed periodic twists,
bounded prefactors, and fixed scale constants. Within each fixed
ray class of $h$, the coefficient sequences and transformed weights
are independent of $h$ by Lemma~\ref{lem:reflection-uniformity}.
Proposition~\ref{lem:reflection} gives the
precise formula, writing $d(\ell)$ for the cusp coefficient at
$\ell=\lambda^{-4}m$.

Since $V_*^\sharp$ decays rapidly at infinity, the effective norm range in \eqref{eq:intro-dual-theta-sum} is $\NK(m)\ll\NK(h)^2/X$, in place of the original range $\NK(nb^3)\asymp X$. These dual sums arise from the theta transformation and involve theta coefficients on the transformed scale, now twisted by the quadratic character $\chi_h^3$. This character arises because, at each prime $p\mid h$, the Fourier and theta factors combine as
\begin{equation}\label{eq:intro-quadratic-twist}
 \chi_p^{-1}\chi_p^{-2}=\chi_p^{-3}=\chi_p^3.
\end{equation}
Now a key point is that since $\chi_p^3$ is quadratic, Goldmakher and Louvel's quadratic large sieve \cite[Theorem~1.1 and Corollary~1.2]{GL} (a generalization of Heath-Brown's quadratic large sieve \cite{HB95} to number fields) bounds the mean square of the sums in \eqref{eq:intro-dual-theta-sum} as $h$ varies. For squarefree quadratic families with row and column norm ranges $M,L$, the quadratic large sieve gives the factor $M+L$, up to $(ML)^\varepsilon$. Crucially, this avoids the additional term $(ML)^{2/3}$ in Blomer, Goldmakher, and Louvel's general higher-order large sieve \cite[Theorem~1.3]{BGL}.

We then apply Cauchy--Schwarz in the cube variable and the quadratic
large sieve in the squarefree column variable to estimate the mean square of $T_h(X)$. The details are given in the proof of Proposition~\ref{prop:R}, which gives, for $\mathcal H,X\ge1$,
\begin{equation}\label{eq:intro-completed-ms}
 \sum_{0<\NK(h)\ll\mathcal H}|T_h(X)|^2
 \ll_{\varepsilon,\xi,U}(\mathcal H X)^\varepsilon
       \Bigl(\mathcal H+\frac{\mathcal H^2}{X}\Bigr).
\end{equation}
At $X=D$ and $\mathcal H\le D$, the heuristic comparison
$B_h(D)\approx\sqrt D\,T_h(D)$ would therefore give the desired
$D^2$ bound. The remaining task is to justify the corresponding
mean-square bound by removing the cube factors.

\subsection{Step 5: Removing the cube factors} 
We now pass from a mean-square bound for $T_h(X)$ to one for $X^{-1/2}B_h(X)$, with both averages taken over $h$. Note that one cannot simply discard the terms with $b\ne1$, because the contributions from different $b$ can cancel. We begin by undoing the addition of cube factors using M\"obius inversion. Related completions appear in Patterson \cite[Theorem~6.1]{Pat77} and Heath-Brown \cite[\S3]{HB00}, with explicit removal of the cube factors by M\"obius inversion in Dunn and Radziwi\l\l\ \cite[Proposition~5.3 and (8.2)]{DR}.

Fix $h$ and $\xi$, and write $P_h(X)=X^{-1/2}B_h(X)$ for the $b=1$ part of $T_h(X)$. Substituting the coefficient formula \eqref{eq:intro-theta-coefficients} into \eqref{eq:intro-completed-theta-sum} gives
\begin{equation}\label{eq:intro-cube-completion}
 T_h(X)=\sum_{\substack{b\in\OO\\b\equiv1\ (3)}}\frac{w_h(b)}{\NK(b)}\,
                  P_h\!\Bigl(\frac{X}{\NK(b)^3}\Bigr),
\end{equation}
where
\begin{equation}\label{eq:intro-cube-weight}
 w_h(b)=\overline{\alpha(b)}^{\,3}\xi(b)^3\chi_b(h)^3.
\end{equation}
Each factor in \eqref{eq:intro-cube-weight} is completely multiplicative in $b$. Thus $w_h(bc)=w_h(b)w_h(c)$ even when $b,c$ share prime factors, and $|w_h(b)|\le1$. M\"obius inversion (cf. \cite[\S1.3]{IK04}) gives
\begin{equation}\label{eq:intro-cube-inversion}
 P_h(X)=\sum_{\substack{d\in\OO\\d\equiv1\ (3)}}\frac{\mu(d)w_h(d)}{\NK(d)}\,
                  T_h\!\Bigl(\frac{X}{\NK(d)^3}\Bigr).
\end{equation}
Indeed, substituting \eqref{eq:intro-cube-completion} into \eqref{eq:intro-cube-inversion} and grouping by the total cube index $b$ gives the factor $w_h(b)\sum_{d\mid b}\mu(d)$, which is $1$ for $b=1$ and $0$ otherwise.


The objective is now an estimate for $P_h$, rather than for the
completed sum $T_h$. For the simplified family and the parameter
ranges arising from Step~3, the required bound is

\begin{equation}\label{eq:intro-cube-target}
 \sum_{0<\NK(h)\ll\mathcal H}|P_h(X)|^2
       \ll_{\varepsilon,\xi,U}(\mathcal H X)^\varepsilon X.
\end{equation}
Since $B_h(X)=\sqrt X\,P_h(X)$, this is equivalent to a bound of size $X^2$, up to the same small power, for the mean square of $B_h(X)$.

For $1\le H_c\le X^{1/3}$, let $P_{h,\le H_c}(X)$ be the part of
\eqref{eq:intro-cube-inversion} with $\NK(d)\le H_c$.
Applying weighted Cauchy--Schwarz for each fixed $h$, then summing over $h$
and using \eqref{eq:intro-completed-ms}, gives
\[
 \begin{aligned}
 &\sum_{0<\NK(h)\ll\mathcal H}|P_{h,\le H_c}(X)|^2\\
 &\quad\le\Bigl(\sum_{\NK(e)\le H_c}\frac1{\NK(e)}\Bigr)
       \Bigl(\sum_{\NK(d)\le H_c}\frac1{\NK(d)}
       \sum_{0<\NK(h)\ll\mathcal H}
       |T_h\!\bigl(X/\NK(d)^3\bigr)|^2\Bigr)\\
 &\quad\precbd\sum_{\NK(d)\le H_c}\frac1{\NK(d)}
       \Bigl(\mathcal H+\frac{\mathcal H^2\NK(d)^3}{X}\Bigr)
       \precbd\mathcal H+\frac{\mathcal H^2H_c^3}{X}.
 \end{aligned}
\]
Thus
\begin{equation}\label{eq:intro-short-cube-ms}
 \sum_{0<\NK(h)\ll\mathcal H}|P_{h,\le H_c}(X)|^2
 \ll_{\varepsilon,\xi,U}(\mathcal H X)^\varepsilon
       \Bigl(\mathcal H+\frac{\mathcal H^2H_c^3}{X}\Bigr).
\end{equation}
For $\mathcal H\le X$, the estimate \eqref{eq:intro-short-cube-ms} is within the target \eqref{eq:intro-cube-target} provided $\mathcal H^2H_c^3/X\le X$. Thus we apply the completed mean-square bound directly only up to the cutoff
\begin{equation}\label{eq:intro-cube-cutoff}
 H_c=\min\!\biggl\{X^{1/3},
                \Bigl(\frac{X}{\mathcal H}\Bigr)^{2/3}\biggr\}.
\end{equation}
At the basic scales from Step~3,
\[
 X\asymp D,\qquad \mathcal H\asymp D^{1-\vartheta},
 \qquad H_c\asymp D^{2\vartheta/3}.
\]
The inverse sum can extend to $\NK(d)\asymp D^{1/3}$, so we must
still control the larger divisors.

Write $\tau_{\mathrm{div}}(b)$ for the number of nonzero integral ideal divisors of $(b)$.
Let $P_{h,>H_c}(X)$ denote the terms with $\NK(d)>H_c$ in
\eqref{eq:intro-cube-inversion}, and put $L_b=X/\NK(b)^3$.
Substituting \eqref{eq:intro-cube-completion} into the truncated inversion formula for $P_{h,>H_c}(X)$ obtained from \eqref{eq:intro-cube-inversion}, and grouping by $b=dc$, gives
\[
 \begin{aligned}
 P_{h,>H_c}(X)
 &=\sum_{\NK(d)>H_c}\sum_c
   \frac{\mu(d)w_h(d)w_h(c)}{\NK(d)\NK(c)}
   P_h\!\Bigl(\frac{X}{\NK(dc)^3}\Bigr)\\
 &=\sum_{\NK(b)>H_c}
   \frac{\beta_0(b)\chi_b(h)^3}{\NK(b)}P_h(L_b),
 \end{aligned}
\]
where all indices are primary and
\[
 \beta_0(b)=\overline{\alpha(b)}^{\,3}\xi(b)^3
       \sum_{\substack{d\mid b\\\NK(d)>H_c}}\mu(d),
 \qquad |\beta_0(b)|\le\tau_{\mathrm{div}}(b).
\]
The support of $U$ restricts $\NK(b)\ll_U X^{1/3}$.
Weighted Cauchy--Schwarz for each $h$, followed by summation, gives
\begin{equation}\label{eq:intro-long-cube-ms}
 \begin{aligned}
 \sum_{0<\NK(h)\ll\mathcal H}|P_{h,>H_c}(X)|^2
 &\le\Bigl(\sum_b\frac{\tau_{\mathrm{div}}(b)}{\NK(b)}\Bigr)
 \sum_b\frac{\tau_{\mathrm{div}}(b)}{\NK(b)}
 \sum_{0<\NK(h)\ll\mathcal H}|P_h(L_b)|^2\\
 &\precbd\sup_b\sum_{0<\NK(h)\ll\mathcal H}|P_h(L_b)|^2.
 \end{aligned}
\end{equation}
Put
$a_\xi(n)=\overline{\alpha(n)}\gamma_2(n)\xi(n)$. Write
\[
 E(\mathcal H,X)
 :=\frac1X\sum_{0<\NK(h)\ll\mathcal H}
 \Bigl|\sum_n^*a_\xi(n)\chi_n(h)U(\NK(n)/X)\Bigr|^2.
\]
A star restricts an index to squarefree primary elements.
By \eqref{eq:intro-gauss-family} and the definition of $P_h$, we have
\[
 E(\mathcal H,X)
 =\frac1X\sum_{0<\NK(h)\ll\mathcal H}|B_h(X)|^2
 =\sum_{0<\NK(h)\ll\mathcal H}|P_h(X)|^2.
\]
At the scales from Step~3, our goal is to prove
\begin{equation}\label{eq:intro-energy-target}
 E(\mathcal H,X)\precbd X.
\end{equation}
At $X=D$, this is precisely the bound \eqref{eq:intro-dual-ms}
for the mean square of $B_h(D)$. Substitution into the Poisson
comparison \eqref{eq:intro-poisson-comparison} then gives the
required original mean-square estimate \eqref{eq:intro-ms}.
Combining \eqref{eq:intro-short-cube-ms} for $P_{h,\le H_c}$
(the terms with $\NK(d)\le H_c$) and \eqref{eq:intro-long-cube-ms}
for $P_{h,>H_c}$ (the terms with $\NK(d)>H_c$), with the cutoff
\eqref{eq:intro-cube-cutoff}, gives
 \[
 E(\mathcal H,X)\precbd X + \sup_{b:\,1\le L_b\le X/H_c^3}E(\mathcal H,L_b).
 \]
It therefore remains to prove $E(\mathcal H,L_b)\precbd X$ for
$1<L_b\le X/H_c^3$. Here the row range $\mathcal H$ stays fixed,
and the required bound is still of size $X$ even though the column
scale has decreased to $L_b$.
 
We now expand the square and apply Poisson summation in $h$ as before; we now record the proof in terms of $E(\mathcal H,L_b)$. The
Gauss-sum coefficients become M\"obius coefficients, and the new
row variable $y$ has norm at most about $L_b^2/\mathcal H$. After
separating the weights, this gives schematically
\[
 \begin{aligned}
 E(\mathcal H,L_b)
 &\precbd X+\frac{\mathcal H}{L_b^2}
 \sum_{0<\NK(y)\ll L_b^2/\mathcal H}|M(y)|^2,\\
 M(y)&=\sum_{\NK(n)\asymp L_b}^*
 \mu(n)\xi_1(n)\overline{\chi_n(y)}U_1(\NK(n)/L_b).
 \end{aligned}
\]
Here $\xi_1$ is another fixed ray class character and $U_1$ is a smooth compactly supported weight produced by separating the variables. The term $X$ includes the diagonal and zero-frequency
contributions, using $\mathcal H\le X$.

Since $L_b=X/\NK(b)^3\le X$, we may enlarge the nonnegative sum over $y$ to
\[
 \NK(y)\ll Y,\qquad
 Y=\frac{XL_b}{\mathcal H}\ge\frac{L_b^2}{\mathcal H}.
\]
The purpose of this enlargement is that a second application of
Poisson summation gives a shorter row range:
\[
 \mathcal H'=\frac{L_b^2}{Y}=\frac{\mathcal HL_b}{X}.
\]
The coefficients return to cubic Gauss-sum coefficients, and
\[
 \sum_{0<\NK(y)\ll Y}|M(y)|^2
 \precbd YL_b+Y E'(\mathcal H',L_b).
\]
Here $E'$ has the same form as $E$, with possibly different smooth
weights and fixed characters; $YL_b$ accounts for the
zero-frequency contribution. Consequently,
\[
 E(\mathcal H,L_b)
 \precbd X+\frac X{L_b} E'\!\Bigl(\frac{\mathcal HL_b}{X},L_b\Bigr),
 \qquad
 \frac{E(\mathcal H,L_b)}X
 \precbd 1+\frac{E'(\mathcal HL_b/X,L_b)}{L_b}.
\]
Thus it suffices to prove
\[
 E'\!\Bigl(\frac{\mathcal HL_b}{X},L_b\Bigr)\precbd L_b.
\]
This is the original type of estimate at smaller parameters:
\[
 (\mathcal H',X')
 =\Bigl(\frac{\mathcal H}{\NK(b)^3},
         \frac{X}{\NK(b)^3}\Bigr),
 \qquad
 \frac{\mathcal H'}{X'}=\frac{\mathcal H}{X}.
\]
Both scales decrease while their ratio stays fixed. The factor
$X/L_b$ in the preceding inequality is exactly what converts the
new target bound $L_b$ into the required bound $X$.
These two Poisson summations give the \emph{transfer estimate}: a bound
for the remaining mean square in terms of new mean squares of the same type.
For related uses of an enlarged summation range, see
Goldmakher--Louvel
\cite[Lemma~4.4 and the proof of Theorem~4.1]{GL}, following
Heath-Brown \cite[Lemma~9]{HB95}.


Combining the bounds \eqref{eq:intro-short-cube-ms} and
\eqref{eq:intro-long-cube-ms} from the first part of Step~5 with
the transfer estimate above gives the recursive bound
\[
 \frac{E(\mathcal H,X)}{X}
 \precbd 1+\sup_{\substack{\NK(b)>H_c\\L_b>1}}\frac{E'(\mathcal H',X')}{X'},
 \qquad X'=L_b,\quad \mathcal H'=\frac{\mathcal H L_b}{X}.
\]
For each $b$ in this supremum, the cutoff \eqref{eq:intro-cube-cutoff}
gives $\mathcal H'<\mathcal H(\mathcal H/X)^2
\ll D^{-2\vartheta}\mathcal H$.
Since $\mathcal H'/X'=\mathcal H/X$, the same contraction applies
at every step. After $O_\vartheta(1)$ steps, every resulting mean
square either has an empty remainder or has row parameter at most $1$,
where counting gives the desired bound at its reduced scales.
Applying the recursive inequality back through these steps proves
\eqref{eq:intro-energy-target} for the original $E(\mathcal H,X)$,
completing the sketch of the proof.

The preceding sketch suppresses auxiliary twists and common factors for the purpose of illustration. To carry out this argument with the auxiliary twists included, we use the family
\[
 \mathcal E(\mathcal H,X,F)
 =\frac1{XF}\sum_{\NK(f)\asymp F}^*
   \sum_{0<\NK(k)\ll\mathcal H}
 \Bigl|\sum_n^*a_\xi(n)\chi_n(k)\chi_n(f)^4
                      U(\NK(n)/X)\Bigr|^2
\]
and prove $\mathcal E(\mathcal H,X,F)\precbd XF$ in the parameter
ranges of Proposition~\ref{prop:canonical}. Unlike the sketch above, the full
argument must also handle the common factors and the resulting dyadic
ranges.
The precise family is defined in \eqref{eq:energy}, and
Proposition~\ref{prop:transfer} states the transfer estimate.
Combining it with the bounds for the two parts of
the inverse sum in Step~5 reduces the row range at each step.
Section~\ref{sec:descent} proves the desired bound by a finite iteration,
following the admissible-exponent method of Heath-Brown
\cite[Lemma~8 and \S8]{HB95}, \cite[Lemma~9]{HB00}; see also
\cite[Theorem~4.1]{GL} and \cite[\S3.2]{BGL}.

\section{From the mean-square estimate to the zero-free region}\label{sec:reduction}

We first carry out Steps~1 and~2 of the outline: extract
cancellation in $A_1(D)$ from a mean-square estimate for the family,
then use a Mellin transform to deduce nonvanishing.
Fix a finite-order Hecke character $\nu$ of $K$ and a finite set $S$
of prime ideals that contains all prime ideals above $2$ or $3$, as
well as all prime ideals dividing the conductor of $\nu$.
For an ideal or element $a$, write $(a,S)=1$ if no prime ideal in $S$
divides $a$.
An ideal is supported on $S$ if all its prime factors belong to $S$.
For $W\in C_c^\infty((0,\infty);\mathbb C)$, recall the family
\[
 A_u(D):=\sum_{(n,S)=1}\mu(n)\nu(n)\chi_n(u)W(\NK(n)/D),
\]
introduced in \eqref{eq:intro-family}. Here and throughout the
original family, $n$ runs over ideals prime to $S$, represented by
their primary generators. The key estimate is the following. 

\begin{proposition}\label{thm:ms}
For every fixed $0<\vartheta\le1/10$ and $\varepsilon>0$, there exists an integer $k=k(\vartheta,\varepsilon)\ge1$
such that, for every compact interval $I\subset(0,\infty)$,
all smooth $W$ supported in $I$, and $D\ge2$,
\begin{equation}\label{eq:ms}
 \sum_{0<\NK(u)\le D^{1+\vartheta}}|A_u(D)|^2
       \ll_{\nu,S,I,\vartheta,\varepsilon}
       \Bigl(\max_{0\le j\le k}\|W^{(j)}\|_\infty\Bigr)^2
       D^{2+\vartheta+\varepsilon}.
\end{equation}
\end{proposition}

We first deduce Theorem~\ref{thm:main} assuming Proposition~\ref{thm:ms}.
The proof of the proposition is completed in Section~\ref{sec:completion}.

\begin{proof}[Proof of Theorem~\ref{thm:main} from Proposition~\ref{thm:ms}]
Fix $0<\vartheta\le1/10$, and put $H=D^{1+\vartheta}$ and
$Y=H^{1/6}=D^{(1+\vartheta)/6}$. For prime ideals
$Y/2<\NK(\mathfrak p)\le Y$, $\mathfrak p\notin S$, let $p$ be their
primary generators, chosen with $p\equiv1\pmod3$. Then
$\chi_n(p^6)=\ind_{\mathfrak p\nmid n}$ and therefore
\[
 |A_1(D)-A_{p^6}(D)|
 \le\|W\|_\infty
       \#\{n:(n,S)=1,\ \mathfrak p\mid n,\ \NK(n)\asymp D\}
 \ll_W D/Y.
\]
For this fixed field, Landau's prime ideal theorem \cite{Lan03}
(see \cite[Theorem~5.33]{IK04}) gives
$J\asymp Y/\log Y$ such primes. Their sixth powers are
distinct rows of norm at most $H$. Apply \eqref{eq:ms} with loss
$\varepsilon/2$ and use $\log Y\ll_\varepsilon D^{\varepsilon/2}$ to obtain
\begin{align}
 |A_1(D)|^2
 &\le \frac2J\sum_{\substack{Y/2<\NK(p)\le Y\\(p,S)=1}}|A_{p^6}(D)|^2+O_W(D^2/Y^2)\nonumber\\
 &\ll_{\nu,S,W,\vartheta,\varepsilon}D^{2+\vartheta+\varepsilon}/Y+D^2/Y^2
   =D^{11/6+5\vartheta/6+\varepsilon}+D^{5/3-\vartheta/3}.
                                                        \label{eq:prime-extract}
\end{align}
Thus $A_1(D)\ll_{\nu,S,W,\vartheta,\varepsilon}
D^{11/12+5\vartheta/12+\varepsilon}$. Given any requested exponent loss,
choose $\vartheta>0$ and then the loss in Proposition~\ref{thm:ms}
sufficiently small. Renaming the resulting loss $\varepsilon$, we obtain
\begin{equation}\label{eq:mobius-saving}
                 A_1(D)\ll_{\nu,S,W,\varepsilon}D^{11/12+\varepsilon}.
\end{equation}

We now use
\eqref{eq:mobius-saving} to rule out a zero of $L_K(s,\nu)$ in
$\Rea s>11/12$. Suppose such a zero $\varrho$ exists.
Choose $0\ne\phi\in C_c^\infty((1,2))$, $\phi\ge0$, and
$W(y)=y^{-\varrho}\phi(y)$.
With the Mellin convention
$\widehat W(s)=\int_0^\infty W(y)y^s\dd y/y$, we have
$\widehat W(\varrho)=\int\phi(y)\dd y/y>0$.
The function
\[
 \mathcal M_W(s)=\int_0^\infty A_1(D)D^{-s}\frac{\dd D}{D}
\]
is holomorphic on $\Rea s>11/12$: the estimate
\eqref{eq:mobius-saving} controls the integral at infinity, and
the compact support of $W$ makes $A_1(D)$ vanish for sufficiently
small $D$. Termwise integration for $\Rea s>1$ gives
\[
 \mathcal M_W(s)
 =\widehat W(s)\sum_{(n,S)=1}\frac{\mu(n)\nu(n)}{\NK(n)^s}
 =\frac{\widehat W(s)}{L_K^S(s,\nu)},
\]
where $L_K^S$ is the Euler product outside $S$:
\[
 L_K^S(s,\nu):=L_K(s,\nu)
 \prod_{\mathfrak p\in S}(1-\nu(\mathfrak p)\NK(\mathfrak p)^{-s}).
\]
By Hecke's meromorphic continuation theorem \cite{Hec20}
(see \cite[\S5.10]{IK04}) and the identity theorem, the identity
$L_K^S(s,\nu)\mathcal M_W(s)=\widehat W(s)$ holds on
$\Rea s>11/12$.
The omitted Euler factors are nonzero here, so evaluation at
$s=\varrho$ gives $0=\widehat W(\varrho)>0$, a contradiction.

To deduce the assertion for Dirichlet $L$-functions in
Theorem~\ref{thm:main}, let $\chi_{-3}$ be the nontrivial character modulo $3$.
For any Dirichlet character $\chi$, quadratic base change gives, up to
Euler factors nonzero in $\Rea s>0$,
\[
 L_K(s,\chi\circ\NK)
       =L(s,\chi)L(s,\chi\chi_{-3}).
\]
The Hecke conclusion excludes zeros of either factor away from $s=1$.
At $s=1$, the only possible pole--zero cancellation is ruled out by
Dirichlet's nonvanishing theorem, which gives
$L(1,\chi_{-3})>0$ (see \cite[Chapters~4 and~6]{Dav00}).
\end{proof}

\section{Poisson summation and the dual mean square}\label{sec:initialization}

We now turn to the mean-square estimate in Proposition~\ref{thm:ms}.
Following Step~3 of the outline, we expand the square and apply
Poisson summation in the row variable $u$. The finite Fourier
transforms of the characters supply Gauss sums. Arithmetic identities
then combine these Gauss sums with the original M\"obius coefficients
to give the normalized cubic Gauss-sum coefficients that appear in the theta
function.

