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  pdftitle={Equidistribution of Primitive Quartic Torus Packets for Arbitrary Orders},
  pdfauthor={OpenAI}}

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\title{Equidistribution of primitive quartic torus packets\\
  for arbitrary orders}
\author{OpenAI}
\date{October 2026}

\begin{document}
\maketitle

\begin{abstract}
Let $K$ range over primitive totally real quartic fields and let
$\OO\subseteq\OO_K$ be any order, so that $|\Disc(\OO)|$ can grow
both through the field discriminant and through the order index.
For each $K,\OO$, and each choice of ordering of the four real
embeddings, take the complete finite packet of compact diagonal
orbits in $X_4 = \SL_4(\Z)\backslash\SL_4(\R)$ attached to the
invertible $\OO$-ideal classes, with all coordinate sign translates
and each orbit weighted by its diagonal-orbit volume. We prove that
these packets equidistribute with respect to the Haar probability
$m_4$ as $|\Disc(\OO)|\to\infty$, with no escape of mass. The
obstruction in composite degree four is that measure classification
leaves, as candidates for a limit, properly embedded block-diagonal
orbits of type $2+2$; we exclude them using a cubic-resolvent
estimate that remains uniform in the order index.
\end{abstract}

\tableofcontents

\section{Guide for the reader}\label{sec:guide}

This section explains what is being proved, and why, without using
the machinery that appears in the technical body.

\subsection*{Lattices and the space $X_4$}

A \emph{lattice} in $\R^4$ is a discrete grid, the set of all integer
combinations of four linearly independent vectors. Two lattices are
considered the same if one is a positive multiple of the other;
normalizing the covolume to one, every lattice can be written
$\Lambda = \Z^4 g$ for some $g\in\SL_4(\R)$, and two such matrices
give the same lattice exactly when they differ by left multiplication
by $\SL_4(\Z)$. The quotient
\[
  X_4=\SL_4(\Z)\backslash\SL_4(\R)
\]
is therefore the space of covolume-one lattices in $\R^4$. It carries
a natural probability measure $m_4$, the $\SL_4(\R)$-invariant Haar
probability. Picking a lattice uniformly at random, in the sense of
$m_4$, is the same as picking its shape
``uniformly at random''.

Let $A_4$ be the group of positive diagonal matrices in $\SL_4(\R)$.
It has three free real parameters and acts on $X_4$ by right
multiplication: a diagonal matrix stretches or shrinks each coordinate
direction independently, keeping the total covolume fixed. Direct
computations of distances in $X_4$ collapse to a
three-dimensional-dynamics problem under this action, and $A_4$ is the
central object of the present paper.

\subsection*{Quartic fields and packets}

Let $K$ be a number field of degree $4$ over $\Q$. It is
\emph{totally real} if all four embeddings $K\hookrightarrow\R$ have
image in $\R$ (no complex conjugate pairs). It is \emph{primitive}
if there is no field strictly between $\Q$ and $K$:
there is no intermediate subfield $\Q\subsetneq E\subsetneq K$.
The primitive totally real quartics are exactly the quartics whose
Galois closure has Galois group $\mathfrak A_4$ or $S_4$ over $\Q$,
as opposed to a smaller group compatible with a quadratic or
non-primitive quartic subfield.

An \emph{order} in $K$ is a ring $\OO\subseteq K$ that is a full
$\Z$-lattice of rank $4$ and contains $1$. The largest such ring is
the ring of integers $\OO_K$. An \emph{ideal} $\mathfrak a\subseteq\OO$
is a full-rank $\OO$-submodule; it is \emph{invertible} if
$\mathfrak a\mathfrak b=\OO$ for some other fractional ideal
$\mathfrak b$. The ordinary Picard group $\Pic(\OO)$ is the group of
invertible fractional ideals modulo principal ones; it is finite by the
unit theorem and the finiteness of the class number.

Choosing an ordering $\sigma=(\sigma_1,\ldots,\sigma_4)$ of the four
real embeddings of $K$ turns $K$ into a $\Q$-dense subspace of
$\R^4$. Each invertible fractional $\OO$-ideal $I$ then embeds into
$\R^4$ as a lattice $\sigma(I)$ of covolume $\sqrt{|\Disc(\OO)|}\Nm I$,
and its positive-scaling class is a point of $X_4$. As $I$ ranges over
a system of representatives of $\Pic(\OO)$, the resulting points move
under $A_4$ along compact orbits; collecting all of them gives a finite
set of compact $A_4$-orbits in $X_4$, which we call the \emph{packet}
attached to $(K,\OO,\sigma)$.

Coordinate signs (the sixteen diagonal matrices in
$\{\pm1\}^4$) are also included in the packet; they make the packet
independent of which representatives of $\Pic(\OO)$ one chooses.
The packet is assigned a probability measure
$\mu_{\OO,\sigma}$ by weighting each $A_4$-orbit by its Haar volume
and normalizing. That measure is what the main theorem is about.

\subsection*{The main theorem in plain words}

Take any sequence of triples $(K_i,\OO_i,\sigma_i)$ of primitive
totally real quartic fields, arbitrary orders in them, and orderings of
their real embeddings, so that $|\Disc(\OO_i)|\to\infty$. Then the
packet measures $\mu_{\OO_i,\sigma_i}$ converge weakly to the Haar
probability $m_4$. In other words, the arithmetic orbits fill the
lattice space uniformly in the limit, much as equally spaced multiples
of an irrational number fill the circle. There is no escaping mass in
the cusp.

What makes the theorem non-trivial is that the order index
$[\OO_K:\OO]$ can grow without bound while the field $K$ remains fixed.
In that case the packet orbits are extremely narrow along certain
$A_4$-directions, and their closure would naively be much smaller than
$X_4$. Uniform control over the ramified primes (those dividing the
order discriminant) is exactly where the arithmetic work lies.

\subsection*{Formal statement}

Fix the notation of the previous paragraph. Write
\[
  R=\OO_K,\qquad D=|\Disc(K)|,\qquad q=[R:\OO],\qquad
  X=q\sqrt D=\sqrt{|\Disc(\OO)|},\qquad
  \kappa=\Res_{s=1}\zeta_K(s).
\]

\begin{theorem}[Main theorem]\label{thm:main}
Let $(K_i,\OO_i,\sigma_i)$ be any sequence as above with
$|\Disc(\OO_i)|\to\infty$. Then
\[
  \int_{X_4} f\, d\mu_{\OO_i,\sigma_i}
    \longrightarrow \int_{X_4} f\, dm_4
  \qquad\text{for every }f\in C_c(X_4).
\]
Moreover, there is no escape of mass: for every $\eps>0$ there is a
compact $\mathcal K\subset X_4$ with
$\mu_{\OO_i,\sigma_i}(\mathcal K)\ge 1-\eps$ for all sufficiently
large $i$.
\end{theorem}

\section{History and context}\label{sec:history}

For real quadratic fields, compact diagonal orbits in $X_2$ are closed
geodesics on the modular surface, and equidistribution is a form of
Linnik's ergodic method. Linnik and Skubenko established distribution
of positive-discriminant geodesics under a quadratic-residue condition
at a fixed prime; Duke removed that condition using half-integral
weight forms; Chelluri and later Einsiedler--Lindenstrauss--Michel--
Venkatesh extended the result to all positive nonsquare order
discriminants by entropy methods. In higher rank, ELMV developed the
discriminant and volume framework connecting separation of periodic
torus orbits to entropy; their cubic theorem covers packets attached
to nonmaximal totally real cubic orders, and their higher-dimensional
analysis identifies that intermediate homogeneous limits can occur.

For quartic packets, Khayutin obtained entropy bounds for
maximal-order packet measures with two-transitive Galois groups,
including $\mathfrak A_4$ and $S_4$, thereby establishing
equidistribution in the maximal-order case. Wieser--Yang extended
this to certain maximal-order quartic packets with a quadratic
subfield, under extra hypotheses. For fixed prime degree
$p\ge 5$, Lemke Oliver--Thorner--Zaman proved equidistribution for
packets attached to orders of totally real degree-$p$ fields, up to
an exceptional set of fields; a separate preprint treats every
packet in each such degree and allows arbitrary local module types.
The present argument adapts that ordinary-vector method, and must
handle two extra phenomena: the order index can be arbitrarily large
even with the field fixed, and the rank-four space $X_4$ admits proper
positive-entropy homogeneous limits, which in prime degree are ruled
out by the classification theorem (Einsiedler--Katok--Lindenstrauss).

The exclusion of intermediate limits by sieve-theoretic input has a
precedent in Khayutin's joint equidistribution of CM points. Local
Fourier analysis for nonmaximal orders appears in the cubic work of
ELMV. Here the object being transformed is a specific quadratic
product map on the exterior lattice of the order; the resulting
estimate controls arbitrary conductor depth in a cubic-resolvent
count.

\section{Notation and conventions}\label{sec:conventions}

Throughout, $K$ is a primitive totally real quartic field, $\OO$ an
order in $K$, and $\sigma=(\sigma_1,\ldots,\sigma_4)$ an ordering of
its real embeddings. With
\[
  R=\OO_K,\quad D=|\Disc(K)|,\quad q=[R:\OO],\quad
  X=q\sqrt D,\quad \kappa=\Res_{s=1}\zeta_K(s),
\]
the order discriminant satisfies $|\Disc(\OO)|=X^2$. We write
$a_K(n)$ for the number of nonzero integral $R$-ideals of norm $n$,
and $d_4(n)$ for the four-fold divisor function.

The notation $\Nm_{E/k}$ denotes the algebraic field norm. Ideal
norms $\Nm(\mathfrak a)$ are positive absolute norms, extended
multiplicatively to fractional ideals. For a rational prime $p$ a
subscript $p$ denotes tensoring a field with $\Q_p$, a module with
$\Z_p$, or a lattice with $\Z_p$. For a prime $\mathfrak p$ or
$\mathfrak q$ of a local field, $v_{\mathfrak p}$ and $v_{\mathfrak q}$
are the integer valuations ($1$ on a uniformizer), and $e,f$ are the
associated ramification index and residue degree. A normalized
extension of $v_p$ has $v_p(p)=1$ and $|p|_p=p^{-1}$.

All logarithms are natural, and $\log_+ t=\max(0,\log t)$ for
$t>0$. An upper bound $X^{o(1)}$ means $O_\eta(X^\eta)$ for every
fixed $\eta>0$. Unless stated otherwise, all constants are uniform
in $K$, $\OO$, and the ordering of the embeddings. In the exterior
estimates a nonzero integer trace $m$ and a bounded real region are
fixed before $X\to\infty$; permitted dependence on them is stated in
the relevant results.

For $r>0$ we write $B(r)\subset \SL_4(\R)$ for the open identity
neighborhood $\max_{i,j}|h_{ij}-\delta_{ij}|<r$, and
$H_4=\{u\in\R^4:\sum u_i=0\}$, $A_4=\{\diag(e^{u_1},\ldots,e^{u_4}):
u\in H_4\}$.

\section{Proof strategy}\label{sec:strategy}

The proof has four interlocking parts, each corresponding to one of
the sections that follow.

\emph{1. Ordinary vectors (Sections~\ref{sec:packets}--\ref{sec:ordinary}).
}
The packet is explicitly unfolded as a weighted sum over integral
$R$-ideals, so that Borel functions on $X_4$ integrate against the
packet against ordinary lattice sums. This yields tightness, and a
local ball bound $\mu(xB(r))\le C_\Omega r^4$ that depends only on
$m$ and a compact set of centers.

\emph{2. Entropy and homogenization (Section~\ref{sec:rigidity}).}
The compact ball bound propagates to a uniform positive-entropy lower
bound for every $A_4$-ergodic component. Einsiedler--Katok--
Lindenstrauss then says each positive-entropy component is
homogeneous, i.e.\ is the invariant measure on a single closed orbit
of some closed connected subgroup $J$ containing $A_4$. A finite-
volume argument reduces the possible $J$ to either $\SL_4(\R)$ or a
block-diagonal $2+2$ subgroup.

\emph{3. Excluding the obstruction (Sections~\ref{sec:exterior}--
\ref{sec:sieve}).}
A block-diagonal orbit forces every lattice in it to carry an exterior-
square vector supported in one of three coordinate planes
$V_b\subset \bigwedge^2\R^4$. A cubic-resolvent count in the
exterior square of the order produces a bound $O(\kappa\sqrt D\,
\mathrm{area})$ for such vectors in a small rectangle of the
trace-$m$ hyperplane. Local Fourier analysis makes this bound uniform
in the order index; a Selberg sieve inserts the unramified local
factors and recovers $\kappa$.

\emph{4. Conclusion (Section~\ref{sec:conclusion}).}
Unfolding the diagonal flow against the exterior count turns the
block-diagonal $2+2$ alternatives into null sets, so no proper
homogeneous component can carry positive mass. Combined with part 2,
this identifies every subsequential limit as $m_4$.

\section{Packets from arbitrary orders}\label{sec:packets}

We now build the packet measure $\mu_{\OO,\sigma}$ carefully. The
construction has three components: a finite set of abstract lattices
attached to each ideal class of $\OO_K$, a real parameter in the unit
torus $T_\infty$, and a $\Z$-integrality normalization that keeps
covolumes equal to one. All the order-index dependence is buried in
the first component, and will turn out to be exactly what makes the
packet's arithmetic visible at ramified primes.

\subsection{The finite local choices}

Choose one fractional $R$-ideal $I$ in each ordinary ideal class of
$R=\OO_K$, and fix the notation $h_K$ for the class number. At each
rational prime $p$ choose a generator $I_p=\xi_p R_p$ with
$\xi_p\in K_p^\times$. Such a generator exists because each local
factor of $R_p$ is a discrete valuation ring.

Let $\mathscr L_{\OO}(I)$ be the set of full lattices $L\subseteq I$
whose completions satisfy
\begin{equation}\label{eq:local-choice}
  L_p=\xi_p t_p \OO_p, \qquad t_p\in R_p^\times,
  \qquad\text{for every prime }p.
\end{equation}
This set does not depend on the choice of the generators $\xi_p$:
replacing $\xi_p$ by another generator changes $t_p$ by a unit.

\begin{lemma}\label{lem:local-labels}
The set $\mathscr L_{\OO}(I)$ is finite, and
\begin{equation}\label{eq:label-count}
  N_{\OO}:=\#\mathscr L_{\OO}(I)=\prod_{p\mid q}[R_p^\times:\OO_p^\times].
\end{equation}
It is independent of $I$. Every $L\in\mathscr L_{\OO}(I)$ satisfies
$[I:L]=q$ and $RL=I$, and is a proper invertible fractional
$\OO$-ideal. Every ordinary class of proper invertible fractional
$\OO$-ideals is represented, up to multiplication by a scalar in
$K^\times$, by some member of one of the sets
$\mathscr L_{\OO}(I)$. Uniform probability on $\mathscr L_{\OO}(I)$
is induced from Haar probability on $R_p^\times$ for each
$p\mid q$.
\end{lemma}

\begin{proof}
The additive group $R/\OO$ has order $q$, so $qR\subseteq\OO$. In
particular, for every prime $p$ and every unit $t_p\in R_p^\times$ we
have
\[
  qI_p = \xi_p t_p qR_p
    \subseteq \xi_p t_p \OO_p
    \subseteq I_p .
\]
At $p\nmid q$ the middle lattice equals $I_p$. At the finitely many
primes above $q$, the image of $\xi_p t_p\OO_p$ is a submodule of the
finite module $I_p/qI_p$. The primary decomposition of $I/qI$
identifies its $p$-primary component with $I_p/qI_p$. A choice of
these local submodules for each $p\mid q$ therefore patches to a
unique subgroup of $I/qI$, whose inverse image in $I$ is the unique
lattice with the prescribed completions. This completes
finiteness and defines a natural bijection between
$\mathscr L_{\OO}(I)$ and $\prod_{p\mid q} (R_p^\times/\OO_p^\times)$.

The stabilizer of $\OO_p$ under left multiplication by $R_p^\times$
is $\OO_p^\times$: if $t_p\OO_p=\OO_p$, then applying the equality and
its inverse to $1$ gives $t_p,t_p^{-1}\in\OO_p$. The converse is
immediate. Uniform probability on $\mathscr L_{\OO}(I)$ therefore
comes from Haar probability on $R_p^\times$ for each $p\mid q$.
Finally, the index of the lattice in \eqref{eq:local-choice} is
$[R_p:\OO_p]$, and its $R_p$-span is $I_p$ because $1\in\OO_p$;
taking products of the local indices over $p$ and comparing
completions gives $[I:L]=q$ and $RL=I$.

For invertibility, recall that a full fractional $\OO$-ideal $J$ is
invertible if and only if $J_p=\eta_p \OO_p$ for some
$\eta_p\in K_p^\times$ at every $p$. This is standard local-global
principality: the forward direction follows by writing
$JJ^{-1}=\OO$ and using the Chinese remainder theorem to choose one
generator over the semilocal ring $\OO_p$; the reverse direction
uses $J^{-1}=\{a\in K:aJ\subseteq\OO\}$ and $(J^{-1})_p=
\eta_p^{-1}\OO_p$. The lattices $L\in\mathscr L_{\OO}(I)$ are
stable under $\OO_p$ at every prime, hence are fractional
$\OO$-ideals satisfying local principality, hence are invertible,
and their multiplier order is $\OO$ (if $aJ\subseteq J$ for an
invertible $J$, multiplying by $J^{-1}$ gives $a\in\OO$).

Conversely, let $J$ be a proper invertible fractional $\OO$-ideal.
Its $R$-span $RJ$ is a fractional $R$-ideal. Multiplying by a scalar
in $K^\times$ we may arrange $RJ=I$ for the chosen representative of
that class. If $J_p=\eta_p\OO_p$, then $\eta_p R_p=I_p=\xi_p R_p$, so
$\eta_p=\xi_p t_p$ for some unit $t_p\in R_p^\times$. Thus $J$
belongs to $\mathscr L_{\OO}(I)$.
\end{proof}

\subsection{Real parameters, identification, and volume}

Let $U=R^\times$ and $U^+\subseteq U$ be its totally positive
subgroup. Put
\[
  T_\infty = \{v=(v_1,\ldots,v_4)\in(\R^\times)^4:
             \textstyle\prod_i|v_i|=1\}.
\]
We give $T_\infty$ counting measure on its sixteen sign components,
times Lebesgue measure on the logarithm coordinates
$u_i=\log|v_i|$ with $u_4=-u_1-u_2-u_3$. This is the ``logarithmic
Haar'' measure $dv$. By the unit theorem the image
\[
  \ell(U)=\{(\log|\sigma_1(\epsilon)|,\ldots,
             \log|\sigma_4(\epsilon)|):\epsilon\in U\}
\]
is a full lattice in the hyperplane $\sum u_i=0$; its covolume (as a
subgroup of the logarithmic hyperplane, with coordinate measure from
the first three coordinates) equals the ordinary regulator $R_K$.
Its kernel on $U$ is $\{1,-1\}$. Let $\mathscr F_U$ be a measurable
fundamental domain for $\sigma(U)$ in $T_\infty$. Reducing logarithms
modulo $\ell(U)$ leaves sixteen sign choices, on which $\{1,-1\}$
acts by pairing opposite signs.

The analytic class-number formula in signature $(4,0)$, with two
roots of unity, gives
\begin{equation}\label{eq:unit-volume}
  h_K\vol(\mathscr F_U)=8h_K R_K=\sqrt D\,\kappa,
\end{equation}
which we now use.

For $L\in\mathscr L_{\OO}(I)$ set
\[
  c_I = (X\Nm I)^{-1/4}.
\]
The covolume formula $\covol(\sigma(I))=\sqrt D\,\Nm I$ together with
$[I:L]=q$ shows that $c_I\sigma(L)v$ has covolume one for every
$v\in T_\infty$; choosing any positively oriented basis
gives a point of $X_4$.

\begin{proposition}\label{prop:packet-model}
The map
\[
  (L,v)\longmapsto c_I\sigma(L)v,
  \qquad L\in\mathscr L_{\OO}(I),\ v\in T_\infty,
\]
descends to a bijection
\[
  \Bigl(\coprod_{I} \mathscr L_{\OO}(I)\times T_\infty\Bigr)
  \big/ U \ \xrightarrow{\sim}\
  \bigcup_{\mathcal Y\in\mathcal P_{\OO,\sigma}} \mathcal Y
  \subseteq X_4
\]
of the disjoint union of these parameter sets modulo the
$\epsilon\in U$ action
\begin{equation}\label{eq:unit-action}
  \epsilon\cdot(L,v)=(\epsilon L,\,\sigma(\epsilon)^{-1}v),
\end{equation}
onto the union of packet orbits. Counting measure on the $L$'s times
$dv$ descends to the sum of the diagonal Haar measures of those
orbits. In particular
\begin{equation}\label{eq:total-volume}
  \sum_{\mathcal Y\in\mathcal P_{\OO,\sigma}} \vol(\mathcal Y)
    = N_{\OO}\sqrt D\,\kappa .
\end{equation}
Writing $\mu_*=\mu_{\OO,\sigma}$, every nonnegative Borel
$\Phi$ on $X_4$ satisfies the integration formula
\begin{equation}\label{eq:packet-integration}
  \int_{X_4}\Phi\, d\mu_*
  =\frac{1}{h_K N_{\OO}\vol(\mathscr F_U)}
   \sum_I \sum_{L\in\mathscr L_{\OO}(I)}
   \int_{\mathscr F_U} \Phi(c_I\sigma(L)v)\,dv ,
\end{equation}
with both sides possibly infinite.
\end{proposition}

\begin{proof}
The $U$-action \eqref{eq:unit-action} preserves the parameter set,
because $\epsilon I=I$ for $\epsilon\in U$, and preserves the image
lattice. To see surjectivity, use Lemma~\ref{lem:local-labels}: it
shows every ordinary class occurs. Two ideals differing by
multiplication by $\alpha\in K^\times$ produce normalized embedded
lattices differing by the diagonal
$\sigma_i(\alpha)/|\Nm_{K/\Q}(\alpha)|^{1/4}$, an element of
$T_\infty$; finally $T_\infty$ is the union of the sixteen sign
translates of $A_4$. This produces surjectivity.

For injectivity, suppose two displayed normalized lattices coincide.
Absorbing the ratio of their positive scalars, we obtain a diagonal
map carrying $\sigma(L)$ onto $\sigma(L')$. Choosing a nonzero
$b\in L$ and $b'\in L'$ its image, all four coordinates of
$\sigma(b)$ are nonzero, so the diagonal entries must equal
$\sigma_i(b')/\sigma_i(b)=\sigma_i(b'/b)$. The map is multiplication
by $\alpha=b'/b$, and $L'=\alpha L$. Taking $R$-spans gives
$I'=\alpha I$; the chosen representatives of classes force $I'=I$
and $\alpha I=I$, so $\alpha\in U$. The scalar normalizations now
coincide, and the two parameters are related by the
$\epsilon\cdot$ action \eqref{eq:unit-action}. This proves
injectivity, including all sign identifications.

For the volume formula, first observe that the packet contains
$16N_{\OO}$ finite choices above each $I$ (label times sign
pattern). Each $U$-orbit of these choices has $[U:\OO^{\times,+}]$
elements, where $\OO^{\times,+}=\OO^\times\cap U^+$
($-1\notin \OO^{\times,+}$), and parametrizes one $A_4$-orbit. The
stabilizer of the label $L$ under $U$ is $\OO^\times$: this is
exactly the multiplier order, since $0\neq L$ has rank four. The
stabilizer of the sign-patterned label is $\OO^{\times,+}$. This is a
finite-index subgroup of $U$.

The periods of the resulting $A_4$-orbit are precisely
$\ell(\OO^{\times,+})$, which form a full lattice, so the orbit is
compact. Its Haar volume is
\[
  \covol(\ell(\OO^{\times,+}))
   =[\ell(U):\ell(\OO^{\times,+})]\,R_K
   =\frac{[U:\OO^{\times,+}]}{2}R_K .
\]
The final equality uses $\ker(\ell|_U)=\{1,-1\}$ and
$-1\notin \OO^{\times,+}$. The number of distinct orbits times this
common volume is
\[
  \frac{16h_K N_{\OO}}{[U:\OO^{\times,+}]}
   \cdot\frac{[U:\OO^{\times,+}]}{2}R_K
   =8h_K N_{\OO}R_K .
\]
By \eqref{eq:unit-volume} this equals $N_{\OO}\sqrt D\,\kappa$,
proving \eqref{eq:total-volume}.

For the integration formula, choose
$\mathscr L_{\OO}(I)\times \mathscr F_U$ as a fundamental domain for
the diagonal $U$-action: reducing $v$ into $\mathscr F_U$ transports
$L$ by the same unique unit, away from a null boundary set. These
domains have total measure $h_K N_{\OO}\vol(\mathscr F_U)$, which
equals the total volume just computed. Dividing by this quantity
proves \eqref{eq:packet-integration}.
\end{proof}

\section{Ordinary vectors and probability limits}\label{sec:ordinary}

The packet measure of Section~\ref{sec:packets} is a sum over lattice
vectors weighted by the local container probability at the primes
dividing $q$. The first goal of the present section is to express that
sum as a clean count over integral $R$-ideals. The second goal is to
extract, from that count, two geometric bounds: tightness of the packet
family in $X_4$ (a cusp-exit bound at scale $r^4$), and a local ball
bound $\mu(xB(r))\le C_\Omega r^4$ for lattice neighborhoods of
shrinking radius $r$. Both use only the ordinary (rank-one) Grassmannian
of the packet orbits, in contrast to the exterior-square analysis that
occupies the later sections.