The column indices produced by expanding the square need not be
coprime. We first extract their common factor, then use M\"obius
inversion to separate the remaining coprimality condition. These
operations introduce an auxiliary twisting index and an exclusion
ideal. The comparison below removes the exclusion without changing
the row range or the product of the column and auxiliary scales.

We record the arithmetic and Poisson identities first, then define
this family. Proposition~\ref{prop:poisson-reduction} states the
estimate for it that suffices to prove \eqref{eq:ms}; the rest of
the section proves that implication.

\subsection{The arithmetic identities}
We first record the identities that convert between M\"obius
coefficients and normalized cubic Gauss sums. Recall that, for a
squarefree primary element $n$ with $(n,S)=1$,
\[
 \alpha(n)=\frac{n}{|n|},\qquad
 \gamma_j(n)=\frac{1}{\sqrt{\NK(n)}}
       \sum_{x\bmod n}\chi_n(x)^j e(x/n)\quad(j\in\mathbb Z),
\]
where $e(z)=\exp(4\pi\mathrm i\,\operatorname{Im}z/\sqrt3)$, as in
\eqref{eq:intro-gauss-sum}. Put
\[
 a_\xi(n)=\overline{\alpha(n)}\gamma_2(n)\xi(n)
 \qquad(n\text{ squarefree}),
\]
where $\xi$ is a fixed ray class character. The character $\xi$ accounts for the original
twist $\nu$ and for the residue-class factors in the identities below.

By character orthogonality \cite[\S3.1]{IK04}, we can absorb functions
on a fixed ray class group into a finite sum of twists $\xi$.
Choose a fixed modulus, supported on $S$, divisible by the conductor
of $\nu$ and sufficiently large that all reciprocity factors below
depend only on the corresponding ray classes. We then expand these
factors in characters of this finite group.

The following lemma collects standard consequences of the Gauss--Jacobi
identities, quadratic Gauss-sum evaluations, and reciprocity.

\begin{lemma}\label{lem:arithmetic}
On a fixed ray class group, there are a function $G$ with values in
$\{z\in\mathbb C:|z|=1\}$ and a symmetric $\{\pm1\}$-valued
bicharacter $\mathcal R$. This means that $\mathcal R(a,b)=\mathcal R(b,a)$ and $\mathcal R$ is
multiplicative in each argument separately:
\[
 \begin{aligned}
 \mathcal R(aa',b)&=\mathcal R(a,b)\mathcal R(a',b),\\
 \mathcal R(a,bb')&=\mathcal R(a,b)\mathcal R(a,b'),
 \end{aligned}
\]
for all classes $a,a',b,b'$ in the ray class group.
These functions have the following properties.
Let $a,b,n$ be primary elements prime to $S$, with $(a,b)=1$ and
$n$ squarefree. Then
\begin{align}
 \chi_b(a)&=\mathcal R(a,b)\chi_a(b),&
 G(ab)&=G(a)G(b)\mathcal R(a,b),                                      \label{eq:recip}\\
 \gamma_2(n)^3&=\mu(n)\alpha(n),&
 \gamma_1(n)\gamma_2(n)&=\mu(n)\alpha(n)G(n),                 \label{eq:gj}\\
 G(n)&=\overline{\chi_n(4)}\gamma_3(n),&
 \gamma_1(n)\gamma_{-1}(n)&=\chi_n(-1).                     \label{eq:g}
\end{align}
Consequently
\begin{align}
 \overline{\alpha(n)}\gamma_2(n)\gamma_1(n)&=\mu(n)G(n),\label{eq:convert1}\\
 \mu(n)\gamma_{-1}(n)
 &=\chi_n(-1)G(n)^{-1}\overline{\alpha(n)}\gamma_2(n),\label{eq:convert2}\\
 a_\xi(ab)&=a_\xi(a)a_\xi(b)\chi_b(a)^4,                     \label{eq:crt-a}\\
 \chi_a(-1)\overline{G(a)}G(b)\mathcal R(a,b)&=G(ba^{-1}).            \label{eq:quotient}
\end{align}
In \eqref{eq:crt-a}, $a,b$ are also squarefree.
The identity for $G(ab)$ in \eqref{eq:recip} extends to all classes of the fixed ray class
group; the quotient in \eqref{eq:quotient} is taken in that group.
\end{lemma}
The proof, including the dependence of $G$ and $\mathcal R$ on fixed ray
classes, is given in Appendix~\ref{app:gauss-identities}.
Under Poisson summation, \eqref{eq:convert2} converts the M\"obius
coefficients to $a_\xi(n)$, up to fixed ray class factors, while
\eqref{eq:convert1} converts them back.

\subsection{Poisson summation with excluded primes}
The row sum to which we apply Poisson summation will have an
additional coprimality restriction. We record the formula with that
restriction included, so that its effect on the Fourier frequencies
and the normalization is explicit.

For a nonzero ideal $\mathfrak r$, write $\rad \mathfrak r$ for the product of
its distinct prime divisors. We use the additive character $e$
introduced in Step~3. In quotients and Gauss sums, use a fixed
generator for each ideal, chosen primary when the ideal is prime to $3$.

\begin{lemma}\label{lem:poisson}
Let $\mathcal H>0$, let $\mathfrak r$ be a nonzero ideal, and let
$\Phi(\NK(k)/\mathcal H)$ be a smooth radial Schwartz weight
on the row lattice. Let $\chi$ be a primitive multiplicative character
of $(\OO/\mathfrak m)^\times$, viewed as a function on $\OO$ by
reduction modulo $\mathfrak m$ and extension by zero on nonunits.
Write 
\[
\gamma(\chi)=\NK(\mathfrak m)^{-1/2}
\sum_{x\bmod\mathfrak m}\chi(x)e(x/\mathfrak m)
\]
for its normalized Gauss sum.
Since $\chi$ is primitive, its modulus $\mathfrak m$ is determined by $\chi$ and is suppressed in the notation. Then
\begin{equation}\label{eq:poisson}
 \begin{aligned}
 &\sum_k\chi(k)\ind_{(k,\mathfrak r)=1}\Phi(\NK(k)/\mathcal{H})\\
 &\quad=\frac{\mathcal{H}\gamma(\chi)}{\sqrt{\NK(\mathfrak m)}}
   \sum_{d\mid\rad \mathfrak r}\frac{\mu(d)\chi(d)}{\NK(d)}
   \sum_h\overline{\chi(h)}
        \widehat\Phi\!\Bigl(\frac{\mathcal{H}\NK(h)}{\NK(d)\NK(\mathfrak m)}\Bigr).
 \end{aligned}
\end{equation}
Here $\widehat\Phi$ is defined by
\[
 \widehat\Phi(|w|^2)=\frac2{\sqrt3}\int_{\mathbb C}
       \Phi(|z|^2)e(-zw)\,dx\,dy,\qquad z=x+\mathrm i y.
\]
The measure is normalized so that $\OO$ has covolume one.
Here $k,h\in\OO$, and $h$ is the Fourier frequency. We have
\[
 \chi\text{ nonprincipal}\quad\Longrightarrow\quad
 \overline{\chi(0)}=0.
\]
For the principal primitive character ($\chi=\mathbf1$, $\mathfrak m=1$),
the zero-frequency contribution is
\[
 \mathcal H\widehat\Phi(0)
   \prod_{\mathfrak p\mid \mathfrak r}\Bigl(1-\frac1{\NK(\mathfrak p)}\Bigr).
\]
\end{lemma}

\begin{proof}
Inclusion--exclusion followed by $k=d\ell$ gives
\[
 \sum_k\chi(k)\ind_{(k,\mathfrak r)=1}\Phi(\NK(k)/\mathcal{H})
 =\sum_{d\mid\rad \mathfrak r}\mu(d)\chi(d)
       \sum_\ell\chi(\ell)\Phi\!\Bigl(\frac{\NK(\ell)}{\mathcal{H}/\NK(d)}\Bigr).
\]
Apply lattice Poisson summation \cite[Theorem~4.5]{IK04} in $\ell$
on residue classes modulo $\mathfrak m$, at scale $\mathcal H/\NK(d)$.
The primitive Gauss-sum identity \cite[(3.12)]{IK04} evaluates the
finite Fourier transform as
$\sqrt{\NK(\mathfrak m)}\gamma(\chi)\overline{\chi(h)}$
for every $h$, giving \eqref{eq:poisson}. For a nonprincipal primitive
character, $\overline{\chi(0)}=0$, so the zero-frequency term vanishes.
For the principal character, sum $\mu(d)/\NK(d)$ over
$d\mid\rad \mathfrak r$ to obtain the displayed zero-frequency contribution.
\end{proof}

\subsection{The dual mean squares} 
We use the following mean square of the column sums with
coefficients $a_\xi(n)$.

\begin{definition}
The parameters $\mathcal H$, $X$, and $F$ are the norm scales of
$k$, $n$, and $f$, respectively. Here $k$ ranges over elements of
$\OO$, and a star restricts a sum to squarefree ideals prime to $S$,
represented by their primary generators. For a smooth compactly
supported weight $W$ on $(0,\infty)$, define the \emph{dual mean square} by
\begin{equation}\label{eq:energy}
 \begin{aligned}
 &\E(\mathcal{H},X,F;\xi,W)\\
 &\qquad=\frac1{XF}
 \sum_{\substack{F\le\NK(f)<2F\\(f,S)=1}}^*\sum_{0<\NK(k)\le \mathcal{H}}
 \Bigl|\sum_{(n,S)=1}^*a_\xi(n)\chi_n(k)\chi_n(f)^4
                  W(\NK(n)/X)\Bigr|^2.
 \end{aligned}
\end{equation}
\end{definition}

Below, $\xi$ ranges over all characters of the fixed ray class group
chosen above, through which $\nu$, $G$, and $\mathcal R$ factor.

Write $\E_{\mathfrak r}$ for the same expression with the additional
restriction $(n,\mathfrak r)=1$. This notation is only needed in the
Poisson reductions; the following comparison returns to $\E$.

\begin{lemma}\label{lem:remove-exclusions}
Let $r_0$ be the product of the primes dividing $\mathfrak r$ outside
$S$. For $\mathcal H,X>0$, $F\ge1$, and smooth compactly supported $W$,
\begin{equation}\label{eq:remove-exclusions}
 \E_{\mathfrak r}(\mathcal H,X,F;\xi,W)
 \le\tau_{\mathrm{div}}(r_0)
 \sum_{d\mid r_0}\E\!\Bigl(\mathcal H,\frac X{\NK(d)},F\NK(d);\xi,W\Bigr).
\end{equation}
\end{lemma}
\begin{proof}
Let $C_{\mathfrak r}(X;k,f)$ denote the inner column sum defining
$\E_{\mathfrak r}$, with $\xi,W$ fixed. Inclusion--exclusion,
$n=dm$, and \eqref{eq:crt-a} give
\[
 \begin{aligned}
 C_{\mathfrak r}(X;k,f)
 &=\sum_{d\mid r_0}\mu(d)
   \sum_{\substack{(n,S)=1\\d\mid n}}^*
   a_\xi(n)\chi_n(k)\chi_n(f)^4W(\NK(n)/X)\\
 &=\sum_{\substack{d\mid r_0\\(d,f)=1}}
   \mu(d)a_\xi(d)\chi_d(k)\chi_d(f)^4
   C_1(X/\NK(d);k,df).
 \end{aligned}
\]
Indeed, $\chi_m(d)^4$ enforces $(m,d)=1$ and combines with
$\chi_m(f)^4$; terms with $(d,f)\ne1$ vanish. Each exterior
coefficient has modulus at most one. Apply Cauchy--Schwarz in $d$,
then enlarge the injective image $f\mapsto df$ to the squarefree
range $F\NK(d)\le\NK(df)<2F\NK(d)$. The normalizing product is
unchanged: $(X/\NK(d))(F\NK(d))=XF$.
\end{proof}


\subsection{Reduction to the dual mean square}
The following proposition gives the dual estimate sufficient for
\eqref{eq:ms}. Its proof applies Poisson summation directly to the
original M\"obius sums.

\begin{proposition}\label{prop:poisson-reduction}
Fix $0<\vartheta\le1/10$ and put $H=D^{1+\vartheta}$.
For each fixed $C\ge1$ and all real $B,F\ge1$, consider the ranges
\begin{equation}\label{eq:initial-scales}
 X=\frac D{BF},\qquad 0<\mathcal{H}\le\frac{CD^2}{HB^2},\qquad
 XF=\frac DB.
\end{equation}
Suppose that for every $\varepsilon>0$ there exists an integer
$J=J(\vartheta,\varepsilon)\ge1$ such that, for every compact interval
$I\subset(0,\infty)$ and every smooth $W$ supported in $I$,
\begin{equation}\label{eq:auxiliary-target}
 \E(\mathcal H,X,F;\xi,W)
 \ll_{\nu,S,I,C,\vartheta,\varepsilon}\|W\|_{C^J(I)}^2D^\varepsilon XF,
\end{equation}
where
\[
 \|W\|_{C^J(I)}=\max_{0\le j\le J}\sup_{x\in I}|W^{(j)}(x)|.
\]
For functions of several variables, the norm uses all partial derivatives
of total order at most $J$.
Then the original mean-square
estimate in Proposition~\ref{thm:ms} holds.
\end{proposition}

\begin{proof}
Write $I=[a_0,b_0]$. Choose a nonnegative radial Schwartz weight
$\Phi$ such that $\Phi(x)\ge1$ on $[0,1]$ and
$\operatorname{supp}\widehat\Phi\subset[0,C_\Phi]$ for some fixed $C_\Phi>0$.
It suffices to prove
$\mathcal M_D\ll_{\nu,S,I,\vartheta,\varepsilon}
\|W\|_{C^{J'}(I)}^2HD^\varepsilon$ for some $J'=J'(\vartheta,\varepsilon)$, where
\begin{equation}\label{eq:expanded-ms}
 \mathcal M_D:=\frac1D\sum_{u\in\OO}\Phi(\NK(u)/H)
 \Bigl|\sum_{(n,S)=1}\mu(n)\nu(n)\chi_n(u)W(\NK(n)/D)\Bigr|^2.
\end{equation}
Conjugate the inner sum, expand the square, and put
$g=(n_1,n_2)$ and $n_j=gz_j$. The squarefree indices satisfy
$(z_1,z_2)=(z_1z_2,g)=1$, and
\[
 \overline{\chi_{n_1}(u)}\chi_{n_2}(u)
 =\ind_{(u,g)=1}\overline{\chi_{z_1}(u)}\chi_{z_2}(u).
\]
Apply Lemma~\ref{lem:poisson} with
$\chi=\overline{\chi_{z_1}}\chi_{z_2}$ and exclusion $g$:
\begin{equation}\label{eq:initial-poisson-rows}
 \begin{aligned}
 &\sum_u\Phi(\NK(u)/H)\ind_{(u,g)=1}\chi(u)\\
 &\quad=\sum_{e\mid g}
 \frac{H\mu(e)\chi(e)\gamma(\chi)}
      {\NK(e)\sqrt{\NK(z_1)\NK(z_2)}}
 \sum_h\overline{\chi(h)}
 \widehat\Phi\!\Bigl(\frac{H\NK(h)}{\NK(e)\NK(z_1)\NK(z_2)}\Bigr).
 \end{aligned}
\end{equation}
The zero frequency occurs only when $z_1=z_2=1$ and contributes
\[
 Z=\frac HD\widehat\Phi(0)
 \sum_{(g,S)=1}^*|W(\NK(g)/D)|^2
       \prod_{p\mid g}\Bigl(1-\frac1{\NK(p)}\Bigr)
 \ll_{\Phi,I}H\|W\|_\infty^2.
\]
Retain all $h\ne0$, including those with $z_1=z_2=1$.

Expand the fixed ray class function
$\overline{\nu(t)}G(t^{-1})=\sum_\xi c_\xi\xi(t)$.
The Chinese remainder theorem and
\eqref{eq:convert2}--\eqref{eq:quotient} give
\begin{equation}\label{eq:initial-paired-gauss}
 \mu(z_1)\mu(z_2)\overline{\nu(z_1)}\nu(z_2)
       \gamma(\overline{\chi_{z_1}}\chi_{z_2}) =\sum_\xi c_\xi a_\xi(z_1)\overline{a_\xi(z_2)}.
\end{equation}
Write $N=\NK$ for the remainder of this proof, and put
$W_0(x)=x^{-1/2}\overline{W(x)}$. Substituting
\eqref{eq:initial-paired-gauss} into \eqref{eq:initial-poisson-rows},
including the normalization $1/D$ in \eqref{eq:expanded-ms}, gives
\[
 \mathcal M_D-Z=\sum_\xi c_\xi\mathcal S_\xi,
 \qquad
 |\mathcal M_D-Z|\ll\max_\xi|\mathcal S_\xi|,
\]
since the character sum is fixed and finite. Fix $\xi$. Using
$\overline{\chi_z(e)}=\chi_z(e^5)$, its contribution is
\[
 \begin{aligned}
 \mathcal S_\xi
 ={}&\frac H{D^2}
 \sum_g^*\sum_{e\mid g}\frac{\mu(e)N(g)}{N(e)}
 \sum_{h\ne0}
 \sum_{\substack{z_1,z_2\\(z_1,z_2)=1\\(z_1z_2,g)=1}}^*
 a_\xi(z_1)\overline{a_\xi(z_2)}
 \chi_{z_1}(he^5)\overline{\chi_{z_2}(he^5)}
 \\
 &\quad{}\times
 W_0\!\Bigl(\frac{N(gz_1)}D\Bigr)
 \overline{W_0\!\Bigl(\frac{N(gz_2)}D\Bigr)}
 \widehat\Phi\!\Bigl(
 \frac{HN(h)}{N(e)N(z_1)N(z_2)}
 \Bigr).
 \end{aligned}
\]
Here and below every starred variable is squarefree, primary, and
prime to $S$, while $h,k$ range over nonzero elements of $\OO$.
We must show
$|\mathcal S_\xi|\ll_{\nu,S,I,\vartheta,\varepsilon}
\|W\|_{C^{J'}(I)}^2HD^\varepsilon$.

Insert the coprimality identity and change variables:
\[
 \ind_{(z_1,z_2)=1}
 =\sum_{v\mid z_1,\ v\mid z_2}\mu(v),
 \qquad z_j=vm_j.
\]
By \eqref{eq:crt-a}, the common factor from the two columns satisfies
\[
 a_\xi(vm)=a_\xi(v)a_\xi(m)\chi_m(v)^4,
 \qquad
 |a_\xi(v)\chi_v(he^5)|^2=\ind_{(v,h)=1},
\]
where $(v,e)=1$. Thus the same sum becomes
\[
 \begin{aligned}
 \mathcal S_\xi
 ={}&\frac H{D^2}
 \sum_{\substack{g,v\\(g,v)=1}}^*
 \sum_{e\mid g}\frac{\mu(e)\mu(v)N(g)}{N(e)}
 \sum_{\substack{h\ne0\\(h,v)=1}}
 \sum_{\substack{m_1,m_2\\(m_1m_2,gv)=1}}^*
 a_\xi(m_1)\overline{a_\xi(m_2)}
 \\
 &\quad{}\times
 \chi_{m_1}(he^5v^4)\overline{\chi_{m_2}(he^5v^4)}
 W_0\!\Bigl(\frac{N(gvm_1)}D\Bigr)
 \overline{W_0\!\Bigl(\frac{N(gvm_2)}D\Bigr)}
 \\
 &\quad{}\times
 \widehat\Phi\!\Bigl(
 \frac{HN(h)}{N(e)N(v)^2N(m_1)N(m_2)}
 \Bigr).
 \end{aligned}
\]
There is now no restriction $(m_1,m_2)=1$.
Make the bijective change of variables
\[
 \begin{gathered}
 b=g/e,\qquad f=ev,\qquad k=eh,\\
 e=(f,k),\qquad v=f/e,\qquad g=be,\qquad h=k/e.
 \end{gathered}
\]
The resulting $b,f$ are coprime and squarefree, $k\ne0$ is arbitrary,
and
\[
 gv=bf,\qquad he^5v^4=kf^4,\qquad
 \mu(e)\mu(v)=\mu(f),\qquad
 \frac{N(g)}{N(e)}=N(b).
\]
Consequently,
\begin{equation}\label{eq:initial-column-output}
 \begin{aligned}
 \mathcal S_\xi
 ={}&\frac H{D^2}
 \sum_{\substack{b,f\\(b,f)=1}}^*\mu(f)N(b)
 \sum_{k\ne0}
 \sum_{\substack{m_1,m_2\\(m_1m_2,b)=1}}^*
 a_\xi(m_1)\overline{a_\xi(m_2)}
 \chi_{m_1}(kf^4)\overline{\chi_{m_2}(kf^4)}
 \\
 &\quad{}\times
 W_0\!\Bigl(\frac{N(bfm_1)}D\Bigr)
 \overline{W_0\!\Bigl(\frac{N(bfm_2)}D\Bigr)}
 \widehat\Phi\!\Bigl(
 \frac{HN(k)}{N(f)^2N(m_1)N(m_2)}
 \Bigr).
 \end{aligned}
\end{equation}
The characters enforce $(m_j,f)=1$, leaving only the displayed
exclusion $(m_j,b)=1$.
The supports of $W_0$ and $\widehat\Phi$ give
\[
 N(b)N(f)\le b_0D,\qquad
 0<N(k)\le\frac{C_\Phi b_0^2D^2}{HN(b)^2}.
\]

Partition $B\le N(b)<2B$ and $F\le N(f)<2F$ into dyadic ranges.
Denote the corresponding contribution to
\eqref{eq:initial-column-output} by $\mathcal S_{\xi;B,F}$, and put
\[
 X=\frac D{BF},\qquad
 \mathcal H=\frac{C_{I,\Phi}D^2}{HB^2},\qquad
 C_{I,\Phi}=\max(2,C_\Phi b_0^2).
\]
These satisfy \eqref{eq:initial-scales}.
In the variables $x_j=N(m_j)/X$, the coupled smooth weight is
\[
 \begin{gathered}
 \mathcal K_{b,f,k}(x_1,x_2)
 =W_0(r x_1)\overline{W_0(r x_2)}
 \widehat\Phi\!\Bigl(\frac{a}{x_1x_2}\Bigr),\\
 r=\frac{N(b)N(f)}{BF}\in[1,4),\qquad
 a=\frac{HN(k)}{N(f)^2X^2}\le C_{I,\Phi}.
 \end{gathered}
\]
For every integer $q\ge0$, these kernels satisfy
\[
 \sup_{b,f,k}\|\mathcal K_{b,f,k}\|_{C^q([a_0/4,b_0]^2)}
 \ll_{I,\Phi,q}\|W\|_{C^q(I)}^2.
\]
For $U\in C_c^\infty(I_*)$, where $I_*=[a_0/8,2b_0]$, put
\[
 S_U(b,f,k)=\sum_{\substack{(n,S)=1\\(n,b)=1}}^*a_\xi(n)
       \chi_n(k)\chi_n(f)^4U(N(n)/X).
\]
The shifts $(X,F)\mapsto(X/N(d),FN(d))$ preserve
\eqref{eq:initial-scales}. Thus Lemma~\ref{lem:remove-exclusions}
and \eqref{eq:auxiliary-target}, with $J=J(\vartheta,\varepsilon/4)$, give
\[
 \begin{aligned}
 \sum_{f,k}|S_U(b,f,k)|^2
 &=XF\,\E_{(b)}(\mathcal H,X,F;\xi,U)\\
 &\le XF\,\tau_{\mathrm{div}}(b)
       \sum_{d\mid b}\E(\mathcal H,X/N(d),FN(d);\xi,U)\\
 &\ll_{\nu,S,I,\vartheta,\varepsilon}
 D^{\varepsilon/2}(XF)^2\|U\|_{C^J(I_*)}^2.
 \end{aligned}
\]
Here $N(b)\ll_I D$, so the divisor factors are absorbed in $D^{\varepsilon/4}$.
We use the following smooth-weight principle, stated and proved in
Lemma~\ref{lem:smooth-mean-square}: a mean-square bound valid for every
common test function, with a $C^J$ norm, also bounds the corresponding
quadratic sum with a kernel depending on the row, at a cost given by
the kernel's $C^{2J+4}$ norm.
For each fixed $b$, apply Lemma~\ref{lem:smooth-mean-square}
with row index $(f,k)$, coefficient $\mu(f)\ind_{(f,b)=1}$,
and kernel $\mathcal K_{b,f,k}$.
The larger row range $0<N(k)\le\mathcal H$ adds only terms
whose original kernel vanishes. With $q=2J+4$, this gives
\begin{equation}\label{eq:weighted}
 \begin{aligned}
 |\mathcal S_{\xi;B,F}|
 &\ll_{\nu,S,I,\vartheta,\varepsilon}
 D^{\varepsilon/2}\|W\|_{C^q(I)}^2
 \sum_{B\le N(b)<2B}^*\frac{HN(b)}{D^2}(XF)^2\\
 &\ll D^{\varepsilon/2}\|W\|_{C^q(I)}^2
 \frac{HB}{D^2}\,B\,(XF)^2
 =D^{\varepsilon/2}\|W\|_{C^q(I)}^2H,
 \end{aligned}
\end{equation}
where $XF=D/B$ and ideal counting gives $O(B)$ choices of $b$.
Summing the $O((\log D)^2)$ nonempty dyadic ranges bounds
$\mathcal S_\xi$ as required. Sum over the fixed finite set of $\xi$,
include $Z$, and restore the factor $D$ to obtain \eqref{eq:ms}.
The required derivative order depends only on $\vartheta,\varepsilon$.