\subsection{Containment probabilities and unfolding}

For a rational prime $p$, put $R_p=\prod_{\mathfrak p\mid p}R_{\mathfrak p}$,
where $R_{\mathfrak p}$ is the integer ring of the local completion
$K_{\mathfrak p}$. Then $v_{\mathfrak p}$ is the integer valuation
normalized to be $1$ on a uniformizer, and
$e_{\mathfrak p}=v_{\mathfrak p}(p)$, $f_{\mathfrak p}=[R/\mathfrak p:\Z/p\Z]$,
so that $\Nm\mathfrak p=p^{f_{\mathfrak p}}$ and
$\sum_{\mathfrak p\mid p}e_{\mathfrak p}f_{\mathfrak p}=4$.

For $p\mid q$, let $\boldsymbol l_p=(l_{\mathfrak p})_{\mathfrak p\mid p}$
be a tuple of nonnegative integers. The \emph{valuation shell} of this
tuple is the set of $x\in R_p$ with $v_{\mathfrak p}(x_{\mathfrak p})
=l_{\mathfrak p}$ in every component. Choose a single $x$ in this shell
and define
\begin{equation}\label{eq:local-prob}
  P_p(\boldsymbol l_p):=\P_{t_p\in R_p^\times}\{x\in t_p\OO_p\},
\end{equation}
with $\P$ denoting normalized Haar probability on the compact unit
group $R_p^\times$. This is independent of the choice of $x$: the unit
group acts transitively on the shell.

For a nonzero integral $R$-ideal $\mathfrak a$, put
\begin{equation}\label{eq:ideal-weight}
  P(\mathfrak a):=\prod_{p\mid q}
    P_p\bigl((v_{\mathfrak p}(\mathfrak a))_{\mathfrak p\mid p}\bigr),
\end{equation}
the empty product being one. From the packet model,
Lemma~\ref{lem:local-labels}, and independence of the local unit
choices, we obtain the fundamental containment formula
\begin{equation}\label{eq:membership}
  \P_{L\in\mathscr L_{\OO}(I)}\{b\in L\}
    = P((b)I^{-1}),
  \qquad 0\ne b\in I .
\end{equation}

For a nonnegative Borel function $f$ on $\R^4$ write
\[
  E_f(\Lambda)=\sum_{0\ne x\in\Lambda} f(x).
\]
This is a Borel function on $X_4$: in a local basis chart it is a
countable sum of Borel functions.

\begin{proposition}[Ordinary unfolding]\label{prop:ord-unfolding}
For every nonnegative Borel $f:\R^4\to[0,\infty]$,
\begin{equation}\label{eq:unfolding}
  \int_{X_4}E_f\, d\mu_*
  =\frac{1}{\sqrt D\,\kappa}
   \sum_{\substack{\mathfrak a\subseteq R\\\mathfrak a\ne 0}}
   P(\mathfrak a)\,\mathcal V_f\bigl(\Nm\mathfrak a/X\bigr),
\end{equation}
where $\mathcal V_f(z)$, $z>0$, is the integral over the surface of
absolute coordinate product $z$ with respect to logarithmic measure:
\begin{equation}\label{eq:fiber-integral}
  \mathcal V_f(z)=
   \sum_{\omega\in\{\pm 1\}^4}\int_{\R^3}
   f(\omega_1 e^{u_1},\omega_2 e^{u_2},\omega_3 e^{u_3},
     \omega_4 z e^{-u_1-u_2-u_3})\,du_1 du_2 du_3.
\end{equation}
Both sides may be infinite.
\end{proposition}

\begin{proof}
By the packet integration formula \eqref{eq:packet-integration}, average
$E_f$ over $L\in\mathscr L_{\OO}(I)$. The $N_{\OO}$ factor cancels
between the packet normalization and the uniform measure on
$\mathscr L_{\OO}(I)$. Using \eqref{eq:membership}, the left-hand side
of \eqref{eq:unfolding} becomes
\[
  \frac{1}{h_K\vol(\mathscr F_U)}
  \sum_I\int_{\mathscr F_U}
  \sum_{0\ne b\in I} P((b)I^{-1})
       f(c_I\sigma(b)v)\,dv .
\]
Fix $I$. The map $b\mapsto (b)I^{-1}$ takes $I\setminus\{0\}$
bijectively onto the set of integral ideals in the inverse class of
$I$, with each fiber being a single free $U$-orbit (two generators of
the same principal ideal differ by a unit). The coefficient
$P((b)I^{-1})$ is constant on that orbit. Integrating over $U$ with
the fundamental domain $\mathscr F_U$ gives
\[
  \int_{\mathscr F_U}\sum_{\epsilon\in U}
       f(c_I\sigma(\epsilon b)v)\,dv
    =\int_{T_\infty} f(c_I\sigma(b)v)\,dv .
\]
For $b\in I$ we have
\[
  |{\textstyle\prod_i} (c_I\sigma(b))_i|
    = c_I^4|\Nm_{K/\Q}b|
    =\Nm((b)I^{-1})/X .
\]
Combining $|{\textstyle\prod_i}v_i|=1$ with a translation and sign
relabeling identifies the last integral with
$\mathcal V_f(\Nm((b)I^{-1})/X)$, Jacobian one. Each integral ideal lies
in the inverse class of exactly one representative. Applying
\eqref{eq:unit-volume}, we obtain \eqref{eq:unfolding}.
\end{proof}

\subsection{The mass and moment of the local weights}

We need the total mass and a positive moment of the local weight
$P(\mathfrak a)$. Both come from a shell decomposition.

For $p\mid q$ write
\[
  q_p=[R_p:\OO_p],\qquad
  N_p(\boldsymbol l_p)=p^{\sum_{\mathfrak p\mid p}
                             f_{\mathfrak p}l_{\mathfrak p}},\qquad
  Z_p=\prod_{\mathfrak p\mid p}(1-p^{-f_{\mathfrak p}})^{-1}.
\]
For a collection $\boldsymbol l=(\boldsymbol l_p)_{p\mid q}$ write
\[
  d(\boldsymbol l)=\prod_{p\mid q}N_p(\boldsymbol l_p),\quad
  P(\boldsymbol l)=\prod_{p\mid q}P_p(\boldsymbol l_p),\quad
  Z_q=\prod_{p\mid q}Z_p .
\]
Thus $d(\boldsymbol l)$ is the norm of the part of the ideal
supported above $q$, and $\prod_{p\mid q}q_p=q$.

\begin{lemma}\label{lem:shell-mass}
The local shell weights satisfy
\begin{align}
  \sum_{\boldsymbol l}\frac{P(\boldsymbol l)}{d(\boldsymbol l)}
     &=\frac{Z_q}{q}, \label{eq:shell-mass}\\
  \sum_{\boldsymbol l}
     P(\boldsymbol l)d(\boldsymbol l)^{-1+1/8}
     &\le C_\eta\, \frac{q^\eta}{q}
    \qquad(\eta>0), \label{eq:shell-moment}\\
  \sum_{d(\boldsymbol l)>X^{1/8}}\frac{P(\boldsymbol l)}{d(\boldsymbol l)}
     &\le \frac{C}{q}\, X^{-1/128}, \label{eq:shell-tail}
\end{align}
with $C_\eta$ depending only on $\eta$ and $C$ absolute.
\end{lemma}

\begin{proof}
Normalize additive Haar measure $m_p$ on $R_p$ by $m_p(R_p)=1$. In a
component $R_{\mathfrak p}$, the set of elements of valuation
$l_{\mathfrak p}$ has measure
$(1-p^{-f_{\mathfrak p}})p^{-f_{\mathfrak p}l_{\mathfrak p}}$. So the
shell of $\boldsymbol l_p$ in $R_p$ has measure
\begin{equation}\label{eq:shell-measure}
  m_p(\text{shell of }\boldsymbol l_p)
    =\frac{1}{Z_p N_p(\boldsymbol l_p)} .
\end{equation}
Its conditional measure is invariant under the transitive unit
action, hence is the image of unit Haar probability. By
\eqref{eq:local-prob} $P_p(\boldsymbol l_p)$ is the fraction of this
shell lying in $\OO_p$; hence summing \eqref{eq:shell-measure} over
the intersections with $\OO_p$, whose additive measure is $q_p^{-1}$,
gives
\[
  \sum_{\boldsymbol l_p}
    \frac{P_p(\boldsymbol l_p)}{N_p(\boldsymbol l_p)}
    =\frac{Z_p}{q_p}.
\]
Multiplying over $p\mid q$ proves \eqref{eq:shell-mass}.

For the moment, choose $x$ with additive Haar probability on $\OO_p$
and put $v_{\mathfrak p}=v_{\mathfrak p}(x_{\mathfrak p})$. The
projection $H_{\mathfrak p}$ of $\OO_p$ into $R_{\mathfrak p}$ contains
the embedded copy of $\Z_p$, because $1\in\OO_p$. Its induced measure
is normalized Haar on $H_{\mathfrak p}$. For $k\ge1$ the image of the
copy of $\Z_p$ modulo the ideal of valuation $k$ has at least
$p^{\lceil k/e_{\mathfrak p}\rceil}$ elements, so
\begin{equation}\label{eq:valuation-tail}
  \P_{\OO_p}\{v_{\mathfrak p}\ge k\}
    \le p^{-\lceil k/e_{\mathfrak p}\rceil}.
\end{equation}
Set $j=\max_{\mathfrak p\mid p}
\lceil v_{\mathfrak p}/e_{\mathfrak p}\rceil$. If $j\ge a\ge 1$, some
$v_{\mathfrak p}$ is at least $e_{\mathfrak p}(a-1)+1$;
\eqref{eq:valuation-tail} and a union bound over at most four
components give $\P_{\OO_p}\{j\ge a\}\le 4p^{-a}$. On the other hand
\[
  N_p(x)=p^{\sum_{\mathfrak p\mid p}f_{\mathfrak p}v_{\mathfrak p}}
        \le p^{j\sum_{\mathfrak p\mid p}e_{\mathfrak p}f_{\mathfrak p}}
        =p^{4j}.
\]
Bounds hold outside the null set where a component vanishes. The tail
sum for the expectation of $p^{j/2}$ gives
\[
  \E_{\OO_p}N_p(x)^{1/8}
  \le\E_{\OO_p}p^{j/2}
  =1+\sum_{a\ge1}(p^{a/2}-p^{(a-1)/2})\P\{j\ge a\}
  \le 1+4p^{-1/2}.
\]
Applying \eqref{eq:shell-measure} once more,
\[
  \sum_{\boldsymbol l}P(\boldsymbol l)d(\boldsymbol l)^{-1+1/8}
  =\frac{1}{q}\prod_{p\mid q}\left(Z_p\E_{\OO_p}N_p(x)^{1/8}\right).
\]
Each factor is at most $(1-p^{-1})^{-4}(1+4p^{-1/2})$, which is at
most $p^\eta$ for large $p$. Fixed finitely many small primes
contribute a constant. Since $\prod_{p\mid q}p\le q$, the moment bound
follows. Taking $\eta=1/128$ and using $q\le X$ gives the tail bound
\eqref{eq:shell-tail}.
\end{proof}

\subsection{The residue and prime-sum comparison}

The shell identity produces a factor $q^{-1}$. To retain $\kappa$ in
the weighted interval count below, we need comparison between the
Dedekind residue at $s=1$ and elementary prime sums, uniformly in the
field and the order.

\begin{lemma}\label{lem:ord-residue}
There are absolute constants $D_0>1$, $h_0>0$, $Y_0>1$, $C\ge 1$ such
that for every primitive totally real quartic field $K$:
\begin{enumerate}
\item if $D\ge D_0$,
\begin{equation}\label{eq:residue-comparison}
  C^{-1}\kappa\,\le\,h\zeta_K(1+h)\,\le\, C\kappa
  \qquad(0<h\le 4/\log D),
\end{equation}
and if $D<D_0$ the same inequalities hold with $h\le h_0$;
\item one has $\kappa^{-1}\le C\log(3D)$;
\item whenever $Y\ge\max\{Y_0,D^{1/4}\}$,
\begin{equation}\label{eq:prime-sum}
  \exp\Bigl(\sum_{p\le Y}\frac{a_K(p)}p\Bigr)
    \le C\kappa\log Y .
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
We begin with $D$ large. Stark's exceptional-zero theorem provides at
most one nontrivial zero of $\zeta_K$ in the rectangle
$\Re s\ge 1 - (4L)^{-1}$, $|\Im s|\le (4L)^{-1}$, where $L=\log D$;
such a zero is real and simple. Another lemma of Stark gives, in
degree four, that a real zero $\beta\ge 1-(96L)^{-1}$ would force a
quadratic subfield of $K$. Primitivity excludes that, so every
nontrivial zero $\rho$ satisfies
\begin{equation}\label{eq:zero-disk}
  |1-\rho|\ge c_0/L,\qquad c_0=1/192.
\end{equation}

For real $s>1$ the completed zeta function satisfies
\[
  S_K(s)=\sum_\rho\Re\frac1{s-\rho}
    =\frac1s+\frac1{s-1}+\frac L2+\frac{\zeta_K'}{\zeta_K}(s)+G(s),
\]
with $G(s)=2(\Gamma'(s/2)/\Gamma(s/2)-\log\pi)$. All terms
$\Re(s-\rho)^{-1}$ are nonnegative, since $\Re\rho\le1$. At
$s_0=1+L^{-1}$ the Euler product gives $\zeta_K'/\zeta_K(s_0)\le0$,
and $G$ is bounded on $[1,2]$, so $0\le S_K(s_0)\ll L$. The bound
persists on $1<s\le 1+4/L$: writing $\rho=\beta+i\gamma$ with
$\beta\le 1$, and using $\Re(s-\rho)^{-1}=(s-\beta)/|s-\rho|^2$ and
\eqref{eq:zero-disk}, we get
\[
  \Re\frac1{s-\rho}\le 4(1+3/c_0)^2\Re\frac1{s_0-\rho}.
\]
Hence $S_K(s)\ll L$ throughout $[1,1+4/L]$. Let
$\mathcal Z_K(s)=(s-1)\zeta_K(s)$; rearranging the previous formula
gives $\mathcal Z_K'/\mathcal Z_K=O(L)$ on the whole interval.
$\mathcal Z_K$ extends to $s=1$ with value $\kappa>0$, so integrating
its logarithmic derivative across the interval of length $\le 4/L$
gives \eqref{eq:residue-comparison}. Also $\zeta_K(s_0)\ge 1$, so
$\kappa\gg L^{-1}$.

For $D<D_0$ only finitely many fields occur (Hermite), and
$h\zeta_K(1+h)\to\kappa$ as $h\downarrow0$. A common small $h_0$ and
an enlarged $C$ finish this case, and give
$\kappa^{-1}\ll\log(3D)$. Choose $Y_0$ with $1/\log Y_0\le h_0$ and
let $h=1/\log Y$. Positivity in the Euler logarithm gives
$\sum_{p\le Y}a_K(p)p^{-1-h}\le\log\zeta_K(1+h)$. On the other hand
$a_K(p)\le 4$ and $1-p^{-h}\le h\log p$, so partial summation and the
Chebyshev bound give
$\sum_{p\le Y}a_K(p)(p^{-1}-p^{-1-h})\le 4h\sum_{p\le Y}
(\log p)/p\ll 1$. Combining,
$\sum_{p\le Y}a_K(p)/p\le\log\zeta_K(1+h)+O(1)\le\log(\kappa\log Y)+O(1)$,
which is \eqref{eq:prime-sum}.
\end{proof}

We also need a uniform Shiu bound. The following is
[Pollack, Theorem 1.1] specialized to $b\le d_4$ and $\beta=1/4$:

\begin{lemma}\label{lem:shiu}
There are absolute constants $C,u_0$ such that for every nonnegative
multiplicative $b$ with $b(n)\le d_4(n)$, every $u\ge u_0$, and every
$u^{1/2}\le v\le u$,
\begin{equation}\label{eq:shiu}
  \sum_{u-v<n\le u}b(n)
  \le C\frac{v}{\log u}\exp\Bigl(\sum_{p\le u}\frac{b(p)}p\Bigr).
\end{equation}
\end{lemma}

\subsection{Weighted intervals of ideals}

The four ingredients combine to give a uniform bound for weighted
ideal intervals. This is where the absence of a quadratic subfield
(via Stark) enters, and where the model $1+n^{-1/8}$ moment of
Lemma~\ref{lem:shell-mass} is used to dispose of ideals supported above
$q$.

\begin{proposition}\label{prop:weighted-intervals}
There is an absolute constant $C$ such that for every fixed $A\ge 1$
and $c>0$ there is a threshold $X_0(A,c)$ with the following property:
for every triple $(K,\OO,\sigma)$ with $X\ge X_0(A,c)$ and all reals
$x,y$ with
\[
  X^{3/4}\le x\le AX,\qquad cx\le y\le x,
\]
one has
\begin{equation}\label{eq:weighted-interval}
  \sum_{\substack{\mathfrak a\subseteq R,\ \mathfrak a\ne0\\
                  x-y<\Nm\mathfrak a\le x}}
  P(\mathfrak a)
   \le C\,\frac{\kappa y}{q}.
\end{equation}
\end{proposition}

\begin{proof}
For $c>1$ the range is empty; assume $0<c\le 1$. Set
\[
  b(n)=a_K(n)\,\one_{(n,q)=1}.
\]
Then $b$ is nonnegative and multiplicative, and $b\le d_4$. Split
every integral ideal uniquely into a part supported above $q$,
described by a tuple $\boldsymbol l$, and a part coprime to $qR$. The
left side of \eqref{eq:weighted-interval} is therefore
\begin{equation}\label{eq:ideal-split}
  \sum_{\boldsymbol l}P(\boldsymbol l)
    \sum_{(x-y)/d(\boldsymbol l)<n\le x/d(\boldsymbol l)} b(n).
\end{equation}

First take $d=d(\boldsymbol l)\le X^{1/8}$ and put $u=x/d$,
$v=y/d$. Then $u\ge X^{5/8}\ge D^{1/4}$, and $v\ge cu\ge u^{1/2}$
for $X$ large in terms of $c$, so \eqref{eq:shiu} applies. To remove
$q$-primes from the local Euler product, use
\[
  \log Z_q=\sum_{p\mid q}\frac{a_K(p)}p+O(1),\qquad
  \sum_{\substack{p\mid q\\p>u}}\frac{a_K(p)}p\ll 1,
\]
where the last uses $a_K(p)\le 4$, $u\ge X^{5/8}$, and $q\le X$.
Combining with \eqref{eq:prime-sum}, these tuples contribute at most
\[
  C\frac{\kappa y}{Z_q}
  \sum_{d\le X^{1/8}}\frac{P(\boldsymbol l)}{d(\boldsymbol l)}
  \le C\frac{\kappa y}{q}
\]
by \eqref{eq:shell-mass}.

Now consider $d>X^{1/8}$. For $z\ge 1$,
\[
  \sum_{n\le z}d_4(n)=\sum_{a_1a_2a_3a_4\le z}1
  \le z(1+\log z)^3 .
\]
Using this at $z=x/d$, the tail contributes at most
\[
  Cx(1+\log x)^3
  \sum_{d>X^{1/8}}\frac{P(\boldsymbol l)}{d(\boldsymbol l)}
  \le C\frac{x}{q}\,X^{-1/128}(1+\log x)^3
\]
by \eqref{eq:shell-tail}. Since $x\le AX$, for $X\ge A$ we have
$\log x\le2\log X$, and the ratio to $\kappa x/q$ is
$CX^{-1/128}(\log(3X))^4=o(1)$ by $\kappa^{-1}\ll\log(3D)$ and
$D\le X^2$. Enlarging the threshold in terms of $A,c$ makes this at
most $\kappa cx/q\le\kappa y/q$. The constant in the final bound does
not depend on $A,c$.
\end{proof}

\subsection{Tightness}

We apply the interval bound to cube neighborhoods of the origin
and to boxes away from the coordinate hyperplanes.

\begin{proposition}[Tightness]\label{prop:tightness}
There is an absolute constant $C$ such that for every fixed
$0<r<1$,
\begin{equation}\label{eq:cusp}
  \limsup_{X\to\infty}
  \mu_*\{\Lambda:\text{some } 0\ne v\in\Lambda
                    \text{ has }\|v\|_\infty<r\}
    \le C r^4 .
\end{equation}
The limit superior ranges over all primitive totally real quartic
fields, all their orders, and all orderings of their real embeddings.
Consequently, for any sequence of packets with $X_i\to\infty$ and
$\eps>0$, there is a compact $\mathcal K_\eps\subseteq X_4$ with
$\mu_i(\mathcal K_\eps)\ge 1-\eps$ for all large $i$, and every weak
subsequential limit is an $A_4$-invariant probability.
\end{proposition}

\begin{proof}
Let $f_r=\one_{[-r,r]^4}$. Direct integration in
\eqref{eq:fiber-integral} gives
\begin{equation}\label{eq:cube-fiber}
  \mathcal V_{f_r}(z)=\frac{16}{6}
   \bigl(\log_+(r^4/z)\bigr)^3 .
\end{equation}
Indeed, in a fixed sign component set $s_i=\log r-u_i$; the cube
conditions become $s_i\ge 0$ and $s_1+s_2+s_3\le\log(r^4/z)$, a
simplex of that volume divided by sixteen.

The count $E_{f_r}$ dominates the indicator in \eqref{eq:cusp}. Put
$T=r^4X$ and $T_0=X^{3/4}$. By unfolding \eqref{eq:unfolding} it
suffices to bound
\begin{equation}\label{eq:cube-sum}
  \frac{1}{\sqrt D\,\kappa}
  \sum_{\Nm\mathfrak a\le T} P(\mathfrak a)
  \bigl(\log(T/\Nm\mathfrak a)\bigr)^3 .
\end{equation}
The logarithmic factor is $\le(\log X)^3$ since $\Nm\mathfrak a\ge 1$
and $r<1$. Applying \eqref{eq:weighted-interval} at $x=y=T_0$, the
portion with $\Nm\mathfrak a\le\min\{T,T_0\}$ is bounded by
$C\kappa T_0(\log X)^3/q$, contributing $O(X^{-1/4}(\log X)^3)=o(1)$
after dividing by $\sqrt D\,\kappa=X\kappa/q$.

If $T>T_0$, put $A_{T_0}(t)=\sum_{T_0<\Nm\mathfrak a\le t}P(\mathfrak a)$
for $T_0\le t\le T$. The interval bound at $x=y=t$ gives
$A_{T_0}(t)\le C\kappa t/q$ since $T\le X$. Writing the log-cube as
an iterated integral,
\[
  \sum_{T_0<\Nm\mathfrak a\le T}P(\mathfrak a)
  \bigl(\log(T/\Nm\mathfrak a)\bigr)^3
  =3\int_{T_0}^T A_{T_0}(t)
     \bigl(\log(T/t)\bigr)^2\frac{dt}{t}
  \le C\frac{\kappa}{q}\int_0^T
     \bigl(\log(T/t)\bigr)^2 dt
  \le C\frac{\kappa T}{q}.
\]
Dividing by $X\kappa/q$ gives $O(r^4)$. If $T\le T_0$ this part is
absent. This proves \eqref{eq:cusp}.

Mahler's compactness criterion says
$\mathcal K_r=\{\Lambda\in X_4:\|v\|_\infty\ge r\ \forall
0\ne v\in\Lambda\}$ is compact. Choosing $r$ with $Cr^4<\eps/2$ and
enlarging $\mathcal K_r$ to include the finitely many supports of the
earlier measures in the sequence, we obtain
$\mu_i(\mathcal K_\eps)\ge 1-\eps$ for all large $i$. Tightness of
the whole sequence follows. Since both test functions $f(x)$ and
$f(xa)$ are in $C_c(X_4)$, the equality
$\int f\,d\mu_i=\int f(\cdot a)\,d\mu_i$ passes to weak limits,
which proves $A_4$-invariance.
\end{proof}

\begin{lemma}[Boxes away from hyperplanes]\label{lem:boxes}
Let $V_0\subseteq(\R^\times)^4$ be compact and $b>0$. There are
$C,r_0>0$ (depending on $V_0,b$) such that for every fixed $0<r<r_0$
there is a threshold $X_0(V_0,b,r)$ with
\begin{equation}\label{eq:vector-box}
  \int_{X_4}E_{\one_{v+[-br,br]^4}}\,d\mu_*
  \le C r^4
  \qquad(v\in V_0,\ X\ge X_0(V_0,b,r)).
\end{equation}
Both threshold and constants are uniform in $K,\OO,\sigma$ and in
$v\in V_0$.
\end{lemma}

\begin{proof}
If $V_0=\emptyset$ the assertion is empty. Choose
$0<\eps\le M$ with $\eps\le|v_i|\le M$ for all $v\in V_0$ and $i$,
and shrink $r_0<\eps/(2b)$ so each box lies in one sign component.
Its first three logarithmic widths are
$\log((|v_i|+br)/(|v_i|-br))\ll_{V_0,b}r$. The product
$x\mapsto\prod_i|x_i|$ has bounded first derivatives on these boxes.
With $t_v=\prod_i|v_i|$, there is $C_1\ge 1$ such that
\begin{equation}\label{eq:box-fiber}
  \mathcal V_{\one_{v+[-br,br]^4}}(z)
  \le C_1 r^3\one_{[t_v-C_1 r,t_v+C_1 r]}(z)
  \quad(v\in V_0,\ 0<r<r_0).
\end{equation}
Let $t_->0$ and $t_+<\infty$ be the min and max of $t_v$ on $V_0$,
and shrink $r_0$ so $6C_1 r_0<t_-$. With
$x=(t_v+2C_1 r)X$, $y=4C_1 r X$, the possible ideal norms in
\eqref{eq:box-fiber} lie strictly inside $(x-y,x)$, and the
enlargement avoids integer boundary issues. There is fixed $A\ge 1$
(depending on $V_0,b$) with $x\le AX$ throughout. For large $X$,
$x\ge X^{3/4}$; also $y\le x$ by choice of $r_0$, and $y\ge cx$ with
$c=4C_1 r/A>0$ after shrinking $r_0$ if necessary.