\end{proof}

\section[Iteration of the dual mean-square estimate]{\texorpdfstring{Iteration of the dual mean-square estimate}{Iteration of the dual mean-square estimate}}\label{sec:descent}

Put $\Sigma=XF$. We prove the following estimate for the family
\eqref{eq:energy}, with exclusions removed by
Lemma~\ref{lem:remove-exclusions}. The two inputs are proved in
Sections~\ref{sec:reflection} and~\ref{sec:transfer-proof}.

\begin{proposition}\label{prop:canonical}
Fix $\kappa>0$ and $C_0\ge1$. Suppose that
\begin{equation}\label{eq:positive-gap}
 \mathcal H,X,F\ge1,\qquad
 \Sigma=XF\le D^{C_0},\qquad
 \mathcal H\le\Sigma D^{-\kappa}.
\end{equation}
For every $\varepsilon>0$ there is an integer
$J=J(\kappa,C_0,\varepsilon)\ge1$ such that
\begin{equation}\label{eq:canonical-bound}
 \E(\mathcal H,X,F;\xi,W)
 \ll_{I,\nu,S,\kappa,C_0,\varepsilon}
 \|W\|_{C^J(I)}^2D^\varepsilon\Sigma
\end{equation}
for every compact interval $I\subset(0,\infty)$ and smooth $W$ supported
in $I$.
\end{proposition}

The proof is given in Section~\ref{sec:canonical-proof}, using
Lemma~\ref{lem:cube-reduction} and Proposition~\ref{prop:transfer}.

\subsection{The completed sums}
Let $\Psi$ be a completely multiplicative $\mathbb C$-valued function on the
nonzero integral ideals of $\OO_K=\mathbb Z[\omega]$ coprime to $3$, vanishing on ideals divisible by a prime in $S$. 
For a primary element $n$, write $\Psi(n)=\Psi((n))$.
Define
\begin{equation}\label{eq:T}
 T(X;\Psi)=
 \sum_{(n,S)=1}^*\sum_{\substack{b\in\OO\\b\equiv1\ (3)\\(b,S)=1}}
 \frac{\overline{\alpha(n)}\gamma_2(n)\Psi(n)
       \overline{\alpha(b)}^{\,3}\Psi(b)^3}
      {\sqrt{\NK(n)}\,\NK(b)}\,V_*(\NK(n)\NK(b)^3/X),
\end{equation}
where $V_*(y)=\sqrt y\,W(y)$. The extra index $b$ supplies the cubes in the Fourier expansion of the
Kubota theta function.
Here $n,b$ are primary elements of $\OO$; only $n$ is required to be
squarefree.

The $b=1$ part is exactly
$X^{-1/2}\sum_{(n,S)=1}^*\overline{\alpha(n)}\gamma_2(n)\Psi(n)W(\NK(n)/X)$.
To identify this with the normalized column sum in \eqref{eq:energy},
fix a ray class character $\xi$. For a nonzero row $k\in\OO$ and a
squarefree primary $f$ prime to $S$, set
\begin{equation}\label{eq:completed-twist}
 \Psi_k(n):=\xi(n)\chi_n(k)\chi_n(f)^4,
 \qquad T(X;k,f):=T(X;\Psi_k).
\end{equation}
The displayed product defines $\Psi_k(n)$ for $(n,S)=1$; set $\Psi_k(n)=0$ otherwise. Thus the zero extension required in \eqref{eq:T} is part of this definition.


\subsection{The completed mean-square estimate}
For the twist \eqref{eq:completed-twist}, the theta transformation converts
the relevant sextic twists into quadratic characters. Combining it with
the quadratic large sieve gives the following estimate, proved in
Section~\ref{sec:reflection}.

\begin{proposition}\label{prop:R}
Fix $\varepsilon>0$, $C_0\ge1$, and a character $\xi$ of the fixed ray
class group. There is $J=J(\varepsilon,C_0)\ge1$ such that
\begin{equation}\label{eq:R}
 \sum_{0<\NK(k)\ll\mathcal H}|T(X;k,f)|^2
 \ll_{I,\xi,S,\varepsilon,C_0}
 D^\varepsilon\|W\|_{C^J(I)}^2
 \Bigl(\mathcal H+\frac{\mathcal H^2\NK(f)}{X}\Bigr)
\end{equation}
whenever
\[
 1\le\mathcal H,X,\NK(f)\le D^{C_0}.
\]
Here $f$ is squarefree and primary with $(f,S)=1$,
$W$ is smooth and supported in a compact interval $I\subset(0,\infty)$,
and $T$ is defined by \eqref{eq:completed-twist}.
\end{proposition}
\subsection{Cube inversion and the remaining sums}
Set
\begin{equation}\label{eq:cube-cutoff}
 H_c^3=\min\!\Bigl(X,\frac{X^2}{\mathcal H^2}\Bigr),
 \qquad L_b=\frac{X}{\NK(b)^3}.
\end{equation}

\begin{lemma}\label{lem:cube-reduction}
Fix $C_0\ge1$ and $\varepsilon>0$. Suppose
$1\le\mathcal H,X,F\le D^{C_0}$ and $\mathcal H\le\Sigma=XF$.
There is an integer $J=J(\varepsilon,C_0)\ge1$ such that
\begin{equation}\label{eq:cube-reduction}
 \E(\mathcal H,X,F;\xi,W)
 \ll_{I,\nu,S,C_0,\varepsilon}D^\varepsilon
 \biggl(\Sigma\|W\|_{C^J(I)}^2+
 \sup_{\substack{b\equiv1\ (3),\ (b,S)=1\\
                  \NK(b)>H_c,\ L_b>1}}
       \E(\mathcal H,L_b,F;\xi,W)\biggr)
\end{equation}
for every compact interval $I\subset(0,\infty)$ and
$W\in C_c^\infty(I)$. An empty supremum is zero; in particular it
is empty when $\mathcal H^2\le X$.
\end{lemma}

\begin{proof}
Write $N=\NK$ and $I=[u,v]$. Complete multiplicativity in
\eqref{eq:T}, including at the zeros of $\Psi_k$, gives
\begin{equation}\label{eq:cube-inverse}
 \begin{aligned}
 &X^{-1/2}\sum_{(n,S)=1}^*a_\xi(n)\chi_n(k)\chi_n(f)^4
 W(N(n)/X)\\
 &\qquad=\sum_{(h,S)=1}
 \frac{\mu(h)\overline{\alpha(h)}^{\,3}\Psi_k(h)^3}{N(h)}
 T(X/N(h)^3;k,f).
 \end{aligned}
\end{equation}
Indeed, after substitution of \eqref{eq:T}, the coefficient of the
total cube index $b$ contains
$\sum_{h\mid b}\mu(h)=\ind_{b=1}$.
This is M\"obius inversion; compare \cite[(1.18)]{IK04} and
\cite[(8.2)]{DR}.

Split the right-hand side of \eqref{eq:cube-inverse} into
$P_{\rm short}(k,f)+P_{\rm long}(k,f)$, where
\[
 \begin{aligned}
 P_{\rm short}(k,f)&=
 \sum_{\substack{(h,S)=1\\N(h)\le H_c}}
 \frac{\mu(h)\overline{\alpha(h)}^{\,3}\Psi_k(h)^3}{N(h)}
 T(X/N(h)^3;k,f),\\
 \E_{\rm short}&=\frac1F\sum_{\substack{F\le N(f)<2F\\(f,S)=1}}^*
 \sum_{0<N(k)\le\mathcal H}|P_{\rm short}(k,f)|^2.
 \end{aligned}
\]
Define $\E_{\rm long}$ in the same way using $P_{\rm long}$.
Weighted Cauchy--Schwarz for each $k,f$, followed by
Proposition~\ref{prop:R}, gives
\begin{equation}\label{eq:short-cube-bound}
 \begin{aligned}
 \E_{\rm short}
 &\le\Bigl(\sum_{N(h)\le H_c}\frac1{N(h)}\Bigr)
 \sum_{N(h)\le H_c}\frac1{N(h)} \frac1F\sum_{\substack{F\le N(f)<2F\\(f,S)=1}}^*
 \sum_{0<N(k)\le\mathcal H}|T(X/N(h)^3;k,f)|^2\\
 &\ll_{I,\nu,S,C_0,\varepsilon}
 D^{\varepsilon/2}\|W\|_{C^J(I)}^2
 \Bigl(\mathcal H+\frac{\mathcal H^2FH_c^3}{X}\Bigr)
 \ll D^{\varepsilon/2}\Sigma\|W\|_{C^J(I)}^2.
 \end{aligned}
\end{equation}
Here $X/N(h)^3\ge1$, and the reciprocal-norm sum contributes
only a logarithm. We use Proposition~\ref{prop:R} with exponent
$\varepsilon/4$ and range $C_0+1$, since $N(f)<2D^{C_0}$.
If $H_c<1$, the sum is empty.

Expand $T$ using \eqref{eq:T} in the remaining terms, with cube index $c$:
\[
 \begin{aligned}
 P_{\rm long}(k,f)
 ={}&\frac1{\sqrt X}
 \sum_{\substack{(h,S)=1\\N(h)>H_c}}\sum_{(c,S)=1}
 \mu(h)\sqrt{N(hc)}\,\overline{\alpha(hc)}^{\,3}\Psi_k(hc)^3\\
 &\quad\times\sum_{(n,S)=1}^*a_\xi(n)\chi_n(k)\chi_n(f)^4
 W\!\Bigl(\frac{N(n)N(hc)^3}{X}\Bigr).
 \end{aligned}
\]
Put $b=hc$; the pairs giving $b$ are exactly $(h,b/h)$ with
$h\mid b$ and $N(h)>H_c$. Since
$\Psi_k(b)^3=\xi(b)^3\chi_b(k)^3\ind_{(b,f)=1}$, this gives
\[
 \begin{aligned}
 P_{\rm long}(k,f)
 ={}&\sum_{\substack{(b,S)=1\\N(b)>H_c}}
 \frac{\beta_0(b,f)\chi_b(k)^3}{N(b)}
 \frac1{\sqrt{L_b}}
 \sum_{(n,S)=1}^*a_\xi(n)\chi_n(k)\chi_n(f)^4
 W(N(n)/L_b),\\
 \beta_0(b,f):={}&\overline{\alpha(b)}^{\,3}\xi(b)^3
 \ind_{(b,f)=1}\sum_{\substack{h\mid b\\N(h)>H_c}}\mu(h),
 \qquad |\beta_0(b,f)|\le\tau_{\mathrm{div}}(b).
 \end{aligned}
\]
The divisor function $\tau_{\mathrm{div}}(b)$ counts ideal divisors.
All these indices are primary and prime to $S$; $b$ need not be
squarefree. The support of $W$ restricts the sum to $N(b)^3\le vX$.
Weighted Cauchy--Schwarz now gives
\[
 \begin{aligned}
 \E_{\rm long}
 &\le\Bigl(\sum_{\substack{H_c<N(b)\le(vX)^{1/3}\\(b,S)=1}}
             \frac{\tau_{\mathrm{div}}(b)}{N(b)}\Bigr)
       \sum_{\substack{H_c<N(b)\le(vX)^{1/3}\\(b,S)=1}}
             \frac{\tau_{\mathrm{div}}(b)}{N(b)}
             \E(\mathcal H,L_b,F;\xi,W).
 \end{aligned}
\]
The reciprocal divisor sum is $\ll_I\log^2(2+X)$, since
\[
 \sum_{N(b)\le R}\frac{\tau_{\mathrm{div}}(b)}{N(b)}
 =\sum_{N(cd)\le R}\frac1{N(c)N(d)}
 \le\Bigl(\sum_{N(c)\le R}\frac1{N(c)}\Bigr)^2
 \ll\log^2(2R)\qquad(R\ge1).
\]
If $L_b\le1$, the column sum is empty unless $L_b\ge1/v$;
otherwise it has $O_I(1)$ terms, so
\[
 \E(\mathcal H,L_b,F;\xi,W)
 \ll_I\frac{\mathcal H}{L_b}\|W\|_\infty^2
 \ll_I\Sigma\|W\|_\infty^2.
\]
Absorb the logarithms in $D^\varepsilon$ and combine the two parts
using $\E\le2\E_{\rm short}+2\E_{\rm long}$.
This proves \eqref{eq:cube-reduction}. If $\mathcal H^2\le X$,
then $H_c=X^{1/3}$, so $N(b)>H_c$ implies $L_b<1$.
\end{proof}

\subsection{The transfer estimate}
Fix a nonnegative radial Schwartz weight $\Phi$ with
$\Phi(t)\ge1$ for $0\le t\le1$ and
$\operatorname{supp}\widehat\Phi\subset[0,C_\Phi]$.
Such a weight is obtained by squaring and rescaling a real radial
Schwartz function with compactly supported Fourier transform.
Write $N=\NK$. For fixed $\mathcal H,L,F,\xi$, put
\begin{equation}\label{eq:smoothed-energy}
 \mathcal A(W):=\frac1{LF}
 \sum_{\substack{F\le N(f)<2F\\(f,S)=1}}^*
 \sum_{k\in\OO}\Phi(N(k)/\mathcal H)
 \Bigl|\sum_{(n,S)=1}^*
 a_\xi(n)\chi_n(k)\chi_n(f)^4W(N(n)/L)\Bigr|^2.
\end{equation}
Thus $\E(\mathcal H,L,F;\xi,W)\le\mathcal A(W)$.
In the following proposition $\Sigma$ is an independent target scale;
we will apply it at column scale $L_b$ with $\Sigma=XF$.

\begin{proposition}\label{prop:transfer}
Assume $1\le\mathcal H,L,F,\Sigma\le D^{C_0}$ and
$\max\{\mathcal H,LF\}\le\Sigma$, for fixed $C_0\ge1$.
For every integer $m\ge0$ and $\varepsilon>0$,
\begin{equation}\label{eq:transfer-summary}
 \mathcal A(W)\ll D^\varepsilon\Sigma\|W\|_{C^{4m+12}(I)}^2
 \Bigl(1+\sup\frac{\E(\mathcal H',X',F';\xi',U)}{\Sigma'}\Bigr)
\end{equation}
where the supremum is over the family \eqref{eq:energy}, with
$\Sigma'=X'F'$, $\mathcal H',X',F'\ge1$, and
\begin{equation}\label{eq:transfer-scales}
 \mathcal H'\le\frac{\mathcal HL}{\Sigma F},\qquad
 \frac{\mathcal H'}{\Sigma'}\le
 \frac{\mathcal H}{\Sigma},\qquad
 \Sigma'\le L.
\end{equation}
For $I=[u,v]$, the tests satisfy
$U\in C_c^\infty(I')$, $I'=[u/16,4v]$, and
$\|U\|_{C^m(I')}\le1$; an empty supremum is zero.
The estimate holds for every compact $I\subset(0,\infty)$ and
$W\in C_c^\infty(I)$, with implied constant depending on
$m,C_0,\varepsilon,I,\nu,S$ and the fixed ray class group.
\end{proposition}

The proof is given in Section~\ref{sec:transfer-proof}, by combining
Lemmas~\ref{lem:first-transfer} and~\ref{lem:second-transfer}.

\subsection{Proof of Proposition~\ref{prop:canonical}}
\label{sec:canonical-proof}
\begin{proof}[Proof of Proposition~\ref{prop:canonical}]
Fix $\kappa>0$ and $C_0\ge1$. We prove by induction on $j\ge0$
that the proposition holds under the additional restriction
$\mathcal H\le D^{j\kappa}$, with the derivative order and implied
constant allowed to depend on $j$. More precisely, for every
$\varepsilon>0$ there is an integer $J=J(j,\kappa,C_0,\varepsilon)\ge1$
such that
\[
 \E(\mathcal H,X,F;\xi,W)
 \ll_{I,\nu,S,j,\kappa,C_0,\varepsilon}
 D^\varepsilon\Sigma\|W\|_{C^J(I)}^2
\]
for every compact interval $I=[u,v]\subset(0,\infty)$ and
$W\in C_c^\infty(I)$, under the hypotheses of the proposition.
The derivative order is independent of $I$; this allows us to apply
the induction hypothesis on the enlarged interval in the transfer estimate.
For the base case $j=0$, we have $\mathcal H\le1$.
In \eqref{eq:energy}, the support of $W$ restricts $n$ to the
$O_I(X)$ ideals with $\NK(n)\in[uX,vX]$.
The factors $a_\xi(n)$, $\chi_n(k)$, and $\chi_n(f)^4$ have absolute
value at most one, so the triangle inequality bounds each inner sum
by $O_I(X\|W\|_\infty)$.
There are $O(F)$ choices of $f$ and $O(\mathcal H)$ choices of $k$.
Consequently,
\[
 \E(\mathcal H,X,F;\xi,W)
 \ll_I\frac{F\mathcal H}{XF}
       \bigl(X\|W\|_\infty\bigr)^2
 =\mathcal H X\|W\|_\infty^2.
\]
Dividing by $\Sigma=XF$ and using $\mathcal H\le1$ and $F\ge1$
proves the assertion for $j=0$, with $J=1$.

Suppose the assertion holds for $j$, and fix $\varepsilon>0$.
Let $m$ be the derivative order supplied by the induction hypothesis
with exponent $\varepsilon/3$. For $\mathcal H\le D^{(j+1)\kappa}$,
apply Lemma~\ref{lem:cube-reduction} with exponent $\varepsilon/3$.
We must bound the mean squares in its supremum uniformly in $b$.
If this supremum is nonempty, then $\mathcal H^2>X$ and
$H_c^3=X^2/\mathcal H^2$.
For each such $b$, recall that $L_b=X/\NK(b)^3>1$.
Since $L_b\le X$ and $\Sigma\ge\max\{\mathcal H,L_bF\}$,
Proposition~\ref{prop:transfer} applies at column scale $L_b$, with derivative
order $m$, and exponent $\varepsilon/3$.
Its estimate also bounds $\E(\mathcal H,L_b,F;\xi,W)$ because
$\E(\mathcal H,L_b,F;\xi,W)\le\mathcal A(W)$.

Let $\mathcal H',X',F'$ be any parameters in the supremum in
\eqref{eq:transfer-summary}, with $\Sigma'=X'F'$ as in that proposition.
By \eqref{eq:transfer-scales} and $\NK(b)>H_c$,
\[
 \begin{aligned}
 \mathcal H'
 &\le\frac{\mathcal H L_b}{\Sigma F}
 =\frac{\mathcal H}{\NK(b)^3F^2}
 <\mathcal H\Bigl(\frac{\mathcal H}{\Sigma}\Bigr)^2
 \le D^{-2\kappa}\mathcal H\le D^{j\kappa},\\
 \frac{\mathcal H'}{\Sigma'}
 &\le\frac{\mathcal H}{\Sigma}\le D^{-\kappa},
 \qquad \Sigma'\le L_b\le\Sigma\le D^{C_0}.
 \end{aligned}
\]
Thus each of these mean squares is covered by the induction hypothesis.
Its weight $U$ is supported in $[u/16,4v]$ and has $C^m$ norm at most one.
Applying the induction hypothesis on this interval gives
\[
 \frac{\E(\mathcal H',X',F';\xi',U)}{\Sigma'}
 \ll_{I,\nu,S,j,\kappa,C_0,\varepsilon}D^{\varepsilon/3}.
\]
The enlarged interval is determined by $I$, so the implied constant
has only the permitted dependence on the support.
Substituting into \eqref{eq:transfer-summary} yields, uniformly in $b$,
\[
 \E(\mathcal H,L_b,F;\xi,W)
 \ll_{I,\nu,S,j,\kappa,C_0,\varepsilon}
 D^{2\varepsilon/3}\Sigma\|W\|_{C^{4m+12}(I)}^2.
\]
Substitution into \eqref{eq:cube-reduction} contributes the remaining
factor $D^{\varepsilon/3}$. Choose $J$ at least $4m+12$ and at least
the derivative order required by Lemma~\ref{lem:cube-reduction}.
We obtain
\[
 \E(\mathcal H,X,F;\xi,W)
 \ll_{I,\nu,S,j,\kappa,C_0,\varepsilon}
 D^\varepsilon\Sigma\|W\|_{C^J(I)}^2.
\]
The choice of $J$ depends only on $j,\kappa,C_0,\varepsilon$.
If the supremum in \eqref{eq:cube-reduction} is empty, the same bound
follows directly from that lemma. This completes the induction.

Finally, take $j=\lceil C_0/\kappa\rceil$.
The hypothesis $\mathcal H\le\Sigma\le D^{C_0}$ ensures
$\mathcal H\le D^{j\kappa}$, so the induction gives the proposition.
\end{proof}

\subsection{The original mean square and the exponent $11/12$}
\label{sec:completion}
\begin{proof}[Proof of Proposition~\ref{thm:ms}]
Fix $0<\vartheta\le1/10$ and put $H=D^{1+\vartheta}$.
The ranges in \eqref{eq:initial-scales} give
\begin{equation}\label{eq:initial-positive-gap}
 \mathcal H\le\frac{CD^{1-\vartheta}}{B^2},\qquad
 \Sigma=XF=\frac DB,\qquad
 \frac{\mathcal H}{\Sigma}\ll D^{-\vartheta}.
\end{equation}
For large $D$, Proposition~\ref{prop:canonical} applies with
$\kappa=\vartheta/2$ and $C_0=2$. If $X<1$, nonempty support forces
$X\gg_I1$, and counting gives $\E\ll_I\mathcal H\|W\|_\infty^2
\ll\Sigma\|W\|_\infty^2$; if $\mathcal H<1$, the sum is empty.
Thus \eqref{eq:auxiliary-target} holds, with derivative order depending
only on $\vartheta,\varepsilon$, and
Proposition~\ref{prop:poisson-reduction} proves \eqref{eq:ms}.
Bounded $D$ is again covered by counting.
\end{proof}


\section{Proof of the completed mean-square estimate}\label{sec:reflection}

We prove Proposition~\ref{prop:R}. We first express the completed sum
\eqref{eq:T} using cubic theta coefficients and state the transformation
formula. We then apply the quadratic large sieve and account for
repeated prime factors in $k$.