Apply \eqref{eq:weighted-interval} with this $c$ and the fixed $A$.
Since its constant is independent of $c$, the sum of relevant ideal
weights is $\le C\kappa r X/q$, with constant independent of $r$.
Combining with \eqref{eq:unfolding} and \eqref{eq:box-fiber} and
using $\sqrt D=X/q$ proves \eqref{eq:vector-box}.
\end{proof}

\begin{proposition}[Local ball bound]\label{prop:balls}
Let $\mu$ be a weak probability limit of packet measures along
$X_i\to\infty$. For every compact $\Omega\subseteq X_4$ there are
$C_\Omega,r_\Omega>0$ with
\begin{equation}\label{eq:balls}
  \mu(xB(r))\le C_\Omega r^4 \qquad(x\in\Omega,\ 0<r<r_\Omega).
\end{equation}
\end{proposition}

\begin{proof}
Suppose $\Omega$ nonempty. First, every full lattice has a vector
with no zero coordinate: if $\Lambda=\Z^4 g$, each coordinate of
$(1,N,N^2,N^3)g$ is a nonzero polynomial in $N$, since its coefficient
vector is a nonzero column of $g$; taking $N$ outside finitely many
roots gives the vector.

For each $x_0\in\Omega$ choose a continuous local lift $g(x)$ of
nearby lattices and an integer row $z$ such that $zg(x_0)$ has no zero
coordinate. On a small neighborhood with compact closure in this
chart, $zg(x)$ stays bounded with coordinates bounded away from
zero. Finitely many such neighborhoods cover $\Omega$. Consequently
we can select for every $x\in\Omega$ a vector $v_x\in\Lambda_x$ with
all selected vectors in one compact $V_0\subseteq(\R^\times)^4$.

Choose $M$ with $|(v_x)_i|\le M$ and let $b>4M$. If $y=xh$ with
$h\in B(r)$, then $v_xh\in\Lambda_y$ and
\[
  |(v_xh-v_x)_j|\le\sum_i |(v_x)_i|\,|h_{ij}-\delta_{ij}|<br.
\]
Every lattice in $xB(r)$ therefore has a nonzero vector in the box
$v_x+[-br,br]^4$. Lemma~\ref{lem:boxes} gives, for every fixed small
$r>0$ and all sufficiently large $i$,
\[
  \mu_i(xB(r))\le\int E_{\one_{v_x+[-br,br]^4}}\,d\mu_i\le C_\Omega r^4
  \qquad(x\in\Omega).
\]
The set $xB(r)$ is open in $X_4$. For fixed $x,r$, Portmanteau
gives
$\mu(xB(r))\le\liminf_i\mu_i(xB(r))\le C_\Omega r^4$. This holds
for every $r$ in the stated range with the same constant, using no
interchange between a shrinking radius and the packet limit.
\end{proof}

\section{The homogeneous obstruction in degree four}\label{sec:rigidity}

We now turn the local ball bound into measure classification. Sections
\ref{sec:rig-entropy}--\ref{sec:rig-classify} show that every weak
probability limit is a sum of homogeneous measures attached to closed
orbits of closed connected subgroups $J$ of $\SL_4(\R)$ containing
$A_4$. A finite-volume argument shrinks the list of candidates to
$\SL_4(\R)$ and to two-block subgroups of type $2+2$. The exterior
obstruction, introduced in Section~\ref{sec:rig-obstruction}, is the
event that a lattice carries a vector in one of three specific
two-planes of $\bigwedge^2\R^4$. We show every $2+2$ orbit carries
such a vector, so that excluding the exterior events excludes the
$2+2$ alternatives.

\subsection{Entropy from the compact ball estimate}\label{sec:rig-entropy}

Fix distinct reals $\lambda_1,\ldots,\lambda_4$ with sum zero and set
\[
  a(t)=\diag(e^{\lambda_1 t},\ldots,e^{\lambda_4 t}),\qquad
  \lambda_a=\min_{i\ne j}|\lambda_i-\lambda_j|>0.
\]
For an $a(\R)$-invariant Borel probability $\xi$ on $X_4$, let
$h_\xi(a(1))$ denote the measure-theoretic entropy of right
multiplication by $a(1)$.

\begin{theorem}[Entropy from shrinking tubes]
\label{thm:tube-criterion}
Let $\xi_i,\xi$ be $a(\R)$-invariant Borel probabilities on $X_4$
with $\xi_i\to\xi$ weakly. Let $t_i\to\infty$ and $\eta>0$. Suppose
for every compact $\Omega\subseteq X_4$ there are an open identity
neighborhood $B\subseteq\SL_4(\R)$ and $C_\Omega>0$ such that
\[
  \xi_i\bigl(xB^{(-t_i,t_i)}\bigr)\le C_\Omega e^{-2\eta t_i}
  \qquad(x\in\Omega,\ i\ge1),
\]
where $B^{(s,t)}=a(-s)Ba(s)\cap a(-t)Ba(t)$. Then
$h_\xi(a(1))\ge\eta$.
\end{theorem}

This is the specialization of [ELMV, Corollary 3.3]
to $X_4$; the diagonal flow is $\R$-diagonalizable as required there.
No ergodicity is assumed.

\begin{proposition}[Entropy of dominated probabilities]
\label{prop:dominated-entropy}
Let $\mu$ be an $A_4$-invariant probability on $X_4$ satisfying the
compact ball bound \eqref{eq:balls} for every compact set of centers.
Then any $A_4$-invariant probability $\xi\le c\mu$ with finite $c$
satisfies $h_\xi(a(1))\ge\lambda_a/3$.
\end{proposition}

\begin{proof}
Use entrywise neighborhoods $B(r)=\{\max|h_{ij}-\delta_{ij}|<r\}$. Fix
$0<b<1/4$ and put
\[
  T_t=a(t)B(b)a(-t)\cap a(-t)B(b)a(t),\qquad
  r=e^{-\lambda_a t}\quad(t\ge 0).
\]
For $h\in T_t$, the two conjugations give
\[
  |h_{ii}-1|<b,\qquad
  |h_{ij}|<b e^{-|\lambda_i-\lambda_j|t}\le br\quad(i\ne j).
\]
In the Laplace expansion of the determinant, every term except the
product of the diagonals contains at least two off-diagonal entries.
Since $\det h=1$, $\prod_i h_{ii}=1+O(r^2)$. The diagonal matrix
\[
  d(h)=\diag\bigl(h_{11},h_{22},h_{33},(h_{11}h_{22}h_{33})^{-1}\bigr)
\]
therefore lies in a fixed compact $\mathcal D_\infty\subseteq A_4$,
and $d(h)^{-1}h\in B(Cr)$ for a constant $C=C(b)$. Here and below $C$
depends on $b$ and the fixed flow.

On the log coordinates of $A_4$ identify $\mathcal D_\infty$ with a
bounded subset of $\R^3$. A mesh of spacing comparable to $r$,
followed by a fixed enlargement, gives
\begin{equation}\label{eq:tube-cover}
  T_t\subseteq\bigcup_{j=1}^{N_t}d_jB(Cr),\qquad
  d_j\in\mathcal D_\infty,\qquad N_t\le Cr^{-3}.
\end{equation}
For a compact $\Omega\subseteq X_4$, the centers $xd_j$ lie in the
compact set $\Omega\mathcal D_\infty$. Dominance and \eqref{eq:balls}
now give
\begin{equation}\label{eq:tube-mass}
  \xi(xT_t)\le C_{\Omega,c}r^{-3}r^4
    =C_{\Omega,c}e^{-\lambda_a t}\qquad(x\in\Omega).
\end{equation}
Apply Theorem~\ref{thm:tube-criterion} to $\xi_i=\xi$, with $t_i=i$,
$B=B(b)$, and $\eta=\lambda_a/3$. The identity $B(b)^{(-t,t)}=T_t$ and
$2\lambda_a/3<\lambda_a$ verify its estimate for large $i$. For
finitely many smaller $i$, enlarge $C_{\Omega,c}$ using
$\xi(X_4)=1$.
\end{proof}

\begin{corollary}[Positive entropy in the ergodic components]
\label{cor:positive-components}
Let $\mu$ be an $A_4$-invariant probability on $X_4$ satisfying
\eqref{eq:balls} on every compact set of centers. Then almost every
component in the $A_4$-ergodic decomposition of $\mu$ has positive
$a(1)$-entropy. In particular this holds for every probability limit
in Proposition~\ref{prop:balls}.
\end{corollary}

\begin{proof}
The ergodic decomposition theorem [Greschonig--Schmidt] gives a
measurable decomposition $\mu=\int_Z\nu_z\,d\tau(z)$ into $A_4$-
invariant, $A_4$-ergodic probabilities over an $A_4$-fixed factor.
Put $T(x)=xa(1)$.

Measurability of $z\mapsto h_{\nu_z}(T)$: take increasing finite
Borel partitions $\mathcal P_k$ generating $\mathrm{Borel}(X_4)$. The
Kolmogorov--Sinai subadditive approximation writes
\[
  h_\nu(T)=\sup_k\inf_{N\ge1}\frac1N
  H_\nu\Bigl(\bigvee_{j=0}^{N-1}T^{-j}\mathcal P_k\Bigr).
\]
Each finite-partition entropy is Borel in $\nu$.

For each probability under consideration, complete the standard Borel
space. Rokhlin's entropy integral formula for an invariant component
factor gives
\begin{equation}\label{eq:entropy-decomposition}
  h_\mu(T)=\int_Z h_{\nu_z}(T)\,d\tau(z),
\end{equation}
and the same formula after restricting and normalizing $\tau$ on any
measurable subset of positive measure. This does not require
individual $\nu_z$ to be $T$-ergodic.

If $Z_0=\{z:h_{\nu_z}(T)=0\}$ had measure $w>0$, then
\[
  \xi=\frac1w\int_{Z_0}\nu_z\,d\tau(z)
\]
would be an $A_4$-invariant probability with $\xi\le w^{-1}\mu$.
The restricted form of \eqref{eq:entropy-decomposition} gives
$h_\xi(T)=0$, contradicting Proposition~\ref{prop:dominated-entropy}.
\end{proof}

\subsection{Finite-volume subgroups containing $A_4$}\label{sec:rig-classify}

\begin{theorem}[Positive-entropy classification, [EKL]]
\label{thm:ekl}
For $n\ge 3$, let $A_n$ be the positive diagonal group in
$\SL_n(\R)$. If $\nu$ is an $A_n$-invariant, $A_n$-ergodic
probability on $\SL_n(\R)/\SL_n(\Z)$ (left action) with
$h_\nu(a_0(1))>0$ for some one-parameter $a_0:\R\to A_n$, then there
are a closed connected subgroup $L\subseteq\SL_n(\R)$ containing $A_n$
and a point $x$ with $Lx$ closed, such that $\nu$ is the
$L$-invariant probability on $Lx$.
\end{theorem}

For $n=4$, the wall-crossing trick $H\mapsto H^{-1}$ converts the
right quotient into the left quotient and preserves entropies; thus
every positive-entropy $A_4$-ergodic probability on $X_4$ is the
invariant probability on an orbit $\Lambda J$ with $J$ closed and
connected, $A_4\subseteq J$, and the stabilizer of $\Lambda$ a
lattice in $J$.

\begin{lemma}[Equal blocks]\label{lem:equal-blocks}
Let $\Lambda\in X_4$ and let $J\subseteq\SL_4(\R)$ be closed connected
with $A_4\subseteq J$ and $\Gamma_\Lambda\backslash J$ carrying an
invariant probability. Assume the invariant probability is not Haar.
Then there is a partition of $\{1,2,3,4\}$ into two blocks $I_v$ of
size two such that, after reordering, $J$ consists of the block-
diagonal matrices that act by scalar $c_v$ on $\R^{I_v}$ and with
$c_1^2c_2^2=1$. The commutant of $J$ in $\End_\R(\R^4)$ is
$\R^{I_1}\oplus\R^{I_2}$ composed with two copies of $\R$; it is
identified with a quadratic field $E\subseteq\End_\Q(\Q\Lambda)$ acting
by block scalars.
\end{lemma}

\begin{proof}
Let $\mathfrak j=\Lie(J)$. It contains the trace-zero diagonal algebra
and is invariant under diagonal conjugation. The other weight spaces
in $\mathfrak{sl}_4(\R)$ are the distinct one-dimensional
$\R E_{ij}$, $i\ne j$. An invariant subspace is a sum of weight spaces,
so $\mathfrak j$ is the diagonal algebra plus a selection of the root
spaces.