\subsection{Realization by the cubic theta function}
With the row $k$ and auxiliary index $f$ fixed, we express
$T(X;k,f)$ as a weighted sum of Fourier coefficients of the cubic theta function.
Its automorphy then expresses this sum in terms of coefficients at
other cusps, as in \cite[\S5 and Appendix~A]{DR}.
We use $\Psi_k$ from \eqref{eq:completed-twist}, suppressing its dependence on the fixed $f$ and $\xi$.

Write $(z,v)\in\mathbb C\times\mathbb R_{>0}$ for upper half-space
coordinates, with horizontal coordinate $z$ and height $v$.
We use Kubota's cubic theta function in the normalization of
\cite[\S5.1, (5.6)]{DR}:
\begin{equation}\label{eq:cubic-theta-definition}
 \theta(z,v)=\frac{3^{5/2}}2v^{2/3}
 +\sum_{\substack{\ell\in\lambda^{-3}\OO\\\ell\ne0}}
 \tau(\ell)vK_{1/3}(4\pi|\ell|v)
 \exp\!\bigl(2\pi\mathrm i(\ell z+\overline{\ell z})\bigr).
\end{equation}
Here $K_{1/3}$ is the modified Bessel function of the second kind,
and $\tau(\ell)$ is the coefficient sequence given explicitly in
\cite[(5.7)--(5.8)]{DR}, following Patterson's calculation
\cite[Theorem~8.1]{Pat77}.

We define $\Theta_k(z,v)$ by twisting the Fourier coefficients of $\overline\theta$. First define $\phi_k:\OO\to\mathbb C$ by
\[
 \phi_k(n)=
 \begin{cases}
  \chi_n(\lambda)^2\Psi_k(n),&n\equiv1\pmod3,\quad(n,S)=1,\\
  0,&\text{otherwise}.
 \end{cases}
\]
Then we define $\Theta_k(z,v)$ by multiplying the Fourier coefficient of
$\overline\theta$ at $\ell$, which is $\overline{\tau(-\ell)}$, by $\phi_k(\lambda^3\ell)$:

\begin{equation}\label{eq:twisted-theta-definition}
 \Theta_k(z,v)=
 \sum_{\substack{\ell\in\lambda^{-3}\OO\\\ell\ne0}}
 \overline{\tau(-\ell)}\phi_k(\lambda^3\ell)
 vK_{1/3}(4\pi|\ell|v)
 \exp\!\bigl(2\pi\mathrm i(\ell z+\overline{\ell z})\bigr).
\end{equation}
Recall that $c_\theta(nb^3)$ is defined in
\eqref{eq:intro-theta-coefficients}. 
Choose a nonzero $q\in\OO$ such that $\phi_k$ is periodic modulo $(q)$, and put
\[
 \widehat\phi_k(h)=\frac1{\NK(q)}\sum_{n\bmod q}\phi_k(n)e(-hn/q).
\]

\begin{lemma}\label{lem:theta-realization}
With these definitions,
\begin{equation}\label{eq:T-theta}
 T(X;k,f)=\frac1{3^{5/2}\sqrt X}
 \sum_{\substack{n,b\in\OO\\n,b\equiv1\ (3)\\n\ {\rm squarefree}\\(nb,S)=1}}
 c_\theta(nb^3)\phi_k(nb^3)\overline{\alpha(nb^3)}
 W\!\Bigl(\frac{\NK(nb^3)}X\Bigr).
\end{equation}
Moreover,
\begin{equation}\label{eq:theta-twist-translates}
 \Theta_k(z,v)=\sum_{h\in\OO/(q)}\widehat\phi_k(h)
       \overline{\theta(z+\lambda^2h/q,v)}.
\end{equation}
\end{lemma}
\begin{proof}
The coefficient formula \eqref{eq:intro-theta-coefficients} gives
\[
 c_\theta(nb^3)\phi_k(nb^3)
 =3^{5/2}|b|\gamma_2(n)\Psi_k(n)\Psi_k(b)^3.
\]
Substitution into \eqref{eq:T}, using $V_*(y)=\sqrt y\,W(y)$,
proves \eqref{eq:T-theta}.
Finite Fourier inversion gives
\[
 \phi_k(n)=\sum_{h\in\OO/(q)}\widehat\phi_k(h)e(nh/q),
 \qquad
 \sum_{h\in\OO/(q)}\widehat\phi_k(h)=\phi_k(0)=0.
\]
Multiplication of the $\ell$th Fourier mode by
$e(\lambda^3\ell h/q)$ translates $z$ to $z+\lambda^2h/q$.
The second identity cancels the constant term, proving
\eqref{eq:theta-twist-translates}.
\end{proof}
The resulting transformation formula is stated in the next subsection;
its automorphy calculation is given in Appendix~\ref{app:fixed-ray}.

\subsection{The theta transformation}
Recall that, for a primary prime $p\notin S$,
$\chi_p(x)=(x/p)_6$ is the sextic residue character on
$(\OO/(p))^\times$. For every integer $j$, write $\chi_p^j$ for
its $j$th power on this group, extended by zero on multiples of $p$;
in particular, $\chi_p^0(x)=\ind_{p\nmid x}$.
Sextic reciprocity \eqref{eq:recip} expresses the factors of
$\Psi_k$ at primes outside $S$ as such powers, with $0\le j\le5$.



For a primary prime $p\notin S$, $j\in\{0,\ldots,5\}$, and $x\in\OO$,
define the local factor
\begin{equation}\label{eq:theta-local-factors}
 B_{p,j}(x)=
 \begin{cases}
  \chi_p(x)^{-j-2},&j\ne0,4,\\
  \NK(p)^{-1/2}(-1+\NK(p)\ind_{p\mid x}),&j=4,\\
  \NK(p)^{-1/2}\chi_p(x)^{-2},&j=0.
 \end{cases}
\end{equation}
The key case is the quadratic character
\[
 B_{p,1}(x)=\chi_p(x)^3.
\]
For the smooth compactly supported weight $V_*$ in \eqref{eq:T},
put $\widehat V_*(s)=\int_0^\infty V_*(x)x^s\dd x/x$.
We write $\int_{(\sigma)}$ for integration upwards along the
vertical line with real part $\sigma$.
With $\Gamma$ denoting Euler's gamma function, the accompanying transform of the weight is
\begin{equation}\label{eq:theta-weight}
 V_*^\sharp(x)=\frac1{2\pi\mathrm i}\int_{(0)}
 \widehat V_*(-t)
 \frac{\Gamma(7/6+t)\Gamma(5/6+t)}
      {\Gamma(7/6-t)\Gamma(5/6-t)}
 \Bigl(\frac{(2\pi)^4x}{27}\Bigr)^{-t}\dd t,\qquad x>0.
\end{equation}
Here $\theta$ is Kubota's cubic theta function, normalized in
\eqref{eq:cubic-theta-definition}. The dual sums use the Fourier
coefficients of $\overline\theta$ at three cusps. Take the representatives
\begin{equation}\label{eq:theta-cusp-representatives}
 \gamma_0=I,\qquad
 \gamma_+=\Bigl(\begin{matrix}1&0\\\omega&1\end{matrix}\Bigr),\qquad
 \gamma_-=\Bigl(\begin{matrix}1&0\\\omega^2&1\end{matrix}\Bigr),
\end{equation}
acting on upper half-space. For $\sigma\in\{0,+,-\}$, define
$d_\sigma(\ell)$ by the Fourier expansion \cite[(5.9), (5.15)]{DR}
\begin{equation}\label{eq:cusp-coefficient-definition}
 \overline{\theta(\gamma_\sigma(z,v))}
 =\frac{3^{5/2}}2\ind_{\sigma=0}v^{2/3} +\sum_{0\ne\ell\in\lambda^{-4}\OO}
 d_\sigma(\ell)vK_{1/3}(4\pi|\ell|v)
 \exp\!\bigl(2\pi\mathrm i(\ell z+\overline{\ell z})\bigr).
\end{equation}
Thus $d_0(\ell)=\overline{\tau(-\ell)}$, with $\tau$ extended by zero
outside $\lambda^{-3}\OO$.
In particular, $c_\theta(nb^3)=d_0(\lambda^{-3}nb^3)$ for the indices in \eqref{eq:intro-theta-coefficients}; the outline uses $d_\theta(m)$ for $d_\sigma(\lambda^{-4}m)$ with one of these three choices of $\sigma$. The arithmetic formulas for all three
sequences are given by \eqref{eq:cusp-fourier-coefficients} and
\eqref{eq:conjugate-cusp-coefficients} in Appendix~\ref{app:fixed-ray}.

We state the formula for a general product of local twists.
Fix a ray class character whose conductor is supported on $S$,
and let $\Psi_0$ be its extension by zero at every prime in $S$.
Let $\mathcal P$ be a finite set of primes outside $S$,
each represented by its primary generator in $\OO$,
and choose integers $0\le j_p\le5$ for each $p\in\mathcal P$.
For primary $n\in\OO$, set
\[
 \Psi(n)=\Psi_0(n)\prod_{p\in\mathcal P}\chi_p^{j_p}(n).
\]


In the transformed sums, $\mathcal A$ will range over subsets satisfying
\[
 \{p\in\mathcal P:j_p\ne0\}\subseteq\mathcal A\subseteq\mathcal P.
\]
Thus only primes with $j_p=0$ may be omitted from $\mathcal A$.
We call the primes in $\mathcal A$ \emph{active} and those in $\mathcal P\setminus\mathcal A$ \emph{inactive}.
For such a subset and $c_0\in\OO\setminus\{0\}$, write
\[
 c=c_0\prod_{p\in\mathcal A}p.
\]


\begin{proposition}\label{lem:reflection}
The quantity $T(X;\Psi)$ defined in \eqref{eq:T} is a sum of $O_{\Psi_0,S}(2^{|\mathcal P|})$ terms of the form
\begin{equation}\label{eq:reflection}
 C\sum_{0\ne\ell\in\lambda^{-4}\OO}
 \frac{d(\ell)\alpha(\ell)}{\sqrt{\NK(\ell)}}\,
 \psi(\lambda^4\ell)
 \prod_{p\in\mathcal A}B_{p,j_p}(\lambda^4\ell)
 V_*^\sharp\!\Bigl(\frac{\NK(\ell) X}{\NK(c)^2}\Bigr).
\end{equation}


Here $\mathcal A$ and $c$ are as above, and $|C|\ll_{\Psi_0,S}1$.
The triples $(d,\psi,c_0)$ belong to a fixed finite family depending
only on $\Psi_0,S$, where $d\in\{d_0,d_+,d_-\}$,
$c_0\in\OO\setminus\{0\}$, and $\psi$ is a unit-modulus additive
character on $\OO$.
\end{proposition}




To average the transformed sums over $k_0$, we need to choose their
cusp coefficients and additive characters consistently as $k_0$ varies.
Fix a finite set $\mathcal P_{\rm fix}$ of primary primes outside $S$
and exponents $j_p\in\{0,\ldots,5\}$, and put
\[
 \Psi_{k_0}(n)=\Psi_0(n)
       \prod_{p\in\mathcal P_{\rm fix}}\chi_p^{j_p}(n)
       \prod_{p\mid k_0}\chi_p(n),
\]
where $k_0$ is primary and squarefree, with
$(k_0,S\prod_{p\in\mathcal P_{\rm fix}}p)=1$.
Let $\mathcal A_{\rm fix}$ range over the subsets satisfying
\[
 \{p\in\mathcal P_{\rm fix}:j_p\ne0\}
       \subseteq\mathcal A_{\rm fix}\subseteq\mathcal P_{\rm fix},
 \qquad
 \mathcal A=\mathcal A_{\rm fix}\cup\{p:p\mid k_0\}.
\]
Every prime dividing $k_0$ belongs to $\mathcal A$, since its exponent is $1$.


\begin{lemma}[Uniformity in the twist]\label{lem:reflection-uniformity}
For the family $\Psi_{k_0}$ above, the summands in
Proposition~\ref{lem:reflection} may be indexed by
$(h,\mathcal A_{\rm fix})$, with $h$ in a fixed finite set depending
only on $\Psi_0,S$. Zero scalar coefficients are permitted.
For each fixed index, the triple $(d,\psi,c_0)$ depends on $k_0$
only through its ray class modulo a fixed ideal supported on $S$
and depending only on $\Psi_0,S$.
\end{lemma}

To estimate the dual sums, we need bounds for the cusp coefficients
and the transformed weight.
\begin{lemma}\label{lem:theta-bounds}
For each $d\in\{d_0,d_+,d_-\}$, the coefficient $d(\ell)$
vanishes unless $\ell$ can be written as
\begin{equation}\label{eq:theta-support}
 \ell=u\lambda^mnb^3,
\end{equation}
where $u\in\OO^\times$, $m\in\mathbb Z$ with $m\ge-4$, and
$n,b\in\OO$ are primary with $n$ squarefree.
\begin{equation}\label{eq:theta-coefficient-bound}
 |d(\ell)|=|d(u\lambda^mnb^3)|\le27\cdot 3^{m/6}|b|.
\end{equation}

For $A>0$, integers $j\ge0$, and $V_*$ supported in a fixed compact
interval $I\subset(0,\infty)$, there is $J=J(A,j)$ such that
\begin{equation}\label{eq:theta-weight-decay}
 |(x\partial_x)^jV_*^\sharp(x)|
 \ll_{A,j,I}\|V_*\|_{C^J(I)}\min(x^{1/4},x^{-A}),\qquad x>0.
\end{equation}
\end{lemma}

Proposition~\ref{lem:reflection}
and Lemmas~\ref{lem:reflection-uniformity} and~\ref{lem:theta-bounds}
are proved in Appendix~\ref{app:fixed-ray}.

\subsection{The quadratic mean square on the dual side}
We record the quadratic large-sieve estimate needed below.

\begin{lemma}\label{lem:quadratic}
Let $\mathcal H,U\ge1$, and let $\beta(n)$ be complex coefficients
on squarefree primary $n$ with $\NK(n)\asymp U$.
For every $\varepsilon>0$,
\begin{equation}\label{eq:Q}
 \sum_{\substack{k\equiv1\ (3),\ (k,S)=1\\\NK(k)\le\mathcal H}}^*
 \Bigl|\sum_{\substack{n\equiv1\ (3),\ n\ {\rm squarefree}\\
                       \NK(n)\asymp U}}
       \beta(n)\chi_k(n)^3\Bigr|^2
 \ll_{S,\varepsilon}(\mathcal HU)^\varepsilon(\mathcal H+U)
       \sum_n|\beta(n)|^2.
\end{equation}
The star restricts $k$ to squarefree primary elements.
\end{lemma}
\begin{proof}
This is Goldmakher and Louvel's quadratic large sieve \cite[Theorem~1.1]{GL}, after
fixing the product of the prime factors of $n$ lying in $S$, and finitely many ray classes.
For completeness, let $e_k\in\{0,1\}$ according as
$\NK(k)\equiv1,3\pmod4$, and let $\kappa_\lambda$ be the nontrivial
character modulo $\lambda$. The character
$x\mapsto(x/k)_2\kappa_\lambda(x)^{e_k}$ is trivial on
units and has primitive conductor $k\lambda^{e_k}$.
When $e_k=0$, the factor $\kappa_\lambda(x)^{e_k}$ is omitted.
Classes modulo $24\OO$ fix the supplementary characters and
reciprocity factors; for coprime $k_1,k_2$ in the same class, the
product character has conductor $k_1k_2$.
These are the hypotheses in \cite[Definition~1 and \S2]{GL}.
\end{proof}

\subsection{Squarefree rows and the completed bound}
Write $N(a)=\NK(a)$. In \eqref{eq:completed-twist}, allow any
$g\in\OO\setminus\{0\}$ in place of $f$, so
$\Psi_k(n)=\xi(n)\chi_n(k)\chi_n(g)^4$.
The zero extension at $S$ is retained.
Choose prime-ideal generators, primary away from $3$, and extend
multiplicatively to all ideals. We first bound squarefree rows.

\begin{lemma}\label{lem:squarefree-completed}
For every $\varepsilon>0$ and $C_0\ge1$ there is
$J=J(\varepsilon,C_0)\ge1$ such that
\begin{equation}\label{eq:squarefree-completed}
 \sum_{\substack{0<N(s)\le\mathcal H\\s\ {\rm squarefree}}}
 |T(X;u_0s,g)|^2
 \ll_{I,\xi,S,\varepsilon,C_0}D^\varepsilon\|W\|_{C^J(I)}^2
 \Bigl(\mathcal H+\frac{\mathcal H^2N(g)}X\Bigr)
\end{equation}
for $1\le\mathcal H,X,N(g)\le D^{C_0}$, $u_0\in\OO^\times$,
every compact interval $I\subset(0,\infty)$, and $W\in C_c^\infty(I)$.
The sum uses the chosen generators of squarefree ideals, including
those meeting $S$.
\end{lemma}
\begin{proof}
By homogeneity assume $\|W\|_{C^J(I)}\le1$, with $J$ chosen below.
Write
\[
 s=t k_0,\qquad
 t=\prod_{\substack{p\mid s\\p\mid g\ \text{or}\ p\in S}}p,\qquad
 N(k_0)\le\mathcal H_0:=\frac{\mathcal H}{N(t)}.
\]
Fix $t$; then $k_0$ is squarefree and primary, with
$(k_0,g)=1$ and $(k_0,S)=1$. Discard empty ranges, so
$\mathcal H_0\ge1$, and retain these restrictions below.
For a coefficient function $A(n,b)$, a positive scale $Y$, and an integer $m\ge-4$, put
\begin{equation}\label{eq:prepared-theta-sum}
 \mathscr S_m[A,Y](k_0):=
 \sum_{\substack{n,b\equiv1\ (3)\\n\ {\rm squarefree}}}
 \frac{A(n,b)\chi_{k_0}(nb)^3}{3^{m/3}\sqrt{N(n)}\,N(b)}
 V_*^\sharp\!\Bigl(\frac{3^mN(n)N(b)^3\mathcal H_0^2}
 {YN(k_0)^2}\Bigr).
\end{equation}

We shall obtain, after partitioning $k_0$ into fixed ray classes,
\[
 T(X;u_0tk_0,g)
 =\sum_{\iota\in\mathcal I}\sum_{m\ge-4}
 c_{\iota,m}(k_0)\mathscr S_m[A_{\iota,m},Y_\iota](k_0),
\]
where $\mathcal I$ is independent of $k_0$, $|\mathcal I|\precbd1$,
$|c_{\iota,m}(k_0)|\ll1$, and $A_{\iota,m}$ is independent of $k_0$.
For some $a_\iota,Y_\iota>0$, the coefficients and lengths satisfy
\begin{equation}\label{eq:auxiliary-conductor-bound}
 |A_{\iota,m}(n,b)|\ll a_\iota,\qquad
 a_\iota^2\le1,\qquad
 a_\iota^2Y_\iota\ll
 \frac{\mathcal H_0^2}{X}N(t)^2N(g).
\end{equation}
All column restrictions are included by extending $A_{\iota,m}$ by zero.
Put $k=u_0tk_0$. For $\mathcal P=\{p\notin S:p\mid kg\}$, sextic
reciprocity \eqref{eq:recip} gives
\begin{equation}\label{eq:theta-row-twist}
 \Psi_k(n)=\Psi_0(n)\prod_{p\in\mathcal P}\chi_p^{j_p}(n),
 \qquad j_p\equiv v_p(k)+4v_p(g)\pmod6,
 \qquad 0\le j_p\le5.
\end{equation}
Even when $j_p=0$, the factor $\chi_p^0(n)=\ind_{p\nmid n}$
retains the zero extension at $p\mid kg$. The fixed factor $\Psi_0$
contains $\xi$, the factors at $S$, the unit factors,
and $n\mapsto\mathcal R(n,\prod_{p\notin S}p^{v_p(k)})$.
The fourth power at $g$ contributes no reciprocity sign, and the
factors at $S$ depend only on exponents modulo six. Thus $\Psi_0$
ranges over a fixed finite family. In the notation of
Lemma~\ref{lem:reflection-uniformity}, take
$\mathcal P_{\rm fix}=\{p\notin S:p\mid tg\}$, so that
$\mathcal P=\mathcal P_{\rm fix}\sqcup\{p:p\mid k_0\}$;
the exponents on $\mathcal P_{\rm fix}$ are fixed with $t,g$.
Partitioning $k_0$ into fixed
ray classes fixes $\Psi_0$ and, by
Lemma~\ref{lem:reflection-uniformity},
the data $d,\psi,c_0$ in each transformed term.

Fix one transformed term, with active primes $\mathcal A_{\rm fix}$
away from $k_0$, and put $c_*=c_0\prod_{p\in\mathcal A_{\rm fix}}p$.
Thus $c=c_*k_0$ in \eqref{eq:reflection}.
By \eqref{eq:theta-support}, write $\ell=u\lambda^mnb^3$,
with $u,m,n,b$ as there.
Every prime dividing $k_0$ has $j_p=1$, and hence
\begin{equation}\label{eq:residual-quadratic-factor}
 B_{p,1}(u\lambda^{m+4}nb^3)
 =\chi_p(u\lambda^{m+4})^3\chi_p(nb)^3.
\end{equation}
Indeed, the transformed exponent is $-1-2\equiv3\pmod6$ and
$\chi_p(b)^9=\chi_p(b)^3$, also when $p\mid b$ by zero extension.
The coefficient bound and rapid decay in Lemma~\ref{lem:theta-bounds}
give absolute convergence of the transformed series, so we may regroup
its terms below. Set
\[
 A_{u,m}(n,b)=
 \frac{d(u\lambda^mnb^3)\alpha(u\lambda^mnb^3)
       \psi(u\lambda^{m+4}nb^3)}{3^{m/6}\sqrt{N(b)}},
 \qquad |A_{u,m}(n,b)|\le27.
\]
The transformed term is therefore
\[
 \begin{aligned}
 &C(k_0)\sum_{u\in\OO^\times}\sum_{m\ge-4}
 \chi_{k_0}(u\lambda^{m+4})^3
 \sum_{\substack{n,b\equiv1\ (3)\\n\ {\rm squarefree}}}
 \frac{A_{u,m}(n,b)\chi_{k_0}(nb)^3}
      {3^{m/3}\sqrt{N(n)}\,N(b)}\\
 &\qquad\times\prod_{p\in\mathcal A_{\rm fix}}
 B_{p,j_p}(u\lambda^{m+4}nb^3)
 V_*^\sharp\!\Bigl(\frac{3^mN(n)N(b)^3X}
 {N(c_*)^2N(k_0)^2}\Bigr),\qquad |C(k_0)|\ll1.
 \end{aligned}
\]
The representation is unique: away from $\lambda$, the prime
exponents of $\ell$ are $3v_p(b)$ or $1+3v_p(b)$. Apart from the
quadratic character and weight, the $k_0$-dependence is a bounded scalar.

Fix $u,m$ and write $q=N(p)$ for an active prime $p\nmid k_0$.
For $j_p=4$, the local identity is
\[
 B_{p,4}(u\lambda^{m+4}nb^3)
 =-q^{-1/2}+q^{1/2}\ind_{p\mid n}
              +q^{1/2}\ind_{p\nmid n,\ p\mid b}.
\]
To display the reindexing in \eqref{eq:prepared-theta-sum}, let
$A^{(p)}(n,b)$ include all the other local factors and set
\[
 A^{(p,n)}(n,b)=\ind_{p\nmid n}A^{(p)}(pn,b),\qquad
 A^{(p,b)}(n,b)=\ind_{p\nmid n}A^{(p)}(n,pb).
\]
Then the changes $n=pn'$ and $b=pb'$ give the exact identity
\[
 \begin{aligned}
 &\mathscr S_m[A^{(p)}(n,b)B_{p,4}(u\lambda^{m+4}nb^3),q^2Y](k_0)\\
 &\quad=-q^{-1/2}\mathscr S_m[A^{(p)},q^2Y](k_0)
 +\chi_{k_0}(p)^3\mathscr S_m[A^{(p,n)},qY](k_0)\\
 &\qquad\quad+q^{-1/2}\chi_{k_0}(p)^3
              \mathscr S_m[A^{(p,b)},Y/q](k_0).
 \end{aligned}
\]
In the second term $p\nmid n'$ preserves squarefreeness; in the
third, $p\nmid n$ is retained and $b'$ is unrestricted at $p$.
Both extracted phases have modulus one. The factors $q^{1/2}$
in the local identity cancel against $\sqrt{N(pn')}$ or leave
$q^{-1/2}$ after division by $N(pb')$.

Let $a$ track a bound $|A(n,b)|\le27a$ for the coefficient in
\eqref{eq:prepared-theta-sum}, and let $Y$ be its scale.
Before inserting the active primes, these are $a=1$ and
$Y=N(c_0)^2\mathcal H_0^2/X$.
Each active prime contributes $q^2$ to the squared conductor norm.
Including this factor, the updates are
\[
 (a^2,Y)\longmapsto
 \begin{cases}
 (a^2,q^2Y),&j_p\notin\{0,4\},\\
 (a^2/q,q^2Y),&j_p=0,\\
 (a^2/q,q^2Y),\ (a^2,qY),\ (a^2/q,Y/q),&j_p=4.
 \end{cases}
\]
Thus $a^2$ never increases. If $p\mid t$ and $p\notin S$, then
$j_p$ is odd, so $a^2Y$ costs at most $N(p)^2$.
If $p\mid g$ and $p\nmid t$, the cost is
$N(p)$ for $j_p=0,4$ and $N(p)^2$ for $j_p=2$; the latter case
requires $v_p(g)\ge2$. Hence
\[
 a^2Y\ll\frac{\mathcal H_0^2}{X}
 \prod_{\substack{p\mid t\\p\notin S}}N(p)^2
 \prod_{\substack{p\mid g\\p\nmid t,\ p\notin S}}N(p)^{v_p(g)}
 \le\frac{\mathcal H_0^2N(t)^2N(g)}X.
\]
Reindexing at distinct primes preserves the form and zero extensions.
By %
Lemma~\ref{lem:reflection-uniformity}, the transformed terms, units,
and at most three choices per prime $p\in\mathcal A_{\rm fix}$ with $j_p=4$ form an index set
of divisor-bounded size in $tg$. Here and below, divisor-bounded in $a$ means bounded by $C\tau_{\mathrm{div}}(a)^A$ for fixed constants $C,A$; in particular, this is $\ll_\varepsilon\NK(a)^\varepsilon$ for every $\varepsilon>0$.
For one index $\iota$, abbreviate
$a=a_\iota$, $Y=Y_\iota$, and $A_m=A_{\iota,m}$.
The local updates give $Y\ll D^{C_1}$ for some fixed $C_1=C_1(C_0)$.
Split $b$ into $B\le N(b)<2B$, and $n$ by a smooth dyadic
partition $V(N(n)/U)$, with $U,B\ge1$ and uniformly bounded cutoffs.
For this part of \eqref{eq:prepared-theta-sum}, write
\[
 \begin{aligned}
 \mathscr S_{m;U,B}(k_0)
 &=3^{-m/3}\sum_{B\le N(b)<2B}\frac{\chi_{k_0}(b)^3}{N(b)}F_b(k_0),\\
 F_b(k_0)
 &=\sum_{n\ {\rm squarefree}}
 \frac{A_m(n,b)\chi_{k_0}(n)^3}{\sqrt{N(n)}}V(N(n)/U)
 V_*^\sharp\!\Bigl(\frac{3^mN(n)N(b)^3\mathcal H_0^2}
                         {YN(k_0)^2}\Bigr).
 \end{aligned}
\]
Here $n,b$ remain primary. Put $z_*=3^mUB^3/Y$ and fix a decay
exponent $A>0$. With $m,U,B,Y$ fixed, Lemma~\ref{lem:smooth},
applied only in $N(n)/U$, gives the following representation in a
real Mellin variable $s$:
\begin{equation}\label{eq:theta-separated-columns}
 \begin{aligned}
 F_b(k_0)&=\int_{\mathbb R}c_{b,k_0}(s)G_{b,s}(k_0)\,ds,\\
 G_{b,s}(k_0)&=\sum_{\substack{n\ {\rm squarefree}\\N(n)\asymp U}}
 \frac{A_m(n,b)\chi_{k_0}(n)^3}{\sqrt{N(n)}}
             (N(n)/U)^{\mathrm i s},\\
 |c_{b,k_0}(s)|&\ll_{I,A}(1+z_*)^{-A}(1+|s|)^{-2}.
 \end{aligned}
\end{equation}
Indeed, the scale $R$ in \eqref{eq:smooth-pointwise} is
$3^mUN(b)^3\mathcal H_0^2/(YN(k_0)^2)\ge z_*$.
Lemma~\ref{lem:theta-bounds} supplies the required derivative
bounds, independently of $b,k_0,U,B,m$.
Since $\sum_{N(n)\asymp U}N(n)^{-1}\ll1$,
Lemma~\ref{lem:quadratic} gives, for any $\sigma>0$,
\[
 \sum_{N(k_0)\le\mathcal H_0}^*|G_{b,s}(k_0)|^2
 \ll_{S,\sigma}a^2(\mathcal H_0U)^\sigma(\mathcal H_0+U).
\]
Apply weighted Cauchy--Schwarz to the Mellin integral, as recorded in
\eqref{eq:integral-mean-square}, with the common majorant in
\eqref{eq:theta-separated-columns}, then weighted Cauchy--Schwarz in $b$, using
$\sum_{B\le N(b)<2B}N(b)^{-1}\ll1$:
\begin{equation}\label{eq:prepared-mean-square}
 \begin{aligned}
 \sum_{N(k_0)\le\mathcal H_0}^*|\mathscr S_{m;U,B}(k_0)|^2
 &\le3^{-2m/3}\Bigl(\sum_{B\le N(b)<2B}\frac1{N(b)}\Bigr)
       \sum_{B\le N(b)<2B}\frac1{N(b)}
       \sum_{N(k_0)\le\mathcal H_0}^*|F_b(k_0)|^2\\
 &\ll_{I,S,A,\sigma}a^2 3^{-2m/3}
       (\mathcal H_0U)^\sigma(\mathcal H_0+U)(1+z_*)^{-2A}.
 \end{aligned}
\end{equation}
Since $m\ge-4$, we have $U\le81Yz_*$ and hence
$\mathcal H_0U\ll D^{C_0+C_1}(1+z_*)$.
Choose $0<\sigma\le1$ small in terms of $\varepsilon_1>0,C_0$ and take
$A=3$. Taking square roots in \eqref{eq:prepared-mean-square} gives
\[
 \Bigl(\sum_{N(k_0)\le\mathcal H_0}^*|\mathscr S_{m;U,B}(k_0)|^2\Bigr)^{1/2}
 \ll aD^{\varepsilon_1}3^{-m/3}
       (\sqrt{\mathcal H_0}+\sqrt U)(1+3^mUB^3/Y)^{-2}.
\]
The infinite dyadic sums converge: for dyadic $U,B\ge1$ and every $Z>0$,
\[
 \sum_{U,B\ {\rm dyadic}}(1+UB^3/Z)^{-2}\ll\log^2(2+Z),
 \qquad
 \sum_{U,B\ {\rm dyadic}}\sqrt U(1+UB^3/Z)^{-2}\ll\sqrt Z.
\]
These estimates make the sum of $\ell^2(k_0)$ norms finite.
The triangle inequality and the geometric sum over
$m\ge-4$ therefore give
\[
 \Bigl(\sum_{N(k_0)\le\mathcal H_0}^*
   \Bigl|\sum_{m\ge-4}c_{\iota,m}(k_0)
       \mathscr S_m[A_m,Y](k_0)\Bigr|^2\Bigr)^{1/2}
 \ll aD^{\varepsilon_1}
       \bigl(\sqrt{\mathcal H_0}\log^2(2+Y)+\sqrt Y\bigr).
\]
Combine the divisor-bounded choices of $\iota$ and use
\eqref{eq:auxiliary-conductor-bound} to obtain
\[
 \sum_{N(k_0)\le\mathcal H_0}^*|T(X;u_0tk_0,g)|^2
 \precbd\max_\iota a_\iota^2(\mathcal H_0+Y_\iota)
 \ll\mathcal H_0+\frac{\mathcal H_0^2N(t)^2N(g)}X
 \le\mathcal H+\frac{\mathcal H^2N(g)}X.
\]
Sum the fixed ray classes and the divisor-bounded choices
$t\mid\rad(g\prod_{\mathfrak p\in S}\mathfrak p)$, choosing
$\varepsilon_1$ and the other small-power losses in terms of
$\varepsilon$. The weight estimates above require a fixed number
$J=J(\varepsilon,C_0)$ of derivatives. Homogeneity restores
$\|W\|_{C^J(I)}^2$ and proves \eqref{eq:squarefree-completed}.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:R}]
Write uniquely $k=u_0sv^2$, with $s$ squarefree and $s,v$ among
the chosen ideal generators. No condition $(s,v)=1$ is imposed.
Since $8\equiv2\pmod6$, including the zero extensions,
\[
 \chi_n(u_0sv^2)\chi_n(f)^4
 =\chi_n(u_0s)\chi_n(fv^2)^4,\qquad
 T(X;u_0sv^2,f)=T(X;u_0s,fv^2).
\]
Apply Lemma~\ref{lem:squarefree-completed} with row bound
$\mathcal H/N(v)^2$ and auxiliary twist $g=fv^2$.
Here $N(g)\le\mathcal H N(f)\le D^{2C_0}$, so use that lemma
with $2C_0$. Summing its bounds gives
\[
 \begin{aligned}
 \sum_{0<N(k)\le\mathcal H}|T(X;k,f)|^2
 &=\sum_{u_0\in\OO^\times}\sum_{N(v)^2\le\mathcal H}
   \sum_{\substack{N(s)\le\mathcal H/N(v)^2\\s\ {\rm squarefree}}}
   |T(X;u_0s,fv^2)|^2\\
 &\ll D^\varepsilon\|W\|_{C^J(I)}^2
   \Bigl(\mathcal H+\frac{\mathcal H^2N(f)}X\Bigr)
   \sum_v\frac1{N(v)^2}.
 \end{aligned}
\]
The last sum is $\zeta_K(2)<\infty$, where $\zeta_K(s)=L_K(s,\mathbf1)$ is the Dedekind zeta function of $K$. This proves \eqref{eq:R}.
\end{proof}


\section{Proof of the transfer proposition}\label{sec:transfer-proof}

We prove Proposition~\ref{prop:transfer} for
$\mathcal A(W)$ defined in \eqref{eq:smoothed-energy}.
Throughout this section, $\mathcal H,L,F,\Sigma,\xi$ satisfy the
hypotheses of that proposition. We retain the fixed weight $\Phi$
and the support bound $C_\Phi$ chosen before \eqref{eq:smoothed-energy}.
The enlargement by $\Sigma/(LF)\ge1$ in the intermediate mean square
makes the second Poisson summation return to \eqref{eq:energy} with
row range at most $\mathcal HL/(\Sigma F)$.
Lemma~\ref{lem:smooth-mean-square} separates the weights, and
Lemma~\ref{lem:remove-exclusions} removes the remaining exclusion.

\subsection{First application of Poisson summation}
Write $N=\NK$, $I=[u,v]$, and $I_*=[u/2,2v]$.
All ideal indices below are prime to $S$ and represented by their
primary generators; a star additionally requires squarefreeness.
The variables $k,h,y$ range over $\OO$.
For squarefree $C,t$ with $(C,t)=1$, and for $d\mid C$, put
\[
 \ell=\frac{L}{N(C)N(t)},\qquad
 w_{C,d}=\frac{\mathcal H N(C)}{N(d)L^2F},\qquad
 Y_{C,d}=c_I\frac{\Sigma LFN(d)}{\mathcal H N(C)^2},
\]
where $c_I\ge\max(1,4C_\Phi v^2)$ is fixed. Retain only
$\ell\ge1/(2v)$. For $U\in C_c^\infty(I_*)$ and a character
$\xi_1$ of the fixed ray class group, define
\begin{equation}\label{eq:P}
 \begin{aligned}
 p_y(n)&=\mu(n)\xi_1(n)\ind_{(n,t)=1}
       \overline{\chi_n(y)}\chi_n(C)^4\chi_n(d),\\
 P_{C,d,t}(y;U)&=\sum_n^*p_y(n)U(N(n)/\ell).
 \end{aligned}
\end{equation}
The notation $p_y$ suppresses its dependence on $C,d,t,\xi_1$.
The nonnegative form required in the second Poisson calculation is
\begin{equation}\label{eq:first-transfer-forms}
 \mathcal Q_{\xi_1}(U)=
 \sum_{\substack{C,t\text{ squarefree}\\(Ct,S)=1, (C,t)=1\\
                  N(C)N(t)\le2vL}}
 \sum_{d\mid C}w_{C,d}
\sum_y\Phi(N(y)/Y_{C,d})|P_{C,d,t}(y;U)|^2.
\end{equation}
The outer domain has no additional restriction coming from the support of $\widehat\Phi$. Here $Y_{C,d}>0$ may be smaller than $1$; the outer
sums are finite, and the $y$-sum converges absolutely.

\begin{lemma}\label{lem:first-transfer}
Under the hypotheses of Proposition~\ref{prop:transfer}, fix an integer
$j\ge0$ and $\varepsilon_0>0$. If $M\ge0$ satisfies
$\mathcal Q_{\xi_1}(U)\le M\|U\|_{C^j(I_*)}^2$
for every $U\in C_c^\infty(I_*)$ and every $\xi_1$, then
\[
 \mathcal A(W)\ll D^{\varepsilon_0}(\Sigma+M)
                    \|W\|_{C^{2j+4}(I)}^2.
\]
\end{lemma}
\begin{proof}
Expand the square defining $\mathcal A(W)$, and write
$n_i=Cu_i$, where $C=(n_1,n_2)$ and
$(u_1,u_2)=(u_1u_2,C)=1$. The row character is
$\chi_{u_1}\overline{\chi_{u_2}}$, primitive modulo $u_1u_2$,
with the extra restriction $(k,C)=1$.
Lemma~\ref{lem:poisson} introduces $d\mid C$ and a frequency $h$.
Let $Z$ denote the zero-frequency contribution. It requires $u_1=u_2=1$, so
\[
 |Z|\ll\frac{\mathcal H}{LF}
       \sum_{F\le N(f)<2F}^*\sum_C^*|W(N(C)/L)|^2
 \ll_I\mathcal H\|W\|_\infty^2.
\]
For the other frequencies, the Chinese remainder theorem,
\eqref{eq:convert1}, and \eqref{eq:quotient} give the paired identity
\begin{equation}\label{eq:paired-first-gauss}
 a_\xi(u_1)\overline{a_\xi(u_2)}
 \gamma(\chi_{u_1}\overline{\chi_{u_2}})
 =\mu(u_1)\mu(u_2)(\xi G)(u_1u_2^{-1}).
\end{equation}
Expand $(\xi G)(z)=\sum_{\xi_1}\ell_{\xi_1}\xi_1(z)$ on the
fixed ray class group. Insert
$\ind_{(u_1,u_2)=1}=\sum_{t\mid u_1,\ t\mid u_2}\mu(t)$ and put
$u_i=tx_i$. The inverse is $x_i=u_i/t$; squarefreeness imposes
$(x_i,t)=1$, but no condition $(x_1,x_2)=1$ remains.
The factors at $C,t$ cancel against their conjugates, leaving
$\mu(d)\mu(t)$ and the restrictions $(f,Ct)=(h,t)=1$.
Thus
\[
 \begin{aligned}
 &\mathcal A(W)-Z
 =\sum_{\xi_1}\ell_{\xi_1}
 \sum_{\substack{C,t\text{ squarefree}\\(Ct,S)=1, (C,t)=1\\
                  N(C)N(t)\le2vL}}
 \sum_{d\mid C}\mu(d)\mu(t)w_{C,d}\\
 &\quad\times\sum_{\substack{F\le N(f)<2F\\(f,Ct)=1}}^*
 \sum_{\substack{h\ne0, (h,t)=1\\
         N(h)\le C_\Phi v^2L^2N(d)/(\mathcal H N(C)^2)}} \sum_{x_1,x_2}^*
 p_{hf^2}(x_1)\overline{p_{hf^2}(x_2)}
 \mathscr K_h(N(x_1)/\ell,N(x_2)/\ell),
 \end{aligned}
\]
where, for each integer $j_0\ge0$,
\[
 \begin{aligned}
 \mathscr K_h(z_1,z_2)
 &=(z_1z_2)^{-1/2}W(z_1)\overline{W(z_2)}
 \widehat\Phi\!\Bigl(
  \frac{\mathcal H N(h)N(C)^2}{L^2N(d)z_1z_2}\Bigr),\\
 \sup_{C,d,h}\|\mathscr K_h\|_{C^{j_0}(I^2)}
 &\ll_{I,\Phi,j_0}\|W\|_{C^{j_0}(I)}^2.
 \end{aligned}
\]
Here $\chi_{x_i}(C)^4$ supplies $(x_i,C)=1$; the other original
zeros are supplied by $p_{hf^2}(x_i)$. The displayed frequency
range follows from the support of $\widehat\Phi$.

For fixed $C,d,t$, the map $(f,h)\mapsto y=hf^2$ has
divisor-bounded multiplicity in $y$. Since $\Sigma/(LF)\ge1$, its
nonzero image satisfies
$N(y)\le4C_\Phi v^2L^2F^2N(d)/(\mathcal H N(C)^2)\le Y_{C,d}$.
Since $\Phi\ge1$ on $[0,1]$ and is nonnegative,
\[
 \sum_{C,t}\sum_{d\mid C}w_{C,d}\sum_{f,h}|P_{C,d,t}(hf^2;U)|^2
 \precbd\mathcal Q_{\xi_1}(U)
 \le M\|U\|_{C^j(I_*)}^2,
\]
with the preceding ranges on the left. Apply
Lemma~\ref{lem:smooth-mean-square} to the full displayed expression
for $\mathcal A(W)-Z$. The kernel bound gives
\[
 |\mathcal A(W)-Z|
 \ll D^{\varepsilon_0/2}M\|W\|_{C^{2j+4}(I)}^2.
\]
Together with $\mathcal H\le\Sigma$, this proves the result after
allocating the divisor losses within $\varepsilon_0$.
\end{proof}

\subsection{Second application of Poisson summation}

Let $C,t$ be coprime squarefree primary elements prime to $S$, and let
$d\mid C$. Fix a character $\xi_1$ of the ray class group.
For arbitrary scales $\ell,Y>0$ and
$U\in C_c^\infty(I_*)$, with
$I_*=[a,b]\subset(0,\infty)$, put
\begin{equation}\label{eq:common-mobius-sum}
 \begin{aligned}
 P(y)&=\sum_{(n,S)=1}^*\mu(n)\xi_1(n)\ind_{(n,t)=1}
 \overline{\chi_n(y)}\chi_n(C)^4\chi_n(d)U(N(n)/\ell),\\
 \mathcal M&=\sum_{y\in\OO}\Phi(N(y)/Y)|P(y)|^2.
 \end{aligned}
\end{equation}
Here $P(y)$ abbreviates $P_{C,d,t}(y;U)$ from \eqref{eq:P}, with the scale $\ell$ now arbitrary.

The second Poisson summation turns the M\"obius coefficients back into
cubic Gauss-sum coefficients. To state the identity, put
$U_0(x)=x^{-1/2}U(x)$ and expand
\[
 \xi_1(z)G(z^{-1})=\sum_{\xi'}c_{\xi'}\xi'(z)
\]
on the fixed ray class group.

On the transformed side, $g$ is the common divisor of the original
columns, $e\mid g$ comes from the row exclusion, and $w$ removes the
remaining coprimality condition. The index $h$ is the nonzero Fourier
frequency. Let $g,w$ range over squarefree primary elements prime to $S$,
$e$ over divisors of $g$, and $h$ over $\OO\setminus\{0\}$, subject to
\[
 (g,Ct)=(w,gCth)=1,\qquad
 N(gw)\le b\ell,\qquad
 N(h)\le\frac{C_\Phi b^2\ell^2N(e)}{YN(g)^2}.
\]
These conditions ensure that $tg/e$ and $Cew$ are coprime and squarefree.
For each such choice and each character $\xi'$, the new column scale,
coefficient, and coupled smooth weight are
\begin{equation}\label{eq:common-poisson-output}
 \begin{gathered}
 X'=\frac{\ell}{N(gw)},\\
 a^\sharp(n)=a_{\xi'}(n)\ind_{(n,tg/e)=1}
                   \chi_n(deh)\chi_n(Cew)^4,\\
 \mathscr K(x_1,x_2)=U_0(x_1)\overline{U_0(x_2)}
               \widehat\Phi\!\Bigl(\frac{YN(h)N(g)^2}{N(e)\ell^2x_1x_2}\Bigr).
 \end{gathered}
\end{equation}
Here $\mathscr K$ depends on $U,Y,\ell,g,e,h$. A star on a sum over
$n_1,n_2$ restricts each index separately to squarefree primary elements.
Define also
\[
 Z=Y\widehat\Phi(0)
 \sum_{\substack{(g,Ct)=1\\(g,S)=1}}^*
 |U(N(g)/\ell)|^2\prod_{p\mid g}\Bigl(1-\frac1{N(p)}\Bigr).
\]


\begin{lemma}[Second Poisson identity]\label{lem:arithmetic-poisson}
With the notation and index ranges above,
\begin{equation}\label{eq:arithmetic-poisson-identity}
 \mathcal M=Z+
 \sum_{g,e,w,h,\xi'}\frac{Y\mu(e)\mu(w)c_{\xi'}N(g)}{N(e)\ell} \sum_{(n_1n_2,S)=1}^*a^\sharp(n_1)\overline{a^\sharp(n_2)}
       \mathscr K\!\Bigl(\frac{N(n_1)}{X'},\frac{N(n_2)}{X'}\Bigr).
\end{equation}
The zero-frequency term satisfies
\[
 |Z|\ll_{I_*,\Phi}Y\ell\|U\|_\infty^2.
\]
\end{lemma}