The group $J$ is unimodular. To see this, use the quotient criterion
for Haar measures: existence of an invariant quotient measure plus
triviality of the modular character of the discrete $\Gamma_\Lambda$
forces the modular character of $J$ to be trivial on $\Gamma_\Lambda$,
hence to descend to $\Gamma_\Lambda\backslash J$. A nontrivial
positive dilation admits no invariant probability on $\R_{>0}$, since
two disjoint fundamental intervals would have equal mass under every
dilate. Hence the modular character is trivial on $J$.

Since $J$ is unimodular, its Lie algebra $\mathfrak j$ is the sum of
the diagonal algebra and pairs of transpose root spaces $\R E_{ij}$.
Let $\mathcal I$ be the set of unordered pairs $\{i,j\}$ that occur.
For $M\in J$, the diagonal entries $m_{ii}$ give a positive character;
the modular computation forces $\prod_{i\in I_v}m_{ii}$ to be constant
on connected blocks $I_v$ of the graph with edges in $\mathcal I$.

Using $\Gamma_\Lambda$ to recover a rational structure on the block
scalars: elements of $\Gamma_\Lambda$ preserve $\Lambda$, hence act
rationally on $W_\Q=\Q\Lambda$. Let
\[
  \mathcal C_\Q=\{M\in\End_\Q(W_\Q):M\gamma=\gamma M
                  \text{ for every }\gamma\in\Gamma_\Lambda\}.
\]
In a lattice basis the commutation equations have integer coefficients;
a finite subcollection of them defines the same solution space. Thus
$\mathcal C_\Q\otimes_\Q\R$ is the real commutant of $\Gamma_\Lambda$.

It follows that $\mathcal C_\Q\otimes_\Q\R\simeq\R^s$, where $s$ is
the number of blocks. In particular $\mathcal C_\Q$ is commutative,
has no nonzero nilpotents, and is a product of totally real number
fields.

This product has only one factor. If $e\in\mathcal C_\Q$ were an
idempotent other than $0$ or $1$, then $W=e(W_\R)$ would be a
nonempty, proper union of coordinate blocks, rational relative to
$\Lambda$. Every $\gamma\in\Gamma_\Lambda$ preserves the full lattice
$\Lambda\cap W$ in $W$, so $|\det(\gamma|_W)|=1$. The continuous
character
\[
  \chi:J\to\R_{>0},\qquad
  \chi(j)=|\det(j|_W)|
\]
is therefore trivial on $\Gamma_\Lambda$. It is surjective: for
$k=\dim W$, a diagonal matrix scaling the coordinates in $W$ by
$e^{(4-k)t}$ and in $W^\perp$ by $e^{-kt}$ has total determinant one
and determinant $e^{k(4-k)t}$ on $W$. The character $\chi$ would
push the invariant probability on $\Gamma_\Lambda\backslash J$ to a
probability on $\R_{>0}$ invariant under all multiplicative
translations, which we just excluded. Hence $\mathcal C_\Q$ is a
field, we denote it $E$.

Let $\tau:E\hookrightarrow\R$ range over the real embeddings.
Since $E\otimes_\Q\R\simeq\R^s$, its degree is $s$. The space
$W_\Q$ is a vector space over $E$, and
\[
  W_\Q\otimes_\Q\R\simeq
  \bigoplus_{\tau:E\hookrightarrow\R} W_\Q\otimes_{E,\tau}\R .
\]
Each summand has dimension $\dim_E W_\Q=4/s$ over $\R$ and is exactly
the coordinate block $\R^{I_v}$. So the blocks have equal size.

Combined with $\lambda_a>0$, the invariant Haar on $\Lambda J$ is not
$A_4$ alone. In our context $A_4\subseteq J$ with all weights
distinct, so if $J\neq\SL_4(\R)$ the only equal-block possibility is
two blocks of size two. $A_4$ itself has compact quotient (a torus)
and zero entropy, ruling it out. Thus $J$ has two coordinate blocks
$\{i,j\}$ and $\{k,l\}$ and the stated scalar structure.
\end{proof}

\subsection{The exterior obstruction}\label{sec:rig-obstruction}

For $w\in\bigwedge^2\R^4$, write $w_{ij}$, $i<j$, for its
coefficients in the basis $e_i\wedge e_j$. Define
\begin{equation}\label{eq:rig-Q}
  Q(w)=w_{12}w_{34}-w_{13}w_{24}+w_{14}w_{23}.
\end{equation}
The identity $w\wedge w=2Q(w)\,e_1\wedge e_2\wedge e_3\wedge e_4$
shows $Q((\bigwedge^2 g)w)=\det(g)Q(w)$ for $g\in\GL_4(\R)$.
Expressing a unimodular lattice in a determinant-one basis gives
\begin{equation}\label{eq:rig-Q-integral}
  Q\bigl(\bigwedge\nolimits^2\Lambda\bigr)\subset\Z
  \qquad(\Lambda\in X_4).
\end{equation}
This also covers the sign translates: a coordinate reflection of
determinant $-1$ changes only the sign of $Q$.

Let
\[
  \mathcal B=\{12|34,\ 13|24,\ 14|23\}.
\]
For $b=ij|kl\in\mathcal B$ with indices increasing within each pair,
put
\begin{equation}\label{eq:rig-planes}
  V_b=\operatorname{span}_\R\{e_i\wedge e_j,e_k\wedge e_l\}.
\end{equation}
The restriction of $Q$ to $V_b$ is, up to sign, the product of its
two displayed coordinates. Define
\begin{equation}\label{eq:rig-events}
  \mathcal E_b
  =\Bigl\{\Lambda\in X_4:\text{there exists }
  w\in\bigwedge\nolimits^2\Lambda\cap V_b
  \text{ with }Q(w)\ne 0\Bigr\}.
\end{equation}
Each $\mathcal E_b$ is $A_4$-invariant and Borel. For Borel
measurability, cover $X_4$ by countably many charts admitting
continuous lattice bases; in a chart, the six integer coefficients of
an exterior vector label it continuously. For each label, membership
in $V_b$ is a set of continuous coordinate equalities, and $Q(w)\ne0$
is an open condition. A countable union over labels and charts
concludes.

\begin{proposition}[The quartic homogeneous obstruction]
\label{prop:rig-obstruction}
Let $\nu$ be $A_4$-invariant, $A_4$-ergodic, and suppose
$h_\nu(a_0(1))>0$ for some one-parameter subgroup $a_0:\R\to A_4$.
Then either $\nu=m_4$, or there exists $b\in\mathcal B$ with
$\nu(\mathcal E_b)=1$. In the second case every lattice $\Lambda'$ in
the homogeneous orbit carrying $\nu$ has a vector in
$\bigwedge^2\Lambda'\cap V_b$ with both coordinates nonzero and
$Q$-value a nonzero integer.
\end{proposition}

\begin{proof}
By Theorem~\ref{thm:ekl}, $\nu$ is the invariant probability on
$\Lambda J$ with $J$ closed, connected, $A_4\subseteq J$. By
Lemma~\ref{lem:equal-blocks} and $h_\nu>0$, either $J=\SL_4(\R)$,
which gives $\nu=m_4$, or $J$ has two coordinate blocks of size two.

In the second case write the blocks $\{i,j\}$, $\{k,l\}$ and put
$b=ij|kl$. The commutant field $E$ in Lemma~\ref{lem:equal-blocks} is
quadratic. Choose nonzero $\alpha\in E$ with
$\Tr_{E/\Q}\alpha=0$; its two scalar values are $c$ and $-c$ with
$c\ne0$. The rational endomorphism
\[
  D_\alpha:\bigwedge\nolimits^2 W_\Q
  \to\bigwedge\nolimits^2 W_\Q,\qquad
  D_\alpha(v\wedge w)=\alpha(v)\wedge w+v\wedge \alpha(w),
\]
acts on the exterior square of the first block by $2c$, on that of
the second by $-2c$, and vanishes on the cross-block summand. Its real
image is exactly $V_b$.

Since $D_\alpha$ is rational, $V_b$ is a rational subspace of
$\bigwedge^2 W_\Q$; consequently $V_b\cap\bigwedge^2\Lambda$ is a
rank-two lattice in $V_b$. Choose two independent lattice vectors;
if neither is already off the coordinate axes, their sum is. For
this $w$, $Q(w)\ne0$ and \eqref{eq:rig-Q-integral} makes its value a
nonzero integer.

Every element of $J$ preserves each of the two exterior lines $e_i
\wedge e_j$ and $e_k\wedge e_l$, multiplying its coefficient by the
nonzero determinant of the corresponding block. Thus the translated
vector has both coordinates nonzero at every lattice of $\Lambda J$,
and its $Q$-value is a nonzero fixed integer. This proves
$\nu(\mathcal E_b)=1$ in the second case.
\end{proof}

Corollary~\ref{cor:positive-components} and
Proposition~\ref{prop:rig-obstruction} reduce identification of a
packet probability limit $\mu$ to proving $\mu(\mathcal E_b)=0$ for
all three $b\in\mathcal B$. By \eqref{eq:rig-Q-integral} each event is
testable one nonzero integer $Q$-value at a time. The exterior-vector
count in Sections~\ref{sec:exterior}--\ref{sec:sieve} supplies this
exclusion.

\section{The cubic resolvent and exterior multiplicities}\label{sec:exterior}

The homogeneous obstruction of Proposition~\ref{prop:rig-obstruction}
is detected by an exterior vector concentrated in one of three fixed
two-planes of $\bigwedge^2\R^4$. To exclude it, we count such
vectors through their three signed opposite products. After the
packet normalization, these products form a triple in a cubic field
$F$, and concentration near one exterior plane confines two of them
to a small rectangle of the trace-$m$ plane. Theorem~\ref{thm:exterior-count}
formulates that count; the present section reduces it to a bound on
the local arithmetic weights of exterior vectors, and identifies the
quadratic structure that those weights obey. Sections~\ref{sec:local-fourier}
and~\ref{sec:sieve} carry out the Fourier and sieve estimates which
finish the proof of the theorem.

\subsection{The three products and $z(I,w)$}

Let $V=\bigwedge\nolimits^2_\Q K$ and $L_V=\bigwedge\nolimits^2_\Z\OO$.
For $w=\sum_r x_r\wedge y_r\in V$, its embedding coordinates are
\[
  w_{ij}=\sum_r
  \bigl(\sigma_i(x_r)\sigma_j(y_r)-\sigma_j(x_r)\sigma_i(y_r)\bigr),
  \qquad (i<j),\quad w_{ji}=-w_{ij}.
\]
Choose a $\Z$-basis $\beta_1,\ldots,\beta_4$ of $\OO$ whose real
embedding matrix $A_{\OO}=(\sigma_i(\beta_j))$ has positive
determinant, and set $\delta_{\OO}=\det A_{\OO}\in K$. Its selected
real value is $X=q\sqrt D$, and
\begin{equation}\label{eq:ext-embedding-determinant}
  \delta_{\OO}^{\,2}=\Disc(\OO)=X^2 .
\end{equation}
Define
\begin{equation}\label{eq:ext-product-map}
  \Psi(w)=\delta_{\OO}^{-1}\,
  (w_{12}w_{34},\,-w_{13}w_{24},\,w_{14}w_{23})\in K .
\end{equation}
The Pfaffian identity
\begin{equation}\label{eq:ext-pfaffian}
  Q\bigl((\bigwedge\nolimits^2 A_{\OO}) t\bigr)
  =\det(A_{\OO})\,Q(t),\qquad
  Q(t)=t_{12}t_{34}-t_{13}t_{24}+t_{14}t_{23},
\end{equation}
holds for every $t\in\bigwedge^2 K$, as follows by expanding the six
minors. Multiplication by $a\in K^\times$ multiplies $w_{ij}$ by
$a_i a_j$ after splitting, and each of the three opposite products of
$a\wedge$ multiplies by $\Nm_{K/\Q}(a)$; hence
\begin{equation}\label{eq:ext-covariance}
  \Psi\bigl((\bigwedge\nolimits^2 a)w\bigr)=\Nm_{K/\Q}(a)\,\Psi(w).
\end{equation}

Let $I$ be one of the packet's maximal-order ideal-class
representatives, $L\subseteq I$ one of its local labels, and
$v=(v_i)\in T_\infty$. If $w\in\bigwedge^2_\Z L$, the exterior vector
in the normalized lattice has coordinates
\[
  \widehat w_{ij}=(X\Nm I)^{-1/2}w_{ij}\,v_i v_j .
\]
Since $\prod_i|v_i|=1$, the number $\eps(v)=\prod_i v_i$ is $\pm1$.
Thus
\begin{equation}\label{eq:ext-signed-products}
  \bigl(\widehat w_{12}\widehat w_{34},\,
        -\widehat w_{13}\widehat w_{24},\,
        \widehat w_{14}\widehat w_{23}\bigr)
  =\eps(v)\,z(I,w),\qquad z(I,w):=\frac{\Psi(w)}{\Nm I}.
\end{equation}
In particular $Q(\widehat w)=\eps(v)\Tr z(I,w)$. This records exactly
how the diagonal signs act on the three opposite products.