\begin{proof}
Expand the square, put $g=(n_1,n_2)$ and $n_i=gz_i$. The nonzero terms satisfy
\[
 (z_1,z_2)=(z_1z_2,gCt)=(g,Ct)=1,\qquad
 \overline{\chi_{n_1}(y)}\chi_{n_2}(y)
 =\ind_{(y,g)=1}\overline{\chi_{z_1}(y)}\chi_{z_2}(y).
\]
Apply Lemma~\ref{lem:poisson} to this last character, with exclusion
$g$ and divisor $e\mid g$. Its zero frequency occurs exactly when
$z_1=z_2=1$ and gives the displayed $Z$; counting $g$ gives its bound.
For the nonzero frequencies, \eqref{eq:convert2}--\eqref{eq:quotient}
and the Chinese remainder theorem give
\[
 \mu(z_1)\mu(z_2)\xi_1(z_1)\overline{\xi_1(z_2)}
 \gamma(\overline{\chi_{z_1}}\chi_{z_2}) =\sum_{\xi'}c_{\xi'}a_{\xi'}(z_1)\overline{a_{\xi'}(z_2)}.
\]
Use also
\[
 \begin{aligned}
 \chi_z(dh)\overline{\chi_z(e)}\chi_z(C)^4
 &=\chi_z(deh)\chi_z(Ce)^4,\\
 \frac{U(N(gz_1)/\ell)\overline{U(N(gz_2)/\ell)}}
      {\sqrt{N(z_1)N(z_2)}}
 &=\frac{N(g)}\ell U_0(N(gz_1)/\ell)\overline{U_0(N(gz_2)/\ell)}.
 \end{aligned}
\]
Thus the full nonzero contribution is
\[
 \begin{aligned}
 \mathcal M-Z={}&\sum_{\xi'}c_{\xi'}
 \sum_{\substack{g\\(g,Ct)=1}}^*\sum_{e\mid g}
 \frac{Y\mu(e)N(g)}{N(e)\ell}\sum_{h\ne0}\\
 &\quad\times\sum_{\substack{z_1,z_2\\(z_1,z_2)=1\\(z_1z_2,gCt)=1}}^*
 a_{\xi'}(z_1)\overline{a_{\xi'}(z_2)} \chi_{z_1}(deh)\overline{\chi_{z_2}(deh)}
 \chi_{z_1}(Ce)^4\overline{\chi_{z_2}(Ce)^4}\\
 &\quad\times
 U_0\!\Bigl(\frac{N(gz_1)}\ell\Bigr)
 \overline{U_0\!\Bigl(\frac{N(gz_2)}\ell\Bigr)}
 \widehat\Phi\!\Bigl(\frac{YN(h)}{N(e)N(z_1)N(z_2)}\Bigr).
 \end{aligned}
\]
All starred ideal indices are prime to $S$. Insert
\[
 \ind_{(z_1,z_2)=1}=\sum_{w\mid z_1,\ w\mid z_2}\mu(w),
 \qquad z_i=wn_i.
\]
Here $(w,gCt)=1$. Equation~\eqref{eq:crt-a} and the zero-extended characters give
\[
 \begin{gathered}
 a_{\xi'}(wn)=a_{\xi'}(w)a_{\xi'}(n)\chi_n(w)^4
       \quad((w,n)=1),\\
 |a_{\xi'}(w)\chi_w(deh)\chi_w(Ce)^4|^2=\ind_{(w,h)=1},\\
 \ind_{(n,gCtw)=1}\chi_n(deh)\chi_n(Cew)^4
 =\ind_{(n,tg/e)=1}\chi_n(deh)\chi_n(Cew)^4.
 \end{gathered}
\]
Consequently the preceding full sum becomes
\[
 \begin{aligned}
 &\mathcal M-Z=\sum_{\xi'}c_{\xi'}
 \sum_{\substack{g,w\\(g,Ct)=(w,gCt)=1}}^*
 \sum_{e\mid g}\frac{Y\mu(e)\mu(w)N(g)}{N(e)\ell}
 \sum_{\substack{h\ne0\\(h,w)=1}}\\
 &\quad\times\sum_{(n_1n_2,S)=1}^*a^\sharp(n_1)\overline{a^\sharp(n_2)}
 U_0\!\Bigl(\frac{N(gwn_1)}\ell\Bigr)
 \overline{U_0\!\Bigl(\frac{N(gwn_2)}\ell\Bigr)} \widehat\Phi\!\Bigl(
   \frac{YN(h)}{N(e)N(w)^2N(n_1)N(n_2)}\Bigr).
 \end{aligned}
\]
There is no remaining condition $(n_1,n_2)=1$. The column support gives
$N(gw)\le b\ell$ and $N(z_1z_2)\le b^2\ell^2/N(g)^2$;
the support of $\widehat\Phi$ therefore imposes the stated bound on
$N(h)$. Substituting \eqref{eq:common-poisson-output} gives
\eqref{eq:arithmetic-poisson-identity}, with only finitely many terms.
\end{proof}

\begin{lemma}\label{lem:second-transfer}
Return to $I=[u,v]$ and $I_*=[u/2,2v]$. For an integer $m\ge0$,
let $S_m$ be the supremum in \eqref{eq:transfer-summary}.
Then, for every $\varepsilon_0>0$,
\[
 \mathcal Q_{\xi_1}(U)\ll D^{\varepsilon_0}\Sigma(1+S_m)
       \|U\|_{C^{2m+4}(I_*)}^2
 \qquad(U\in C_c^\infty(I_*)).
\]
The test functions in $S_m$ are supported in $I'=[u/16,4v]$ and have $C^m(I')$ norm at most one; an empty supremum is zero.
\end{lemma}
\begin{proof}
Write $N=\NK$.
Apply Lemma~\ref{lem:arithmetic-poisson} with input length $\ell$,
row scale $Y_{C,d}$, and the same $C,d,t$, using the character $\xi_1$.
Here $h$ is the new Fourier variable for the sum over $y$.
Its nonzero output has squarefree $g,w$, $e\mid g$, and
\[
 (g,Ct)=(w,gCth)=1,\qquad
 r=tg/e,\quad f'=Cew,\quad k'=deh.
\]
Thus $r,f'$ are coprime and squarefree, and the coefficient $a^\sharp$ from \eqref{eq:common-poisson-output} becomes
\[
 a^\sharp(n)
 =a_{\xi'}(n)\ind_{(n,r)=1}\chi_n(k')\chi_n(f')^4.
\]
Write $X'_0$ for the individual column scale $X'$ in
\eqref{eq:common-poisson-output}; below, $X'$ will denote a common
dyadic scale. The length, coefficient, and Fourier parameter simplify to
\[
 \begin{aligned}
 X'_0&=\frac{\ell}{N(g)N(w)}=\frac L{N(r)N(f')},\\
 w_{C,d}\frac{Y_{C,d}N(g)}{N(e)\ell}
 &=\frac{c_I\Sigma N(r)}{L^2},\\
 \frac{Y_{C,d}N(h)N(g)^2}{N(e)\ell^2}
 &=\frac{c_I\Sigma F N(k')N(r)^2}{\mathcal H L}.
 \end{aligned}
\]
Put $U_0(x)=x^{-1/2}U(x)$. The coupled kernel is therefore
\[
 \mathscr K_{r,f',k'}(x_1,x_2)
 =U_0(x_1)\overline{U_0(x_2)}
       \widehat\Phi\!\Bigl(\frac{c_I\Sigma F N(k')N(r)^2}{\mathcal H Lx_1x_2}\Bigr),
\]
with both columns evaluated at $x_j=N(n_j)/X'_0$.
In particular its nonzero support requires
\[
 N(r)N(f')\le 2vL,\qquad
 0<N(k')\le\frac{4C_\Phi v^2}{c_I}
              \frac{\mathcal H L}{\Sigma F N(r)^2}
          \le\frac{\mathcal H L}{\Sigma F N(r)^2},
\]
since $C_\Phi(2v)^2/c_I\le1$.

For fixed $r,f',k'$, all preimages are obtained by choosing
\[
 t\mid r,\qquad f'=Cew,\qquad e\mid k',\qquad
 (w,k')=1,\qquad d\mid(C,k'),
\]
and setting $g=e(r/t)$, $h=k'/(de)$.
The factorization $f'=Cew$ is disjoint and squarefree.
All these preimages have the same kernel: $Ctgw=rf'$, and the
Fourier parameter displayed above depends only on $r,k'$.
Thus the support restrictions discard only zero kernels.
The choices $t\mid r$ contribute $\tau_{\mathrm{div}}(r)$.
At a prime $p\mid f'$, the choices $p\mid C$, $p\mid e$, and
$p\mid w$, respectively, contribute
\[
 \bigl(1+\ind_{p\mid k'}\bigr)
       -\ind_{p\mid k'}-\ind_{p\nmid k'}=\ind_{p\mid k'}
\]
to the sum of $\mu(e)\mu(w)$ over the preimages; the two choices
inside the first term are $p\nmid d$ and $p\mid d$.
Let $\mathcal Z$ be the total zero-frequency contribution: the sum of the terms $Z$ from Lemma~\ref{lem:arithmetic-poisson}, with the outer weights $w_{C,d}$ and ranges in \eqref{eq:first-transfer-forms}.
Consequently, regrouping before taking absolute values gives
\[
 \begin{aligned}
 \mathcal Q_{\xi_1}(U)-\mathcal Z
 ={}&\frac{c_I\Sigma}{L^2}
 \sum_{\xi'}c_{\xi'}
 \sum_{\substack{r,f'\ \mathrm{squarefree}\\(r,f')=1}}
 N(r)\tau_{\mathrm{div}}(r)\sum_{\substack{k'\ne0\\f'\mid k'}}\\
 &\quad\times
 \sum_{n_1,n_2}^*a^\sharp(n_1)\overline{a^\sharp(n_2)}
 \mathscr K_{r,f',k'}\!\Bigl(\frac{N(n_1)}{X'_0},
                            \frac{N(n_2)}{X'_0}\Bigr).
 \end{aligned}
\]

Fix $R\le N(r)<2R$, $F'\le N(f')<2F'$ dyadically, and put
\[
 X'=\frac L{RF'},\qquad \Sigma'=X'F'=\frac LR,\qquad
 \mathcal H'=\frac{\mathcal H L}{\Sigma F R^2}.
\]
Retain only those $r$ for which a nonzero term occurs in these ranges.
Since $f'\mid k'$, each such $r$ satisfies
\[
 F'\le N(f')\le N(k')\le
 \frac{\mathcal H L}{\Sigma F N(r)^2}.
\]
In particular $\mathcal H'\ge1$. For every $j\mid r$, using $\mathcal H\le\Sigma$, we also have
\begin{equation}\label{eq:child-column-lower-bound}
 \frac{X'}{N(j)}
 \ge\frac{\Sigma F N(r)^2}{\mathcal H R N(j)}
 \ge\frac{F\Sigma}{\mathcal H}\ge1.
\end{equation}
Moreover $1\le X'/X'_0<4$. The rescaled kernels
\[
 \mathcal K_{r,f',k'}(x_1,x_2)
 =\mathscr K_{r,f',k'}\!\Bigl(\frac{X'}{X'_0}x_1,
                              \frac{X'}{X'_0}x_2\Bigr)
\]
have support in $[u/8,2v]^2\subset\operatorname{int}(I'^2)$ and satisfy
\[
 \sup_{r,f',k'}\|\mathcal K_{r,f',k'}\|_{C^{2m+4}(I'^2)}
 \ll_{I_*,m,\Phi}\|U\|_{C^{2m+4}(I_*)}^2,
\]
where the supremum is over the retained indices in the dyadic block.
This follows from the Schwartz bounds for $\widehat\Phi$.
The common parameters satisfy exactly
\[
 \frac{\mathcal H'}{\Sigma'}=
 \frac{\mathcal H}{\Sigma FR}\le\frac{\mathcal H}{\Sigma},\qquad
 \Sigma'\le L,\qquad
 \mathcal H'\le\frac{\mathcal H L}{\Sigma F}.
\]

Fix $\delta>0$, to be chosen in terms of $\varepsilon_0$ at the end.
For each retained $r$, enlarge the $f',k'$ ranges by positivity,
keeping this set of $r$ fixed. Lemma~\ref{lem:remove-exclusions}
then gives, for $V\in C_c^\infty(I')$,
\[
 \begin{aligned}
 \sum_{F'\le N(f')<2F'}^*\sum_{0<N(k')\le\mathcal H'}
 \Bigl|\sum_n^*a^\sharp(n)V(N(n)/X')\Bigr|^2
 &=\Sigma'\E_{(r)}(\mathcal H',X',F';\xi',V)\\
 &\ll_\delta D^\delta(\Sigma')^2S_m
                  \|V\|_{C^m(I')}^2.
 \end{aligned}
\]
Indeed exclusion removal replaces $(X',F')$ by
$(X'/N(j),F'N(j))$, $j\mid r$, preserving $\mathcal H'$ and $\Sigma'$.
Equation~\eqref{eq:child-column-lower-bound} gives $X'/N(j)\ge1$,
and $F'N(j)\ge1$ as well. Thus every resulting mean square lies in
the defining supremum.

There are $O(R)$ ideals $r$ in the dyadic range.
Apply Lemma~\ref{lem:smooth-mean-square} with row index
$(r,f',k')$, the preceding mean-square bound, and the uniform kernel bound.
The absolute contribution of this dyadic block is at most
\[
 D^{2\delta}\frac{\Sigma R}{L^2}\,
 R(\Sigma')^2S_m\|U\|_{C^{2m+4}(I_*)}^2
 =D^{2\delta}\Sigma S_m\|U\|_{C^{2m+4}(I_*)}^2,
\]
because $(\Sigma R/L^2)R(L/R)^2=\Sigma$.

Finally, the zero-frequency bound in Lemma~\ref{lem:arithmetic-poisson} gives
\[
 \begin{aligned}
 |\mathcal Z|
 &\ll_{I_*,\Phi}\|U\|_\infty^2
     \sum_{N(Ct)\le 2vL}\sum_{d\mid C}w_{C,d}Y_{C,d}\ell\\
 &\ll_{I_*}\Sigma\|U\|_\infty^2
     \sum_C\frac{\tau_{\mathrm{div}}(C)}{N(C)^2}
     \sum_{N(t)\le 2vL}\frac1{N(t)}
 \ll_{I_*,\delta}D^\delta\Sigma\|U\|_\infty^2.
 \end{aligned}
\]
This uses only $Y_{C,d}>0$, not $Y_{C,d}\ge1$.
Sum the $O_{I_*,C_0}((\log(2D))^2)$ dyadic blocks and the fixed
finite character set, and take $\delta=\varepsilon_0/4$.
The stated bound follows, with implied constant depending only on
$I_*,m,C_0,\varepsilon_0$, the fixed cutoffs and arithmetic data.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:transfer}]
Use $S_m$ as in Lemma~\ref{lem:second-transfer}, with $m$ as in the proposition.
Lemma~\ref{lem:second-transfer}, with loss $D^{\varepsilon/4}$,
supplies the hypothesis of Lemma~\ref{lem:first-transfer} with
$j=2m+4$ and $M\ll D^{\varepsilon/4}\Sigma(1+S_m)$.
Applying that lemma with the same loss gives
\[
 \mathcal A(W)\ll D^{\varepsilon/2}\Sigma(1+S_m)
       \|W\|_{C^{4m+12}(I)}^2,
\]
which proves \eqref{eq:transfer-summary}.
\end{proof}


\appendix
\section{Arithmetic identities and theta calculations}\label{sec:arithmetic}

\subsection{Reciprocity and Gauss sums}\label{app:gauss-identities}

\begin{proof}[Proof of Lemma~\ref{lem:arithmetic}]
\par\noindent\textit{Proof of \eqref{eq:gj}.}\quad
For squarefree $n$, set $G(n)=\overline{\chi_n(4)}\gamma_3(n)$.
Its dependence on a fixed ray class will be verified below.
Fix a primary prime $p\equiv1\pmod3$ outside $S$. For multiplicative characters $A,B$ of $(\OO/(p))^\times$,
extended by zero at $0$, write
\[
J(A,B)=\sum_{x\bmod p}A(x)B(1-x).
\]
For each $y\in\OO/(p)$, the substitution $u=2x-1$ gives
\[
 \#\{x\bmod p:4x(1-x)=y\} =\#\{u\bmod p:u^2=1-y\} =1+\chi_p^3(1-y),
\]
since $p\nmid2$ and $\chi_p^3$ is the quadratic character, extended
by zero at $0$. Hence
\[
 \chi_p(4)J(\chi_p,\chi_p) =\sum_{x\bmod p}\chi_p\bigl(4x(1-x)\bigr) =\sum_{y\bmod p}\chi_p(y)\bigl(1+\chi_p^3(1-y)\bigr) =J(\chi_p,\chi_p^3).
\]
The Gauss--Jacobi identity (cf. \cite[(3.18)]{IK04}) gives
\[
 \frac{\gamma_1(p)\gamma_3(p)}{\gamma_4(p)}
 =\frac{J(\chi_p,\chi_p^3)}{\sqrt{\NK(p)}}
 =\chi_p(4)\frac{J(\chi_p,\chi_p)}{\sqrt{\NK(p)}}
 =\chi_p(4)\frac{\gamma_1(p)^2}{\gamma_2(p)}.
\]
For $j\not\equiv0\pmod6$, the normalized Gauss sums satisfy
\[
 \gamma_j(p)\gamma_{-j}(p)=\chi_p(-1)^j.
\]
Taking $j=2$ gives $\gamma_2(p)\gamma_4(p)=\chi_p^2(-1)=1$,
since $\chi_p^2$ is cubic.
Combining this with the Gauss--Jacobi relation above gives, after
cancelling $\gamma_1(p)$ and rearranging,
\[
\gamma_1(p)\gamma_2(p)=\overline{\chi_p(4)}\gamma_3(p)\gamma_2(p)^3.
\]

To finish the prime case of \eqref{eq:gj}, it remains to evaluate
$\gamma_2(p)^3$. We do so by computing $J(\chi_p^2,\chi_p^2)$.
Put $q=\NK(p)$. We have $J(\chi_p^2,\chi_p^2)\in\OO$ and $
\NK\bigl(J(\chi_p^2,\chi_p^2)\bigr)=q$.
For $0\le\ell\le(q-1)/3$, the exponent $(q-1)/3+\ell$ lies
strictly between $0$ and $q-1$, so
$\sum_{x\bmod p}x^{(q-1)/3+\ell}=0$ in $\OO/(p)$.
Reducing modulo $p$ and expanding the binomial therefore gives
\[
 \begin{aligned}
 J(\chi_p^2,\chi_p^2)
 &\equiv\sum_{x\bmod p}x^{(q-1)/3}(1-x)^{(q-1)/3}\\
 &=\sum_{\ell=0}^{(q-1)/3}(-1)^\ell\binom{(q-1)/3}{\ell}
       \sum_{x\bmod p}x^{(q-1)/3+\ell}
 =0\pmod p.
 \end{aligned}
\]
 Thus $J(\chi_p^2,\chi_p^2)/p\in\OO$ has norm one,
so $J(\chi_p^2,\chi_p^2)$ is a unit multiple of $p$.
To determine the unit, write
$\chi_p^2(x(1-x))=\omega^{j_x}$ with $j_x\in\{0,1,2\}$
for $x\in\OO/(p)\setminus\{0,1\}$. Then
\[
 \prod_{x\ne0,1}x(1-x)=1
 \quad\Longrightarrow\quad
 \sum_{x\ne0,1}j_x\equiv0\pmod3.
\]
Since $(\omega-1)^2$ generates $(3)$, expansion modulo this ideal gives
\[
 J(\chi_p^2,\chi_p^2) =\sum_{x\ne0,1}\omega^{j_x} \equiv(q-2)+(\omega-1)\sum_{x\ne0,1}j_x \equiv-1\pmod3.
\]
Together with $p\equiv1\pmod3$, this fixes the unit:
\[
 J(\chi_p^2,\chi_p^2)=-p.
\]
Consequently
\[
 \gamma_2(p)^3
 =\frac{\gamma_2(p)^2}{\gamma_4(p)}
 =\frac{J(\chi_p^2,\chi_p^2)}{\sqrt{\NK(p)}}
 =-\alpha(p).
\]
The Chinese remainder theorem extends these prime-modulus identities
to \eqref{eq:gj} for squarefree $n$, with one minus sign for each prime
factor accounting for $\mu(n)$.
\par\noindent\textit{Proof of \eqref{eq:crt-a}.}\quad
For coprime squarefree primary $a,b$ prime to $S$, the Chinese
remainder theorem gives
\[
 \gamma_2(ab)=\gamma_2(a)\gamma_2(b)
                    \chi_a(b)^2\chi_b(a)^2,
\]
and cubic reciprocity gives $\chi_a(b)^2=\chi_b(a)^2$.
Multiplying by $\overline{\alpha(ab)}\xi(ab)$ therefore proves
\eqref{eq:crt-a}.

\par\noindent\textit{Proof of \eqref{eq:recip}.}\quad
We now prove \eqref{eq:recip} and the formula for $G(n)$ in
\eqref{eq:g}, including their dependence on a fixed ray class.
We begin by evaluating the normalized quadratic Gauss sum.
For $c\in\OO$ coprime to $2$, we will show that
\begin{equation}\label{eq:quadratic-gaussian}
 \Gamma_{\rm quad}(c):=|c|^{-1}\sum_{x\bmod c}e(x^2/c)
       =\frac12\sum_{y\bmod2\OO}e(-cy^2/4).
\end{equation}
To justify \eqref{eq:quadratic-gaussian}, apply Poisson summation over
$z\in\OO$ to $e(z^2/c)e^{-\pi\eta|z|^2}$, with $\eta>0$.
The Gaussian makes the sum convergent. The Fourier transform of the product is
\[
 \frac{|c|}{2\sqrt{r_\eta}}
 e^{-\pi\eta \NK(c)|y|^2/(4r_\eta)}e(-cy^2/(4r_\eta)),
 \qquad r_\eta=1+3\NK(c)\eta^2/16.
\]
Divide both sides by the Gaussian mass
\[
 \sum_{z\in\OO}e^{-\pi\eta|z|^2}\asymp\eta^{-1}.
\]
On the Fourier side, replacing $r_\eta$ by $1$ has total error
\[
 O_c\!\biggl(\eta^2\sum_{y\in\OO}|y|^2e^{-C_c\eta|y|^2}\biggr)
 =O_c(1),
\]
where $C_c>0$ is fixed. The normalized error is therefore
$O_c(\eta)\to0$. Grouping the original sum modulo $c$ and the
Fourier sum modulo $2\OO$, then letting $\eta\downarrow0$, gives
\eqref{eq:quadratic-gaussian}.

For $c=a+b\omega$, formula \eqref{eq:quadratic-gaussian} becomes
\[
 \Gamma_{\rm quad}(a+b\omega)=\frac{1+\mathrm i^{-b}+\mathrm i^a+\mathrm i^{b-a}}2.
\]
Thus $\Gamma_{\rm quad}(c)$ depends only on $c\bmod4\OO$ and is invariant under
multiplication by a square in $(\OO/4\OO)^\times$. Representatives
for the four square classes and their values are
\[
 \begin{array}{c|rrrr}
 c&1&-1&\lambda&-\lambda\\ \hline
 \Gamma_{\rm quad}(c)&1&1&\mathrm i&-\mathrm i
 \end{array}.
\]
Consequently the function
\[
 \mathcal R(c_1,c_2):=\frac{\Gamma_{\rm quad}(c_1c_2)}{\Gamma_{\rm quad}(c_1)\Gamma_{\rm quad}(c_2)}
\]
on these square classes satisfies
\[
 \mathcal R((-1)^e\lambda^f,(-1)^g\lambda^h)=(-1)^{eh+fg+fh},
 \qquad e,f,g,h\in\{0,1\},
\]
and is a symmetric bicharacter.
For a prime $p$, each $y\in\OO/(p)$ has $1+\chi_p^3(y)$ square roots.
Grouping by $y=x^2$ gives
\[
 \Gamma_{\rm quad}(p)
 =\frac1{|p|}\sum_{y\bmod p}\bigl(1+\chi_p^3(y)\bigr)e(y/p)
 =\gamma_3(p),
\]
since $\sum_{y\bmod p}e(y/p)=0$.
The Chinese remainder theorem extends this equality to squarefree
$n$. Together with cubic reciprocity, it identifies the above $\mathcal R$
with the quotient $\chi_b(a)/\chi_a(b)$ for coprime primary $a,b$.
Furthermore $\chi_c(4)=(-2/c)_3=(c/(-2))_3$ is a character modulo
$2$. Consequently $G(c)=\overline{\chi_c(4)}\Gamma_{\rm quad}(c)$ factors through a fixed ray class group
and satisfies \eqref{eq:recip}, with $\mathcal R(c,c)=\chi_c(-1)$.

\par\noindent\textit{Proof of \eqref{eq:quotient}.}\quad
In the fixed ray class group, the multiplicative relation for $G$ gives
\[
 G(a^{-1})=\chi_a(-1)\overline{G(a)}.
\]
Using the symmetry and multiplicativity of $\mathcal R$, we obtain
\[
 G(ba^{-1}) =G(b)G(a^{-1})\mathcal R(b,a^{-1}) =\chi_a(-1)\overline{G(a)}G(b)\mathcal R(a,b),
\]
which is \eqref{eq:quotient}.

\par\noindent\textit{Proof of \eqref{eq:convert1}--\eqref{eq:convert2}.}\quad
The second identity in \eqref{eq:g} follows from the inverse-character
Gauss identity:
$\gamma_1(n)\gamma_{-1}(n)=\chi_n(-1)$.
It remains to prove \eqref{eq:convert1}--\eqref{eq:convert2}.
Multiplying the second identity in \eqref{eq:gj} by
$\overline{\alpha(n)}$ gives \eqref{eq:convert1}, and combining it
with this inverse-character identity gives \eqref{eq:convert2}.
\end{proof}

\subsection{The cubic theta transformation with fixed ray class twists}\label{app:fixed-ray}

We prove the transformation formula of Proposition~\ref{lem:reflection}
for the completed sum $T(X;\Psi)$ in \eqref{eq:T}, together with
the uniformity assertion of Lemma~\ref{lem:reflection-uniformity} and 
the coefficient and weight bounds of Lemma~\ref{lem:theta-bounds}.
The proof adapts the theta-transformation method of Dunn and
Radziwi\l\l\ \cite[\S5 and Appendix~A]{DR}, which extends Patterson
\cite{Pat77} and Yoshimoto \cite{Yos87}. The treatment of the fixed ray
class twists and the uniformity assertions are supplied below.
We use the setup preceding the proposition and the dependence of
constants specified in these three statements: $\Psi_0,S$ are fixed,
while $\mathcal P$ is the varying finite set of primes outside $S$
and $j_p$ are the exponents of their character factors in $\Psi$.
Character powers follow the zero-extension convention preceding
\eqref{eq:theta-local-factors}; in particular,
$\chi_p^0(x)=\ind_{p\nmid x}$. This convention also applies when
we write $\chi_p(x)^j$.

Write $w=(z,v)\in\mathbb C\times\mathbb R_{>0}$, and use $\theta$
from \eqref{eq:cubic-theta-definition}. The three sequences
$d_\sigma$, $\sigma\in\{0,+,-\}$, are defined by the Fourier expansions
of $\overline{\theta(\gamma_\sigma w)}$ in
\eqref{eq:cusp-coefficient-definition}, with the representatives
\eqref{eq:theta-cusp-representatives}.
For the additive characters, write
$\breve e(z)=\exp(2\pi \mathrm i(z+\bar z))$; thus
$e(z)=\breve e(z/\lambda)$, with $e$ as in \eqref{eq:intro-gauss-sum}.


\begin{proof}[Proof of Proposition~\ref{lem:reflection} and Lemmas~\ref{lem:reflection-uniformity} and~\ref{lem:theta-bounds}]

\noindent\textit{Finite Fourier expansion.}\quad
Define $\phi:\OO\to\mathbb C$ by
\[
 \phi(n)=
 \begin{cases}
  \chi_n(\lambda)^2\Psi_0(n),&n\equiv1\pmod3,\quad(n,S)=1,\\
  0,&\text{otherwise}.
 \end{cases}
\]
For $\Psi(n)=\Psi_0(n)\prod_{p\in\mathcal P}\chi_p^{j_p}(n)$, define
\begin{equation}\label{eq:general-twisted-theta-definition}
 \Theta_\Psi(z,v)=
 \sum_{\substack{\ell\in\lambda^{-3}\OO\\\ell\ne0}}
 \overline{\tau(-\ell)}\phi(\lambda^3\ell)
 \Bigl(\prod_{p\in\mathcal P}\chi_p^{j_p}(\lambda^3\ell)\Bigr)
 vK_{1/3}(4\pi|\ell|v)\breve e(\ell z).
\end{equation}
When $\Psi=\Psi_k$, the multiplier of $\overline{\tau(-\ell)}$
is $\phi_k(\lambda^3\ell)$, so this definition recovers
$\Theta_k$ in \eqref{eq:twisted-theta-definition}.
We first write $\Theta_\Psi$ as a finite sum of translates of
$\overline\theta$ and determine their reduced denominators.
Choose once and for all a nonzero $L\in\OO$, with prime divisors in
$S$, divisible by the conductor of $\Psi_0$, every prime in $S$,
and sufficiently high powers of the primes above $2$ and $3$.
The supplementary law of cubic reciprocity for $\lambda$
\cite[(1.5)]{DR}, which evaluates $\chi_n(\lambda)^2=(\lambda/n)_3$,
makes $\phi$ periodic modulo $L$.
As before \eqref{eq:theta-twist-translates}, define
\[
 \widehat\phi(h_0)=\frac1{\NK(L)}\sum_{x\bmod L}\phi(x)e(-h_0x/L),
 \qquad h_0\in\OO/(L).
\]
Since $|\phi|\le1$, we have $|\widehat\phi(h_0)|\le1$.

For $p\in\mathcal P$ and $0\le j\le5$, we expand $\chi_p^j$ on $\OO/(p)$
in additive characters. Its Fourier coefficients are defined,
for $h_p\in\OO/(p)$, by
\begin{align}
 C_{p,j}(h_p)&:=\frac1{\NK(p)}\sum_{y\bmod p}\chi_p^j(y)e(-h_py/p).
 \notag\\
 \intertext{Finite Fourier inversion then gives, for $x\in\OO$,}
 \chi_p^j(x)&=\sum_{h_p\in\OO/(p)}C_{p,j}(h_p)e(h_px/p).
 \label{eq:theta-local-fourier}
\end{align}

Multiplying \eqref{eq:theta-local-fourier} over the primes in
$\mathcal P$ gives, for $x\in\OO$,
\[
 \prod_{p\in\mathcal P}\chi_p^{j_p}(x)
 =\sum_{\substack{h_p\in\OO/(p)\\p\in\mathcal P}}
  \Bigl(\prod_{p\in\mathcal P}C_{p,j_p}(h_p)\Bigr)
  e\!\Bigl(x\sum_{p\in\mathcal P}\frac{h_p}{p}\Bigr).
\]
Each summand is indexed by a tuple $(h_p)_{p\in\mathcal P}$.
At $x=\lambda^3\ell$, its additive character gives the shift
$\lambda^2\sum_{p\in\mathcal P}h_p/p$ of $\overline\theta$.
Writing $\boldsymbol h=(h_0,(h_p)_{p\in\mathcal P})$ with
$h_0\in\OO/(L)$, we claim that
\[
\Theta_\Psi(z,v)=\sum_{\boldsymbol h} c_{\mathrm F}(\boldsymbol h)\overline{\theta(z+z_{\boldsymbol h},v)},
\]
where
\begin{equation}\label{eq:theta-factorized-shifts}
 z_{\boldsymbol h}=\lambda^2\Bigl(h_0/L+\sum_{p\in\mathcal P}h_p/p\Bigr),
 \qquad c_{\mathrm F}(\boldsymbol h)=\widehat\phi(h_0)\prod_{p\in\mathcal P} C_{p,j_p}(h_p).
\end{equation}

To verify this identity, expand the fixed factor $\phi$ as well.
The finite Fourier expansions identify all nonzero Fourier
coefficients, and the constant terms of the translates cancel because
\[
 \sum_{\boldsymbol h} c_{\mathrm F}(\boldsymbol h)=\phi(0)\prod_{p\in\mathcal P}\chi_p^{j_p}(0)=0.
\]
\begin{samepage}
For $j\not\equiv0\pmod6$, we use $\gamma_j(p)$ from \eqref{eq:intro-gauss-sum}. Changing $y$ to $-y$ gives
\[
 \frac1{\sqrt{\NK(p)}}\sum_{y\bmod p}\chi_p(y)^j e(-y/p)
 =\chi_p(-1)^j\gamma_j(p),\qquad |\gamma_j(p)|=1.
\]
\end{samepage}

For $h_p\ne0$, substitute $y=h_p^{-1}u$ in the definition of
$C_{p,j}(h_p)$. For $h_p=0$ and $j\ne0$, use character orthogonality;
for $j=0$, sum the additive character over nonzero residues directly.
These calculations give
\begin{equation}\label{eq:ray-fourier}
C_{p,j}(h_p)=
 \begin{cases}
 \NK(p)^{-1/2}\chi_p(-1)^j\gamma_j(p)\chi_p(h_p)^{-j},
       &j\ne0,\ h_p\ne0,\\
 0,    &j\ne0,\ h_p=0,\\
 -\NK(p)^{-1},&j=0,\ h_p\ne0,\\
 1-\NK(p)^{-1},&j=0,\ h_p=0.
 \end{cases}
\end{equation}

For the tuple $\boldsymbol h$, define its set of \emph{active primes} by
$\mathcal A(\boldsymbol h):=\{p\in\mathcal P:h_p\ne0\}$.
By \eqref{eq:ray-fourier}, $c_{\mathrm F}(\boldsymbol h)\ne0$ implies
$\{p\in\mathcal P:j_p\ne0\}\subseteq\mathcal A(\boldsymbol h)$.
Thus only primes with $j_p=0$ can be inactive, as asserted in the proposition.
Put $r=\prod_{p\in\mathcal A(\boldsymbol h)}p$, and let $c_0$ be the reduced
denominator of $\lambda^2h_0/L$. At each active prime $p$,
\[
 v_p(\lambda^2h_p/p)=-1,
 \qquad v_p(z_{\boldsymbol h}-\lambda^2h_p/p)\ge0,
\]
so the factor $p$ cannot cancel from the denominator. At primes
dividing $L$, all terms $\lambda^2h_p/p$ are integral. Thus the reduced denominator is $c=c_0r$ up to a unit, including
when $(h_0,L)\ne1$. The finite Fourier expansion has therefore
expressed $\Theta_\Psi$ as $O_L(2^{|\mathcal P|})$ groups of
translates $\overline{\theta(z+z_{\boldsymbol h},v)}$, with shifts $z_{\boldsymbol h}$ and
coefficients $c_{\mathrm F}(\boldsymbol h)$ given by \eqref{eq:theta-factorized-shifts}.
The groups are indexed by $h_0\in\OO/(L)$ and the active set
$\mathcal A$. Each group has a common reduced denominator
$c=c_0\prod_{p\in\mathcal A}p$ up to a unit. For each translate
$\overline{\theta(z+z_{\boldsymbol h},v)}$, we next identify which of the three
Fourier expansions in \eqref{eq:cusp-coefficient-definition} will
be used after changing coordinates at $z_{\boldsymbol h}$.

\noindent\textit{Reduced denominators and cusp coefficients.}\quad
We next express each translate $\overline{\theta(z+z_{\boldsymbol h},v)}$
using one of the three cusp expansions identified above.
Our representatives $\gamma_0,\gamma_+,\gamma_-$ correspond to
$\gamma_1,\gamma_{10},\gamma_{19}$, respectively, in the numbering
of \cite[\S5.1]{DR}, which follows \cite[Table~II]{Pat77}.
We allow
$(h_0,L)\ne1$ and keep the reduced denominator $c_0$ of
$\lambda^2h_0/L$ in the calculation; this is the extension beyond
\cite[Corollary~5.1]{DR}.


Set $M=\lambda^{12}L^4$. Since $(p,M)=1$ for $p\in\mathcal A(\boldsymbol h)$,
the Chinese remainder theorem lets us choose representatives
$h_p\in\OO$ with $h_p\equiv0\pmod{M^2}$. Changing representatives
modulo $p$ changes $z_{\boldsymbol h}$ by an element of $\lambda^2\OO=3\OO$,
under which $\theta$ is periodic.
To track dependence on the primes in $r$, restrict $r$ to a residue
class modulo $M^2$. For fixed $h_0$, active set, and this residue
class, write $z_{\boldsymbol h}=a/c$ in lowest terms,
normalizing \(a\equiv1\pmod3\) if \(\lambda\mid c\), and
\(c\equiv1\pmod3\) otherwise.  Since
\[
 a=\lambda^2c\Bigl(h_0/L+\sum_{p\mid r}h_p/p\Bigr),
\]
these choices fix the residue of $a$ modulo $Mc_0$.
Since $(a,c)=1$ and $(r,M)=1$, the Chinese remainder theorem gives
$\delta'\in\OO$ satisfying the following congruences, where
$q$ ranges over prime divisors of the indicated elements:
\begin{align*}
 a\delta'&\equiv1\pmod{q^{v_q(Mc_0)}}&& (q\mid c_0),\\
 \delta'&\equiv0\pmod{q^{v_q(M)}}&& (q\mid M,\ q\nmid c_0),\\
 a\delta'&\equiv1\pmod r.
\end{align*}
Put
\[
 b_g=\frac{a\delta'-1}{c},
 \qquad g=\begin{pmatrix}a&b_g\\c&\delta'\end{pmatrix}\in\mathrm{SL}_2(\OO).
\]
The congruences for $\delta'$ give
\[
 b_g\equiv
 \begin{cases}
 0,&q\mid c_0,\\
 -c^{-1},&q\mid M,\ q\nmid c_0,
 \end{cases}
 \pmod{q^{v_q(M)}}.
\]

To choose the cusp expansion at $a/c$, put
\[
 H=
 \begin{cases}
  I,&3\mid c,\\[2pt]
  \bigl(\begin{smallmatrix}1&0\\u_0&1\end{smallmatrix}\bigr),
     &v_\lambda(c)=1,\quad u_0\in\{\lambda,-\lambda\},\quad u_0\equiv c\pmod3,\\[2pt]
  \bigl(\begin{smallmatrix}u_0&-1\\1&0\end{smallmatrix}\bigr),
     &(c,\lambda)=1,\quad u_0\equiv a\pmod3.
 \end{cases}
\]
In the last case choose $u_0\in\OO$ from a fixed set of representatives
modulo $3$.
In each case, put $g_1=gH^{-1}$. Then $g=g_1H$ and $g_1\equiv I\pmod3$, as in \cite[\S5.1]{DR}. The invariances
\[
 \theta(\gamma w)=\theta(w)\quad(\gamma\in\mathrm{SL}_2(\mathbb Z)),
 \qquad \theta(z+t,v)=\theta(z,v)\quad(t\in\mathbb Z+3\OO)
\]
reduce $\overline\theta(Hw)$ to one of the three functions
$\overline{\theta(\gamma_\sigma w)}$, $\sigma\in\{0,-,+\}$, with the matrices $\gamma_\sigma$ from
\eqref{eq:theta-cusp-representatives}.
These representatives give the infinity expansion and the two
additional cusp expansions needed here:
translating by \(\omega\) or \(-\omega\), then applying inversion,
gives the representatives labeled $-$ and $+$, respectively
\cite[(5.9)--(5.15), Appendix~A]{DR}.
Write \(t_\sigma(\ell)\) for the coefficient of
\(vK_{1/3}(4\pi|\ell|v)\breve e(\ell z)\) in $\theta(\gamma_\sigma w)$.
Let $\tau_1,\tau_2:\lambda^{-4}\OO\setminus\{0\}\to\mathbb C$
be the coefficient sequences defined in \cite[(5.13), (5.14)]{DR}.
Patterson's cusp calculation \cite[Theorem~8.1 and Table~III]{Pat77}
then gives
\begin{equation}\label{eq:cusp-fourier-coefficients}
 t_0(\ell)=\tau(\ell),\qquad
 t_-(\ell)=\omega^2\tau_1(\omega^2\ell)\breve e(\ell),
 \qquad
 t_+(\ell)=\omega\tau_2(\omega\ell)\breve e(\ell).
\end{equation}
Complex conjugation changes the Fourier frequency from $\ell$ to
$-\ell$. Thus the coefficients in the normalization
\eqref{eq:cusp-coefficient-definition} are
\begin{equation}\label{eq:conjugate-cusp-coefficients}
 d_\sigma(\ell)=\overline{t_\sigma(-\ell)},\qquad \sigma\in\{0,+,-\}.
\end{equation}
For indices $\ell=u\lambda^mnb^3$ as in \eqref{eq:theta-support},
the coefficient magnitudes satisfy
\[
 |t_0(\ell)|\le
 \begin{cases}
 3^{k/2+2}|b|,&m=3k-4,\ k\ge1,\\
 3^{k/2+5/2}|b|,&m=3k-3,\ k\ge0.
 \end{cases}
\]
The $m=-4$ coefficients of $t_-$ and $t_+$ have magnitude at most $9|b|$. 
For the chosen $H$, let $\sigma\in\{0,+,-\}$ index the corresponding expansion in \eqref{eq:cusp-coefficient-definition}.
For each translate $\overline{\theta(z+a/c,v)}$, we have identified
the Fourier expansion in $z$ of $\overline{\theta(H(z,v))}$ as one
of the three expansions in \eqref{eq:cusp-coefficient-definition}:
\[
 \overline{\theta(H(z,v))}
 =\frac{3^{5/2}}2\ind_{\sigma=0}v^{2/3}
 +\sum_{0\ne\ell\in\lambda^{-4}\OO}
 d_\sigma(\ell)vK_{1/3}(4\pi|\ell|v)\breve e(\ell z).
\]
These coefficients satisfy the support restriction
\eqref{eq:theta-support} and bound \eqref{eq:theta-coefficient-bound},
proving the coefficient assertions of Lemma~\ref{lem:theta-bounds}.
The next step uses $g=g_1H$ to express the original translate through
this Fourier series evaluated at $g^{-1}(z+a/c,v)$, and computes
the accompanying multiplier and additive phase. The matrix $g$
maps $\infty$ to $a/c$, which is why this is called an expansion
at the cusp $a/c$.

\noindent\textit{The local character transformation.}\quad
The matrix factorization $g=g_1H$ lets us apply the automorphy law
\[
 \theta(g_1w)=\kappa(g_1)\theta(w),\qquad
 \kappa(g_1)=(c_1/a_1)_3,\qquad
 g_1=\Bigl(\begin{matrix}a_1&b_1\\c_1&d_1\end{matrix}\Bigr)\equiv I\pmod3,
\]
where $\kappa$ is Kubota's cubic character \cite[(5.4), (5.6)]{DR}.
We will combine its conjugate with the finite Fourier coefficients
$C_{p,j}(h_p)$ in \eqref{eq:ray-fourier} to obtain the factors
$B_{p,j}$ of \eqref{eq:theta-local-factors}.
For the translated theta function, the automorphy law gives
\begin{equation}\label{eq:theta-cusp-automorphy}
 \overline{\theta(z+a/c,v)}
 =\overline{\kappa(g_1)}\,\overline{\theta\bigl(Hg^{-1}(z+a/c,v)\bigr)},
\end{equation}
where
\begin{equation}\label{eq:theta-cusp-coordinates}
 g^{-1}(z+a/c,v)
 =\Bigl(-\frac{\delta'}c-\frac{\bar z}{c^2(v^2+|z|^2)},
          \frac{v}{\NK(c)(v^2+|z|^2)}\Bigr).
\end{equation}
At $z=0$, the term with Fourier index $\ell$ in the expansion of
$\overline\theta(Hw)$ therefore acquires the phase
$\breve e(-\delta'\ell/c)$.
We claim that the multiplier is given by
\begin{equation}\label{eq:ray-multiplier}
\kappa(g_1)=
 \begin{cases}
 (c_0/a)_3(a/r)_3,&3\mid c,\\
 (-u_0/(a-u_0b_g))_3\,((c_0/u_0)/a)_3(a/r)_3,
        &v_\lambda(c)=1,\\
 (a/c_0)_3(a/r)_3,&(c,\lambda)=1.
 \end{cases}
\end{equation}
To verify \eqref{eq:ray-multiplier}, use the determinant equation
and cubic reciprocity. For the case $v_\lambda(c)=1$, the required
congruences are
\[
 a(c-u_0\delta')\equiv-u_0\pmod{a-u_0b_g},
 \qquad b_gc\equiv-1\pmod a.
\]
For $(c,\lambda)=1$, the congruence
\[
 -b_gc=1-a\delta'\equiv1\pmod9
\]
removes the supplementary factors; reciprocity at the remaining
primes then gives $(\delta'/(-b_gc))_3=1$.
The factors other than $(a/r)_3$ depend only on the fixed residues
at primes in $S$. We may therefore write
\[
 \kappa(g_1)=\kappa_0(a/r)_3
          =\kappa_0\prod_{p\mid r}\chi_p(a)^2,
 \qquad |\kappa_0|=1,
\]
where $\kappa_0$ is fixed once $h_0$, the active set, and
$r\bmod M^2$ are fixed.

Fix $h_0$ and the active set, so that the denominator $c=c_0r$
is fixed while the nonzero residues $h_p$ vary. Put
$D_0=\lambda^3c_0$. For each active prime $p$, define in $\OO/(p)$
\[
 \sigma_p=\lambda^2c/p,\qquad
 \epsilon_p=-\bigl((\lambda^3c/p)\sigma_p\bigr)^{-1}.
\]
The inverse in $\epsilon_p$ is taken in the field $\OO/(p)$;
it exists because $r$ is squarefree and $p\nmid\lambda c_0$.
The expression for $a$ gives \(a\equiv\sigma_ph_p\pmod p\).
For a dual Fourier index
$\ell\in\lambda^{-4}\OO$, the additive form of the Chinese remainder theorem yields
\begin{equation}\label{eq:ray-additive-crt}
\breve e(-\delta'\ell/c)
 =\psi(\lambda^4\ell)\prod_{p\mid r}e(\epsilon_ph_p^{-1}\lambda^4\ell/p),
\end{equation}
where
\[
 \psi(\lambda^4\ell):=e(-\delta'_0r^{-1}\lambda^4\ell/D_0),
 \qquad \delta'_0=\delta'\bmod D_0.
\]
The residues $\delta'_0$ and $r^{-1}\bmod D_0$ are fixed by the
choices above, so $\psi$ ranges over a fixed finite family of
additive characters of $\OO$.
We claim the local transformation identity
\begin{equation}\label{eq:ray-local-transform}
\sum_{h_p\ne0} C_{p,j}(h_p)\chi_p(a)^{-2}
 e(\epsilon_ph_p^{-1}\lambda^4\ell/p)
 =\chi_p(\sigma_p)^{-2}\omega_{p,j}B_{p,j}(\lambda^4\ell),
\end{equation}
where $B_{p,j}$ is defined in \eqref{eq:theta-local-factors}, and
\[
 \omega_{p,j}=
 \begin{cases}
 \chi_p(-1)^j\gamma_j(p)\gamma_{j+2}(p)\chi_p(\epsilon_p)^{-j-2},&j\ne0,4,\\
 \gamma_4(p),&j=4,\\
 -\gamma_2(p)\chi_p(\epsilon_p)^{-2},&j=0\ {\rm active}.
 \end{cases}
\]
The indices of $\gamma_j(p)$ are read modulo six; in the second case, $\chi_p(-1)^4=1$.
In each case $|\omega_{p,j}|=1$. To prove
\eqref{eq:ray-local-transform}, combine \eqref{eq:ray-fourier}, the
conjugate of \eqref{eq:ray-multiplier}, and
\eqref{eq:ray-additive-crt}. Setting $y=h_p^{-1}$ changes the
character exponent as follows:
\[
 \chi_p(h_p)^{-j}\chi_p(a)^{-2} =\chi_p(\sigma_p)^{-2}\chi_p(h_p)^{-j-2} =\chi_p(\sigma_p)^{-2}\chi_p(y)^{j+2}, \qquad y=h_p^{-1}.
\]
For $j\ne0,4$, the character $\chi_p^{j+2}$ is nontrivial, and its
Gauss sum gives the first case of $B_{p,j}$.
For $j=4$, it is trivial on nonzero residues; after rescaling by
$\epsilon_p\ne0$, the sum is
\[
 \sum_{y\ne0}e(\lambda^4\ell y/p)
 =-1+\NK(p)\ind_{p\mid\lambda^4\ell}.
\]
For active $j=0$, the coefficient $-\NK(p)^{-1}$ combines with a
cubic Gauss sum to give
\[
 B_{p,0}(\lambda^4\ell)
 =\NK(p)^{-1/2}\chi_p(\lambda^4\ell)^{-2},
\]
with the unit factors included in $\omega_{p,0}$. This proves
\eqref{eq:ray-local-transform} in every case. An inactive
prime has $j=0$ and contributes only the scalar $1-\NK(p)^{-1}$.
Consequently, for each fixed $h_0$ and active set $\mathcal A$,
combining the Fourier coefficients $d_\sigma(\ell)$ of
$\overline{\theta(H(z,v))}$ with the sums over $h_p\ne0$ gives
\[
 d_\sigma(\ell)\psi(\lambda^4\ell)
 \prod_{p\in\mathcal A}B_{p,j_p}(\lambda^4\ell),
\]
up to a scalar independent of $\ell$. These are the arithmetic
factors in the dual sum \eqref{eq:reflection}. In particular,
$B_{p,1}(\lambda^4\ell)=\chi_p(\lambda^4\ell)^3$ is the quadratic
factor used in \eqref{eq:residual-quadratic-factor}.