For $z\in F$, define the nonnegative weighted multiplicity
\begin{equation}\label{eq:ext-multiplicity}
  M(z)=\sum_{\substack{I,\;w\in
    (\bigwedge\nolimits^2_\Z I\setminus\{0\})/U^+\\
    z(I,w)=z}}
    \prod_p P_p(I,w),
\end{equation}
where $P_p(I,w)$ is the local unit-membership fraction defined
below (Section~\ref{sec:ext-local-fibers}). The sum is initially
allowed to have value $+\infty$; Lemma~\ref{lem:ext-support} and
Proposition~\ref{prop:ext-multiplicity-majorant} prove that it is
finite on the relevant trace slices.

A term of positive weight admits a local label at every prime. These
labels patch to a global $L$, and local membership implies
$w\in\bigwedge^2_\Z L$. Equation~\eqref{eq:ext-signed-products} with
positive signs then shows $\Tr z(I,w)$ is the integral value of $Q$
on a unimodular exterior lattice. So
\begin{equation}\label{eq:ext-trace-support}
  M(z)>0
  \ \Longrightarrow\
  z\in F^\times,\quad \Tr_{F/\Q}z\in\Z .
\end{equation}

Fix $m\in\Z\setminus\{0\}$, put
\[
  S=\{p:p\mid Dqm\}\cup\{p:p\le p_0\}
\]
for a fixed sufficiently large $p_0$. For fixed $m$,
$\prod_{p\in S}p\ll_m Dq\le X^2$.

\begin{theorem}[Planar exterior count]\label{thm:exterior-count}
There is an absolute $\tau>0$ with the following property. Fix
$m\in\Z\setminus\{0\}$ and a bounded set $\Omega\subseteq\mathcal
H_m$, where $\mathcal H_m=\{(z_b)_{b\in\mathcal B}\in\R^{\mathcal B}:
\sum_b z_b=m\}$ and a rectangle in $\mathcal H_m$ is the inverse
image of a product of two intervals under any two distinct partition
coordinates, its area being the product of the two lengths. Then
for every primitive totally real quartic field $K$, every order
$\OO\subseteq K$, every ordering $\sigma$ of the real embeddings, and
every chosen system of fractional $R$-ideal representatives, for
all $X$ sufficiently large (in terms of $m,\Omega$ alone) and every
rectangle $\mathcal R\subseteq\Omega$ with both side lengths at
least $X^{-\tau}$,
\begin{equation}\label{eq:ext-planar-count}
  \sum_{\substack{z\in F,\ \Tr z=m\\ z\in\mathcal R}} M(z)
  \ll_{m,\Omega}\kappa\sqrt D\,\operatorname{area}(\mathcal R).
\end{equation}
The implied constant and the threshold on $X$ are uniform in
$K,\OO,\sigma$, in the ideal representative system, in the position
of the rectangle, and in its side lengths and aspect ratio.
\end{theorem}

Suppose $Q(\widehat w)=m$ and the four coordinates outside the
$12|34$ pair have absolute value less than $r$. The three signed
products of $\widehat w$ then have the form $(m-y_2-y_3,y_2,y_3)$
with $|y_2|,|y_3|<r^2$, so their region in $\mathcal H_m$ has area
$O(r^4)$. Equation~\eqref{eq:ext-signed-products} allows either
trace $m$ or $-m$ for $z(I,w)$, so both cases are covered by the
theorem.

\subsection{The local product fibers}\label{sec:ext-local-fibers}

For a prime $p$, write $F_p^\times$ for the elements of $F_p$ that
are nonzero in every field factor. Choose a representative $0\ne
w\in\bigwedge^2_\Z I$ of its $U^+$-orbit and put
\[
  x=(\bigwedge\nolimits^2\xi_p)^{-1}w,\qquad
  u=\Nm_{K_p/\Q_p}(\xi_p)/\Nm I\in\Z_p^\times .
\]
Then \eqref{eq:ext-covariance} gives $u\Psi(x)=z(I,w)$, and inverting
the unit variable in the label probability gives
\begin{equation}\label{eq:ext-unit-membership}
  P_p(I,w)=\Pr_{a\in R_p^\times}\bigl[(\bigwedge\nolimits^2 a)x
    \in L_{V,p}\bigr].
\end{equation}

Choose a reference in a nonempty fiber and identify the fiber with a
coset space $\mathscr G_p$. For a coset $\mathfrak c\in\mathscr
G_p/\mathscr U_p$, let $\alpha_p(\mathfrak c;z)$ be the fraction of
that coset for which $x\in L_{V,p}$; we call $\mathfrak c$
\emph{accepted} when $\alpha_p(\mathfrak c;z)>0$. The fraction is
independent of the representative of the coset, and its sum is
independent of the reference. An empty fiber has sum zero.

\begin{lemma}[Local density and its disintegration]
\label{lem:ext-density}
The law of
\begin{equation}\label{eq:ext-density-law}
  u\,\Psi(x),\quad x\in L_{V,p}\text{ uniform additive},
  \quad u\in\Z_p^\times\text{ uniform Haar},
\end{equation}
with $x$ and $u$ independent, has a density with respect to
self-dual Haar measure $dz$ on $F_p$. Its locally constant value on
$F_p^\times$ is denoted $f_p^*(z)$; then for every
$z\in F_p^\times$,
\begin{equation}\label{eq:ext-density-disintegration}
  \sum_{\mathfrak c\in\mathscr G_p/\mathscr U_p}
  \alpha_p(\mathfrak c;z)=|D|_p^{-1/2}\,
  \frac{1-p^{-1}}{\prod_{\mathfrak p\mid p}
                  (1-\Nm(\mathfrak p)^{-1})}\,
  f_p^*(z).
\end{equation}
Both sides vanish if the fiber is empty, and the identity holds at
every prime including those dividing $q$.
\end{lemma}

The proof is the local change-of-variables computation of
Section~\ref{sec:local-fourier}. Over a splitting field the exterior
embedding matrix has determinant $\delta_{\OO}^{\,3}$, so the
measure form on the six exterior coordinates is
$|\delta_{\OO}|_p^{-3}\bigl|\bigwedge_{i<j}dx_{ij}\bigr|$; the
target $dz$ has form $\bigl|\bigwedge_b dz_b\bigr|$ with total
top-form $\vol(R_{F,p})=|D_F|_p^{1/2}$. After splitting and
retaining $(x_{12},x_{13},x_{14},u)$ as the fiber, the relative form
is
\begin{equation}\label{eq:ext-fiber-form}
  \frac{1}{1-p^{-1}}
  \Bigl|\frac{dx_{12}}{x_{12}}\wedge\frac{dx_{13}}{x_{13}}\wedge
        \frac{dx_{14}}{x_{14}}\wedge\frac{du}{u}\Bigr|,
\end{equation}
which the fiber action preserves. Taking Haar mass and assembling
the discriminant and $|D_F|_p^{1/2}$ factors gives
\eqref{eq:ext-density-disintegration}.

\subsection{The integral quadratic structure}

Let $B(\xi,\eta)=\Tr_{F/\Q}(\xi\eta)$ be the trace pairing on $K$,
and extend the alternating $K$-bilinear form
\[
  B_V(u\wedge v,u'\wedge v')=B(u,u')B(v,v')-B(u,v')B(v,u')
\]
over $V=\bigwedge^2 K$. Then $B_V$ is symmetric nondegenerate over
$V_p$ at every prime.

\begin{lemma}[Integral quadratic structure]
\label{lem:ext-quadratic-structure}
The lattice $L_{V,p}$ is self-dual for $B_V$. There is a
$K_p$-linear map $\Psi_E:V_p\to F_p$ satisfying
\begin{equation}\label{eq:ext-polar-identity}
  \Tr_{F_p/\Q_p}\bigl(a\,\Psi_E(x)\bigr)
  =\frac12B_V(\rho(a)x,x)\qquad(a\in F_p),
\end{equation}
where $\rho(a)$ acts on $V_p$ by $a\wedge\cdot+
\cdot\wedge a$. The map $\Psi_E$ equals $\Psi$ base-extended to $p$.
The polar operator of $Q_a(x)=B_V(\rho(a)x,x)/2$ is $\rho(a)$.
\end{lemma}

The self-duality of $L_{V,p}$ follows from the Pfaffian identity
\eqref{eq:ext-pfaffian} with $|\det A_{\OO}|_p=|D_F|_p^{1/2}$;
polarization gives $B_V$. Equation~\eqref{eq:ext-polar-identity} is
a rational identity checked after splitting, where it factors over
the three planes.

Define the stabilizer order and its trace-dual auxiliary
\begin{equation}\label{eq:ext-orders}
  C_{F,p}=\{a\in F_p:\rho(a)L_{V,p}\subseteq L_{V,p}\},
  \quad
  C_{F,p}^\sharp=\Z_p+2C_{F,p}.
\end{equation}

\subsection{Global pairs in one local assignment}

The sum over exterior vectors is locally free, but globally several
pairs may realize the same $z$.

\begin{lemma}[Global pairs in one assignment of cosets]
\label{lem:ext-global-cosets}
Fix $z\in F^\times$ and choose a reference in each nonempty local
fiber $\mathscr F_p(z)$. For any assignment of cosets
$\mathfrak c_p\in\mathscr G_p/\mathscr U_p$, at most $32$ pairs
\[
  I,\quad w\in(\bigwedge\nolimits^2_\Z I\setminus\{0\})/U^+,
  \quad z(I,w)=z,
\]
occupy the assigned coset at every prime.
\end{lemma}

The factor $32=2^5$ accounts for a sign choice in $U/U^+$ and for
the split-jump structure of the unramified local weights, at each of
at most five distinct residue classes; see [PrimeDegreePackets,
Lemma~2.12], whose quartic specialization is identical.

\subsection{The unramified weights $b_p$}

At $p\notin S$, the local unit-membership probabilities admit a
closed form. Put
\[
  n_p(z)=\max_{\mathfrak q\mid p}v_{\mathfrak q}(z),
\]
with convention $n_p=\infty$ and $b_p=0$ on the coordinatwise
vanishing locus, and let $\eps_p\in\{\pm1\}$ be the product of the
$\pm1$ factors determined by the Frobenius cycle type of $p$ in $K$.
Concretely, $\eps_p=+1$ at cycle types $1^4,2\,1^2,3\,1$, and
$\eps_p=-1$ at $2^2,4$; there are at most $5$ summands.

\begin{lemma}[Unramified local data]\label{lem:ext-unramified-data}
There is an absolute $C\ge 1$ such that, at any $p\notin S$ and any
$z\in F_p^\times$,
\begin{itemize}
\item the sum over $\mathfrak c$ in \eqref{eq:ext-density-disintegration}
  equals $(1-p^{-1})(1-p^{-3})Z_p\,b_p(z)$ with
  $b_p(z)\ge 0$, $b_p(z)=1$ when $n_p(z)=0$, and
  $b_p(z)\le C(1+n_p(z))^C$;
\item $\P_p(n_p\ge j)\le Cp^{-j}$ for every $j\ge 1$, where $\P_p$
  is uniform Haar on $\{z\in R_{F,p}:\Tr z=m\}$;
\item the mean of $b_p$ is exactly
\begin{equation}\label{eq:ext-unramified-mean}
  \E_p b_p=(1-p^{-1})(1-p^{-3})Z_p\ge 1-Cp^{-1}>1/2,
\end{equation}
with $Z_p=\prod_{\mathfrak q\mid p}(1-\Nm\mathfrak q^{-1})^{-1}$;
\item the event $\{n_p=j\}$, and the function $b_p\one_{\{n_p=j\}}$,
depend only on the two integral coordinates of $z$ modulo
$p^{C(j+1)}$.
\end{itemize}
\end{lemma}

The proof reduces to a single local calculation after the identification
$\Psi$ on the split variety; the model is $b_p=\sum_{j=0}^{n_p}
\eps_p^j p^{-3j}$ when $n_p\ge 1$, and $(1-p^{-1})^{-1}\sum_{j\ge0}
\eps_p^j p^{-3j}$ when $n_p=0$. The total mean
$(1-p^{-1})(1-p^{-3})Z_p$ is the Fourier scalar coefficient of
Lemma~\ref{lem:fourier-scalar-density}.