\noindent\textit{The archimedean transform.}\quad
We now derive the transformed weight $V_*^\sharp$ in
\eqref{eq:theta-weight} and the scalar multiplying the dual sum. For $\Rea s>1$, introduce the Dirichlet series
associated with the completed sum \eqref{eq:T}:
\[
 \mathcal T(s,\Psi)=
 \Bigl(\sum_{\substack{n\in\OO\\n\equiv1\ (3)\\(n,S)=1}}^*
   \overline{\alpha(n)}\gamma_2(n)\Psi(n)\NK(n)^{-s}\Bigr) \Bigl(\sum_{\substack{b\in\OO\\b\equiv1\ (3)\\(b,S)=1}}\overline{\alpha(b)}^{\,3}
   \Psi(b)^3\NK(b)^{-3s+1/2}\Bigr).
\]
With $V_*$ from \eqref{eq:T} and its Mellin transform defined
before \eqref{eq:theta-weight}, Mellin inversion gives
\begin{equation}\label{eq:theta-mellin-inversion}
 T(X;\Psi)=\frac1{2\pi \mathrm i}\int_{(\sigma)}
 \widehat V_*(s-\tfrac12)\mathcal T(s,\Psi)X^{s-1/2}\dd s,
 \qquad \sigma>1.
\end{equation}
The identity \eqref{eq:T-theta}, with $\Psi$ in place of $\Psi_k$,
identifies these coefficients with those of $\Theta_\Psi$ after
inserting the factor $\overline{\alpha(nb^3)}$. We now compute the
normalization relating $\mathcal T(s,\Psi)$ to the Mellin transform
of the derivative of $\Theta_\Psi$.

For $z=x+\mathrm i y$, use $\partial_{\bar z}=(\partial_x+\mathrm i\partial_y)/2$ and $\partial_z=(\partial_x-\mathrm i\partial_y)/2$. Differentiation in $\bar z$ supplies a factor $\bar\ell$ in each Fourier mode. Combined with the factor $|\ell|^{-1}$ from the
Bessel integral below, this produces the required angular factor
$\overline{\alpha(\ell)}$. We therefore define, initially for $\Rea s>1$,
the Mellin transform
\[
 \mathcal J(s)=\int_0^\infty
 \partial_{\bar z}\Theta_\Psi(z,v)\Bigr|_{z=0}
 v^{2s-1}\dd v.
\]
For each nonzero Fourier mode, differentiation and the Mellin
integral for $K_{1/3}$ (see \cite[(10.43.19)]{DLMF}) give the following formulas:
\begin{align}
 \partial_{\bar z}\breve e(\ell z)
   &=2\pi \mathrm i\bar\ell\,\breve e(\ell z),\label{eq:fourier-derivative}\\
 \int_0^\infty v^{2s}K_{1/3}(4\pi|\ell|v)\dd v
   &=\frac{2^{2s-1}\Gamma(s+1/3)\Gamma(s+2/3)}
          {(4\pi|\ell|)^{2s+1}} \label{eq:bessel-mellin} \qquad \Rea s>0.
\end{align}

For $\Rea s>1$, the coefficient formula
\eqref{eq:intro-theta-coefficients} and \eqref{eq:bessel-mellin}
show that the sum of the integrals of the absolute values is finite.
We may therefore integrate the differentiated Fourier series
term by term.

For \(\ell=\lambda^{-3}nb^3\), the phase and scale simplify to
\[
 \mathrm i\overline{\alpha(\ell)}|\ell|^{-2s}
  =27^s\overline{\alpha(n)}\,
   \overline{\alpha(b)}^{\,3}\NK(n)^{-s}\NK(b)^{-3s}.
\]
Combining this with \eqref{eq:intro-theta-coefficients} gives
\begin{equation}\label{eq:ray-mellin}
\mathcal J(s)=\frac{3^{5/2}}4
       \Bigl(\frac{27}{(2\pi)^2}\Bigr)^s
       \Gamma(s+1/3)\Gamma(s+2/3)\mathcal T(s,\Psi).
\end{equation}
We next show that $\mathcal J(s)$ is entire and use
\eqref{eq:ray-mellin} to continue $\mathcal T(s,\Psi)$.

Write $(z',v')=g^{-1}(z+a/c,v)$. Differentiating
\eqref{eq:theta-cusp-coordinates} at $z=0$ gives
\[
 \frac{\partial z'}{\partial\bar z}\bigg|_{z=0}
   =-\frac1{c^2v^2},
 \qquad
 \frac{\partial\overline{z'}}{\partial\bar z}\bigg|_{z=0}=0,
 \qquad
 \frac{\partial v'}{\partial\bar z}\bigg|_{z=0}=0.
\]
Thus the derivative in cusp coordinates is
$-(cv)^{-2}\partial_{z'}$, with no height-derivative term. The defining Fourier series
\eqref{eq:general-twisted-theta-definition}
for $\Theta_\Psi$ controls $v\to\infty$. For $v\to0$, use
\eqref{eq:theta-cusp-automorphy} and the cusp expansions
\eqref{eq:cusp-coefficient-definition}.
In each expansion the horizontal derivative removes the constant
mode. The remaining modes decay exponentially as $v\to\infty$;
at $v\to0$, the transformed height $1/(\NK(c)v)$ tends to infinity,
giving exponential decay in $1/v$. Thus the integral defining $\mathcal J(s)$
converges for every $s$ and defines an entire function.
Equation~\eqref{eq:ray-mellin}, after division by its gamma factors,
also continues $\mathcal T(s,\Psi)$ to an entire function.

For the term indexed by $\boldsymbol h$ in \eqref{eq:theta-factorized-shifts},
write $c_{\boldsymbol h},\delta'_{\boldsymbol h},g_{1,\boldsymbol h},H_{\boldsymbol h}$ for the corresponding choices above.
Let $\sigma_{\boldsymbol h}$ be the cusp index determined by $H_{\boldsymbol h}$.
Define its cusp Mellin transform by
\begin{equation}\label{eq:dual-cusp-mellin}
 \mathcal J_{\boldsymbol h}^\vee(s)=\int_0^\infty
 \partial_z\bigl\{\overline{\theta(H_{\boldsymbol h}(z,v))}\bigr\}
 \Bigr|_{z=-\delta'_{\boldsymbol h}/c_{\boldsymbol h}}v^{2s-1}\dd v.
\end{equation}
Substituting $v\mapsto(\NK(c_{\boldsymbol h})v)^{-1}$ in each translated term
of $\mathcal J(s)$, using \eqref{eq:theta-cusp-automorphy}, gives
\begin{equation}\label{eq:theta-mellin-functional-equation}
 \mathcal J(s)=-\sum_{\boldsymbol h} c_{\mathrm F}(\boldsymbol h)\overline{\kappa(g_{1,\boldsymbol h})}\,
       \overline{\alpha(c_{\boldsymbol h})}^{\,2}\NK(c_{\boldsymbol h})^{1-2s}\mathcal J_{\boldsymbol h}^\vee(1-s).
\end{equation}
Here $c_{\mathrm F}(\boldsymbol h)$ is the coefficient in \eqref{eq:theta-factorized-shifts}.
The cusp coefficients $d_{\sigma_{\boldsymbol h}}(\ell)$ and \eqref{eq:bessel-mellin}
give, for $\Rea s>1$,
\begin{equation}\label{eq:dual-cusp-mellin-series}
 \mathcal J_{\boldsymbol h}^\vee(s)=\frac{\mathrm i\,\Gamma(s+1/3)\Gamma(s+2/3)}{4(2\pi)^{2s}}
 \sum_{0\ne\ell\in\lambda^{-4}\OO}
 \frac{d_{\sigma_{\boldsymbol h}}(\ell)\alpha(\ell)}{\NK(\ell)^s}
 \breve e(-\delta'_{\boldsymbol h}\ell/c_{\boldsymbol h}).
\end{equation}
The Dirichlet series $\mathcal T(s,\Psi)$ converges absolutely
for $\Rea s>1$, while the series in \eqref{eq:dual-cusp-mellin-series}
at $1-s$ converges absolutely for $\Rea s<0$. The functional
equation \eqref{eq:theta-mellin-functional-equation} and Stirling's
formula \cite[\S5.A.4]{IK04} therefore give polynomial bounds in
$|\operatorname{Im}s|$ for $\mathcal T(s,\Psi)$ on both sides of
the strip $0\le\Rea s\le1$.
Splitting the integral defining $\mathcal J(s)$ at $v=1$ gives finite order;
Phragm\'en--Lindel\"of \cite[Theorem~5.53]{IK04} then gives the same
type of bound inside the strip (compare the arguments of Dunn and Radziwi\l\l\ in
\cite[Propositions~5.1--5.2]{DR}).
Together with the rapid decay of $\widehat V_*$ on vertical lines,
these bounds justify moving the $s$-contour in
\eqref{eq:theta-mellin-inversion} to $\Rea s<0$. No poles are crossed,
since $\mathcal T(s,\Psi)$ is entire. Setting $t=\tfrac12-s$
then gives a line $\Rea t>1/2$, where the dual coefficient series
converges absolutely.

For fixed $h_0$ and active set, insert
\eqref{eq:dual-cusp-mellin-series} into
\eqref{eq:theta-mellin-functional-equation} and use
\eqref{eq:ray-local-transform} to sum over the nonzero $h_p$.
After the normalization in \eqref{eq:ray-mellin}, the scalar $C$
in \eqref{eq:reflection} for this group of translates is
\[
 C=-\frac{\mathrm i}{81}\overline{\alpha(c)}^{\,2}
 \widehat\phi(h_0)\overline{\kappa_0}
 \prod_{p\ {\rm inactive}}(1-\NK(p)^{-1})
 \prod_{p\ {\rm active}}\chi_p(\sigma_p)^{-2}\omega_{p,j_p},
 \qquad |C|\le1/81 .
\]
Dividing \eqref{eq:ray-mellin} by its gamma factors and setting
$t=\tfrac12-s$ produces the gamma quotient in
\eqref{eq:theta-weight}. Its numerator gamma factors have their
first pole at $t=-5/6$, and its reciprocal denominator gamma
factors are entire. Thus no poles lie between the current contour
$\Rea t>1/2$ and $\Rea t=0$. The rapid decay of $\widehat V_*$,
together with Stirling's formula, allows us to shift each kernel
contour to $\Rea t=0$. This gives the weight $V_*^\sharp$ defined in
\eqref{eq:theta-weight}, evaluated at $\NK(\ell)X/\NK(c)^2$.
Together with \eqref{eq:ray-local-transform}, this gives the dual
sum \eqref{eq:reflection}.

For $k_0$ as in %
Lemma~\ref{lem:reflection-uniformity}, fix $h_0$, the factors of $\Psi$
other than $\chi_{k_0}$, and the active/inactive choices at their
primes. Every prime dividing $k_0$ has exponent $j_p=1$ and is
therefore active, so
\[
 r=k_0\prod_{p\in\mathcal A_{\rm fix}}p.
\]
Thus $r/k_0$ is fixed, and fixing $k_0\bmod M^2$ fixes $r\bmod M^2$.
The choices of $H,c_0$ and the residues in
\eqref{eq:ray-additive-crt} are therefore fixed in each such
class. Thus $d=d_\sigma$, $\psi$, and $c_0$ have precisely the asserted
dependence on $k_0$.

It remains to bound the transformed weight.
The first numerator pole of the gamma quotient in
\eqref{eq:theta-weight} is at $t=-5/6$.
We may therefore shift the kernel contour to $\Rea t=-1/4$
for $0<x\le1$, obtaining the factor $x^{1/4}$, and to $\Rea t=A$
for $x\ge1$, obtaining $x^{-A}$. Each application of $x\partial_x$
introduces a factor $-t$. Stirling's formula on
$-1/4\le\Rea t\le A$ then gives, for every $A>0$ and $j\ge0$,
\begin{equation}\label{eq:ray-kernel}
\begin{aligned}
|(x\partial_x)^jV_*^\sharp(x)|
 &\ll_{A,j}\min\{x^{1/4},(1+x)^{-A}\}\\
 &\quad\times\sup_{-A\le\eta\le1/4}\int_{\mathbb R}
 (1+|u|)^{\lceil4A\rceil+j+2}|\widehat V_*(\eta+\mathrm i u)|\dd u.
\end{aligned}
\end{equation}
When $V_*$ is supported in a fixed compact interval
$I\subset(0,\infty)$, repeated integration by parts in its Mellin
transform gives rapid decay in $|u|$, uniformly for
$-A\le\eta\le1/4$. Thus, for a sufficiently large $J=J(A,j)$, the supremum of integrals in \eqref{eq:ray-kernel} is
$\ll_{A,j,I}\|V_*\|_{C^J(I)}$. This proves \eqref{eq:theta-weight-decay}.
Thus \eqref{eq:reflection} expresses $T(X;\Psi)$ as
$O_{\Psi_0,S}(2^{|\mathcal P|})$ dual sums with $|C|\le1/81$,
the asserted ray class dependence of $(d,\psi,c_0)$, and the
weight decay just established. Together with the support and
coefficient bounds proved above, this completes the proof of Proposition~\ref{lem:reflection}
and Lemmas~\ref{lem:reflection-uniformity} and~\ref{lem:theta-bounds}.
\end{proof}

\section{Separating variables in smooth weights}\label{app:weights}
This appendix justifies the separation of smooth weights in the
completed mean-square estimate of Section~\ref{sec:reflection}, and
the treatment of kernels depending on the row in the Poisson
reductions of Sections~\ref{sec:initialization} and~\ref{sec:transfer-proof}.

\subsection{Separating the variables}
We use Mellin inversion to separate the variables of a smooth
weight, with coefficient bounds controlled by finitely many derivatives.

\begin{lemma}\label{lem:smooth}
Fix a box $\mathcal I=I_1\times\cdots\times I_d$, where each $I_j$ is a
compact interval in $(0,\infty)$. Every $\mathcal K\in C_c^\infty((0,\infty)^d)$
supported in $\mathcal I$ has a representation
\begin{equation}\label{eq:smooth-separation}
 \mathcal K(\mathbf x)=\int_{\mathbb R^d}b(\mathbf t)
       \prod_{j=1}^d x_j^{\mathrm i t_j}\dd\mathbf t,
 \qquad x_j>0.
\end{equation}
For $J\ge0$ and every even integer $q>J+d$, the coefficient satisfies
\[
 \int_{\mathbb R^d}|b(\mathbf t)|(1+|\mathbf t|)^J\dd\mathbf t
 \ll_{\mathcal I,J,q,d}\|\mathcal K\|_{C^q(\mathcal I)}.
\]
\end{lemma}
\begin{proof}
Take the Fourier transform in logarithmic coordinates:
\[
 b(\mathbf t)=\frac1{(2\pi)^d}\int_{\mathbb R^d}
 \mathcal K(e^{y_1},\ldots,e^{y_d})
 e^{-\mathrm i\mathbf t\cdot\mathbf y}\dd\mathbf y.
\]
Applying Fourier inversion to this gives \eqref{eq:smooth-separation}. Since the intervals $I_j$ are fixed and bounded away from zero, the $C^q$ norm in logarithmic coordinates is bounded by a constant times
$\|\mathcal K\|_{C^q(\mathcal I)}$. Applying
$(1-\Delta_{\mathbf y})^{q/2}$ under the integral therefore gives
\[
 |b(\mathbf t)|\ll_{\mathcal I,q,d}
 (1+|\mathbf t|)^{-q}\|\mathcal K\|_{C^q(\mathcal I)}
 \qquad(q\ge0\text{ even}).
\]
Multiplication by $(1+|\mathbf t|)^J$ and integration proves the
bound when $q>J+d$. See also the multivariable Mellin formulation
in \cite[\S10.1, (130)--(131)]{PY19}.
\end{proof}

In our applications, the weight has the more specific form
\[
 \mathcal K_R(\mathbf x)=\prod_{j=1}^dW_j(x_j)
 F\Bigl(R\prod_{j=1}^d x_j^{a_j}\Bigr),\qquad R>0,
\]
where the exponents $a_j$ are fixed real numbers and
$W_j\in C_c^\infty((0,\infty))$ is supported in $I_j$.
Assume that $F\in C^\infty((0,\infty))$ satisfies
\[
 \|F\|_{A,q}:=\max_{0\le m\le q}\sup_{u>0}
 (1+u)^A\bigl|(u\partial_u)^mF(u)\bigr|<\infty
 \qquad(A\ge0,\ q\in\mathbb Z_{\ge0}).
\]
On the fixed box $\mathcal I$, the argument of $F$ is comparable
to $R$. The chain rule and the pointwise bound in the proof give
coefficients $b_R$ satisfying, for $A\ge0$ and even $q\ge0$,
\begin{equation}\label{eq:smooth-pointwise}
 |b_R(\mathbf t)|
 \ll (1+R)^{-A}(1+|\mathbf t|)^{-q}\|F\|_{A,q}
       \prod_{j=1}^d\|W_j\|_{C^q(I_j)}.
\end{equation}
Consequently, for $J\ge0$ and even $q>J+d$,
\begin{equation}\label{eq:smooth-weight-bound}
 \int_{\mathbb R^d}|b_R(\mathbf t)|(1+|\mathbf t|)^J\dd\mathbf t
 \ll (1+R)^{-A}\|F\|_{A,q}
       \prod_{j=1}^d\|W_j\|_{C^q(I_j)}.
\end{equation}
The constants depend only on $A,J,q,d$, the intervals $I_j$, and
the exponents $a_j$. Thus separation preserves the decay in $R$,
and its cost is controlled by finitely many derivatives of the
original weights.

\subsection{Weights depending on the row}
The next lemma extends mean-square bounds for a common test
function to smooth kernels depending on the row, with losses controlled
by uniform bounds on their derivatives.

Fix compact intervals $I\subset\operatorname{int}I_*$ in $(0,\infty)$
and an integer $m\ge0$.

\begin{lemma}\label{lem:smooth-mean-square}
Consider two families of finite sums
\[
 S_{j,r}(U)=\sum_n a_{j,r}(n)U(x_{j,r,n}),\qquad
 j=1,2,\quad x_{j,r,n}>0,
\]
with a finite set of rows $r$ and nonnegative weights $w_r$.
Suppose that, for every $U\in C_c^\infty(I_*)$,
\[
 \sum_r w_r|S_{j,r}(U)|^2\le M_j\|U\|_{C^m(I_*)}^2,
 \qquad j=1,2.
\]
Then for any $|c_r|\le w_r$ and smooth kernels $\mathcal K_r$
supported in $I^2$,
\begin{equation}\label{eq:coupled-mean-square}
 \Bigl|\sum_r c_r\sum_{n_1,n_2}
 a_{1,r}(n_1)\overline{a_{2,r}(n_2)}
 \mathcal K_r(x_{1,r,n_1},x_{2,r,n_2})\Bigr|
 \ll_{I,I_*,m}\sqrt{M_1M_2}\,
       \sup_r\|\mathcal K_r\|_{C^{2m+4}(I^2)}.
\end{equation}
\end{lemma}
\begin{proof}
Choose a real $V\in C_c^\infty(I_*)$ equal to one on $I$, and set
$U_t(x)=V(x)x^{\mathrm i t}$.
Apply \eqref{eq:smooth-separation} of Lemma~\ref{lem:smooth}
to each $\mathcal K_r$ with $d=2$ and $\mathcal I=I^2$.
Replacing the second Mellin variable by its negative and multiplying
by $V(x)V(y)$, which equals one on the support of $\mathcal K_r$, gives
\[
 \mathcal K_r(x,y)=\int_{\mathbb R^2}b_r(s,t)
 U_s(x)\overline{U_t(y)}\,ds\,dt,
\]
with the following uniform bound, obtained from the pointwise
coefficient estimate in the proof of Lemma~\ref{lem:smooth} with
$d=2$ and $\mathcal I=I^2$:
\[
 \sup_r|b_r(s,t)|\ll_{I,q}
 \frac{\sup_r\|\mathcal K_r\|_{C^q(I^2)}}{(1+|s|+|t|)^q}
 \qquad(q\ge0\text{ even}).
\]
The hypothesis applies to each $U_t$, and
$\|U_t\|_{C^m(I_*)}\ll_{I,I_*,m}(1+|t|)^m$. Taking the common bound for $b_r$ before integrating and applying Cauchy--Schwarz in $r$ therefore bounds the absolute value in \eqref{eq:coupled-mean-square} by a constant depending on $I,I_*,m,q$ times
\begin{equation}\label{eq:kernel-bilinear-integral}
 \sqrt{M_1M_2}\sup_r\|\mathcal K_r\|_{C^q(I^2)}
 \int_{\mathbb R^2}
 \frac{(1+|s|)^m(1+|t|)^m}{(1+|s|+|t|)^q}\,ds\,dt.
\end{equation}
Taking $q=2m+4$ makes the integral converge and proves the claim.
\end{proof}

\subsection{Recombining the separated sums}
We use weighted Cauchy--Schwarz to pass from mean-square bounds
for the separated sums to a bound for their weighted integral.

\begin{lemma}\label{lem:recombine}
Let $r,\iota$ range over finite sets, let $d\ge1$, and let $w_r\ge0$.
Suppose the measurable coefficients $c_{\iota,r}$ satisfy
\[
 |c_{\iota,r}(\mathbf t)|\le b_\iota(\mathbf t),\qquad
 M:=\sum_\iota\int_{\mathbb R^d}b_\iota(\mathbf t)\dd\mathbf t<\infty,
\]
where $b_\iota:\mathbb R^d\to[0,\infty)$.
For measurable $F_{\iota,\mathbf t}(r)$, whenever the right-hand
side is finite, one has
\begin{equation}\label{eq:integral-mean-square}
 \sum_r w_r\Bigl|\sum_\iota\int c_{\iota,r}(\mathbf t)
 F_{\iota,\mathbf t}(r)\dd\mathbf t\Bigr|^2
 \le M\sum_\iota\int b_\iota(\mathbf t)
       \sum_r w_r|F_{\iota,\mathbf t}(r)|^2\dd\mathbf t.
\end{equation}
The same assertion holds for finite sums with the integrals omitted.
\end{lemma}
\begin{proof}
For each fixed $r$, weighted Cauchy--Schwarz gives
\[
 \Bigl|\sum_\iota\int c_{\iota,r}(\mathbf t)
 F_{\iota,\mathbf t}(r)\dd\mathbf t\Bigr|^2
 \le M\sum_\iota\int b_\iota(\mathbf t)
 |F_{\iota,\mathbf t}(r)|^2\dd\mathbf t.
\]
Multiply by $w_r$ and sum over $r$ to obtain
\eqref{eq:integral-mean-square}. The same argument with sums in
place of integrals proves the finite version.
\end{proof}

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\end{document}
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