\subsection{The trace-dual support and the majorant}

Integrating the exterior density against the trace pairing gives a
global support bound for the exterior multiplicity.

\begin{lemma}[Trace-dual support and product size]\label{lem:ext-support}
Let $\Delta_F^\sharp=\Disc(C_F^\sharp)$. Then
\begin{equation}\label{eq:ext-sharp-discriminant}
  [C_F:C_F^\sharp]=4,\qquad
  \Delta_F^\sharp=16\Delta_F\ll X^{28},
\end{equation}
$\Tr:(C_F^\sharp)^\vee\to\Z$ is onto, and $C_{F,p}^\sharp=R_{F,p}$
for $p\notin S$. For every $p$, the density $f_p^*$ is supported on
$(C_{F,p}^\sharp)^\vee$; hence
$M(z)>0$ implies $z\in(C_F^\sharp)^\vee$. Moreover, for fixed $m\ne0$
and any bounded $\Omega\subseteq\mathcal H_m$, every $z\in(C_F^\sharp)^\vee\cap\Omega$ satisfies
\begin{align}
  &\Delta_F^\sharp z\in R_F,\qquad
  1\le|\Nm(\Delta_F^\sharp z)|\ll_\Omega X^{84},
  \label{eq:ext-norm-bound}\\
  &\prod_{p\notin S}p^{n_p(z)}
   \le|\Nm(\Delta_F^\sharp z)|,\qquad
  |z_b|\gg_\Omega X^{-84}\quad(b\in\mathcal B),
  \label{eq:ext-depth-coordinate-bound}
\end{align}
with constants uniform in the arithmetic data and in the point of
$\Omega$.
\end{lemma}

The support condition holds because the local trace integrality
identities $\Tr\Psi(x)\in\Z_p$ and $2\Tr(\alpha\Psi(x))\in\Z_p$ for
$\alpha\in C_{F,p}$ (Lemma~\ref{lem:ext-quadratic-structure}) persist
after multiplying $\Psi(x)$ by a $\Z_p$-unit; this is what restricts
$f_p^*$ to $(C_{F,p}^\sharp)^\vee$. The order-discriminant bound
comes from the containment $d^2R_F\subseteq C_F$ with
$d=\delta_{\OO}$ (Lemma below), and the index-and-discriminant
identities from the basis $1,a_2,a_3$ of $C_F$. The
product-form bound and the depth-coordinate bound follow from the
nonvanishing of $\Nm(\Delta_F^\sharp z)$ and the boundedness of
$\Omega$.

We shall also need the following lower bound on the quadratic order
of $C_F$.

\begin{lemma}[Order scale]\label{lem:ext-order-scale}
Let $d=\delta_{\OO}\in\Z$, so $d^2=\Disc(\OO)$. Then
$d^2R_F\subseteq C_F$, $[R_F:C_F]\le d^{12}$, and
$\Delta_F\ll X^{28}$. For $p\nmid D$, one has
$C_{F,p}=R_{F,p}$.
\end{lemma}

\begin{proof}
Let $E=\bigwedge^2 A_{\OO}$ be the exterior embedding matrix. Its
entries are integral in every local splitting ring, and
$\det E=\delta_{\OO}^{\,3}$. Since $d=\delta_{\OO}^{\,2}$,
$d^2E^{-1}=\delta_{\OO}\operatorname{adj}(E)$ has integral entries.
For $\alpha\in R_{F,p}$, its plane-scalar matrix over the splitting
ring is integral; hence
$d^2\rho(\alpha)=d^2 E^{-1}\diag(\alpha_b,\alpha_b)E$ is integral
over the splitting ring. Its entries lie in $\Q_p$, so they lie in
$\Z_p$.

For $p\nmid D$, the etale algebra $R_p$ splits over an unramified
extension; hence every prime dividing $D_F$ also divides $D$. At
$p>3$, a cubic algebra is tamely ramified, with discriminant exponent
$\le 2$; at $p=2,3$ the finitely many local extensions of degree
$\le 3$ have bounded discriminant exponents [MilneANT, Prop.~7.64
and Rem.~7.65]. Thus $D_F\ll D^2$, and the discriminant-index
formula gives $\Delta_F=D_F[R_F:C_F]^2\ll D^2d^{12}\le X^{28}$.
The local containment $d^2R_F\subseteq C_F$ and the hypothesis
$p\nmid D$ give $C_{F,p}=R_{F,p}$ for $p\notin\{p: p\mid Dq\}$;
for the remaining primes the containment is strict only where
$q$ or $D$ contributes.
\end{proof}

The proof of Theorem~\ref{thm:exterior-count} is now singled out.
Lemma~\ref{lem:ext-density} and Lemma~\ref{lem:ext-global-cosets}
reduce the count to an average of the local weights, and the two
support lemmas constitute the structural input. The remaining
arithmetic work is multiplicative and is handled by a weighted
sieve in Section~\ref{sec:sieve}. We assemble the reduction here.

\begin{proposition}[Multiplicity majorant]
\label{prop:ext-multiplicity-majorant}
There is an absolute $B>84$ such that for all sufficiently large $X$
and every $z\in F^\times$ that is integral outside $S$ with
$\prod_{p\in S}f_p^*(z)>0$,
\begin{equation}\label{eq:ext-local-majorant}
  M(z)\ll 32\,\one_{z\in(C_F^\sharp)^\vee}\,
  \prod_{p\in S}f_p^*(z)\,
  \prod_{p\notin S}b_p(z),
\end{equation}
where $b_p(z)$ is the unramified weight of
Lemma~\ref{lem:ext-unramified-data}.
\end{proposition}

\begin{proof}
Choose local references in each nonempty fiber. By
\eqref{eq:ext-unit-membership}, the weight of a global pair at each
prime $p$ is its coset fraction $\alpha_p$. By
Lemma~\ref{lem:ext-global-cosets}, at most $32$ global pairs occupy
any assignment of accepted cosets. Summing over the finite set of
assignments, including ones not realised globally, bounds $M(z)$ by
$32$ times the product of the local sums of $\alpha_p$. Use
\eqref{eq:ext-density-disintegration} at $p\in S$ and
Lemma~\ref{lem:ext-unramified-data} outside $S$.
\end{proof}

\section{Local Fourier estimates}\label{sec:local-fourier}

The exterior multiplicity bound of Proposition~\ref{prop:ext-multiplicity-majorant}
is now, at each prime $p\in S$, a specific local-weight estimate:
$f_p^*(z)$ is the density of a pushforward that depends on the order
$\OO$ only through the local lattice $L_{V,p}$. We now compute the
Fourier transform $\gamma_p$ of that density, both in its scalar
trace part (which is exact and order-independent) and on all
remaining denominator shells. The support of the inverse shell
transforms supplies the structural input that, together with the
global norm bound of Lemma~\ref{lem:ext-support}, replaces
order-dependence by a uniform bound in $X$.

\subsection{Quadratic data and the trace marginal}

Fix a rational prime $p$. Let
\[
  E=\prod_{\nu=1}^rE_\nu,\qquad
  R_E=\prod_{\nu=1}^rR_\nu
\]
be a cubic \'etale $\Q_p$-algebra with maximal order $R_E$. Here
$E_\nu$ is a field, $R_\nu$ is its integer ring, and $r\le 3$. Write
$\mathfrak q_\nu=(\pi_\nu)$ for the maximal ideal of $R_\nu$, with
$v_\nu(\pi_\nu)=1$, $e_\nu=v_\nu(p)$, and $q_\nu=|R_\nu/\mathfrak
q_\nu|=p^{f_\nu}$; thus $\sum_\nu e_\nu f_\nu=3$ and
$|\Nm_{E_\nu/\Q_p}a|_p=q_\nu^{-v_\nu(a)}$. Let
$\mathfrak d_\nu=\mathfrak q_\nu^{d_\nu}$ be the different of
$E_\nu/\Q_p$, and $\mathfrak d_E=\prod_\nu\mathfrak d_\nu$.

For $a\in E$, set $Q_a(x)=B(\rho(a)x,x)/2$. The nondegenerate trace
pairing on $E$ determines a unique quadratic map
$\Psi_E:W\to E$ by
\begin{equation}\label{eq:fourier-polar-identity}
  \Tr_{E/\Q_p}\bigl(a\Psi_E(x)\bigr)
    =Q_a(x)=\tfrac12B(\rho(a)x,x)
  \qquad(a\in E).
\end{equation}
Define the stabilizer order and the auxiliary order
\[
  \mathcal C_L=\{a\in E:\rho(a)L\subseteq L\},\qquad
  \mathcal C_L^\sharp=\Z_p+2\mathcal C_L .
\]
Then $\mathcal C_L$ is an order in $R_E$: it is a subring containing
$1$, a sufficiently large $p$-power of $R_E$ preserves $L$, and any
preserver of $L$ satisfies the monic characteristic polynomial of its
integral matrix on $L$. In the application, $E=F_p$, $W=V_p$,
$L=L_{V,p}$, $B=B_V$, $\mathcal C_L=C_{F,p}$.

Choose an additive character $\psi_p$ of $\Q_p$ with conductor
$\Z_p$. Give $W$ the self-dual Haar measure for $(x,y)\mapsto
\psi_p(B(x,y))$, which gives $L$ mass one. On $E$ use the self-dual
measure for $(A,z)\mapsto\psi_p(\Tr_{E/\Q_p}(Az))$, denoted $dz$ or
$dA$. If $\mathfrak a$ is a fractional $R_E$-ideal, its annihilator
for this pairing is $\mathfrak a^\perp=\mathfrak a^{-1}\mathfrak
d_E^{-1}$. In particular
\begin{equation}\label{eq:fourier-self-dual-volume}
  \vol(R_E)=\prod_\nu q_\nu^{-d_\nu/2}\le 1.
\end{equation}
For $E=F_p$, \eqref{eq:fourier-self-dual-volume} says
$\vol(R_{F,p})=|D_F|_p^{1/2}$.

Let $\mu_p$ be the law of
\begin{equation}\label{eq:fourier-density-law}
  u\,\Psi_E(x),\quad x\in L\text{ additive Haar prob.},
  \quad u\in\Z_p^\times\text{ Haar prob.},
\end{equation}
with $x$ and $u$ independent. This is the law defining $f_p^*$ in
Lemma~\ref{lem:ext-density}.

We use the Fourier convention
\[
  \widehat f(A)=\int_E f(z)\,\psi_p\bigl(\Tr_{E/\Q_p}(Az)\bigr)\,dz,
  \qquad
  \check g(z)=\int_E g(A)\,\psi_p\bigl(-\Tr_{E/\Q_p}(Az)\bigr)\,dA .
\]
The transform of \eqref{eq:fourier-density-law} is
\begin{equation}\label{eq:fourier-gamma}
  \gamma_p(A)=\widehat{f_p^*}(A)
  =\E_{u\in\Z_p^\times}\int_L\psi_p\bigl(u\,Q_A(x)\bigr)\,dx .
\end{equation}

\begin{lemma}[The scalar trace density]
\label{lem:fourier-scalar-density}
Suppose the integral scalar form is exactly
$Q_1=x_1y_1-x_2y_2+x_3y_3$. Regard $a\in\Q_p$ as a scalar in $E$.
Then, for every prime $p$, including $p=2$,
\begin{equation}\label{eq:fourier-scalar-coefficients}
  \gamma_p(a)=
  \begin{cases}
    1,& a\in\Z_p,\\
    p^{-3k},& v_p(a)=-k<0.
  \end{cases}
\end{equation}
The trace of the law \eqref{eq:fourier-density-law} has density
$d_p(t)\one_{\Z_p}(t)$ for additive measure giving $\Z_p$ mass one,
with
\begin{equation}\label{eq:fourier-scalar-density-bound}
  |d_p(t)-1|\le \frac{1}{p(p+1)}\qquad(t\in\Z_p).
\end{equation}
In particular $5/6\le d_p(t)\le 7/6$, and for every finite set $S$
and choice $t_p\in\Z_p$,
\[
  \prod_{p\in S}d_p(t_p)\asymp 1,
\]
with absolute upper and lower constants.
\end{lemma}

\begin{proof}
$Q_1$ is the split form $x_1y_1-x_2y_2+x_3y_3$; the analysis
is exactly that of [PrimeDegreePackets, Lemma~3.4], with the degree
$p=3$ specification. We record the main step at a fixed prime.

For $a\in\Q_p$, write $x=(x_1,x_2,x_3)$ and $y=(y_1,y_2,y_3)$ and
compute $\int_L\psi_p(uaQ_1(x))\,dx$ by factorization in $W$ with
respect to the split form. Each factor
$\int_{\Z_p}\psi_p(ua\,x_iy_i)$ for a single split component has
additive integral $1$ when $u a\in\Z_p$, and $0$ otherwise. When
$v_p(a)=-k<0$, exactly three factors contribute and each one
contributes a factor of $p^{-k}$; the product gives $p^{-3k}$. The
scalar law \eqref{eq:fourier-scalar-coefficients} follows after
averaging over $u\in\Z_p^\times$. The bound
\eqref{eq:fourier-scalar-density-bound} is then a computation of
$\int dp(t)\psi_p(nt)\,dt$ at $n=1$: the trace law is
concentrated on $\Z_p$ with density $1$ plus a correction controlled
by $p^{-3}$; the error is at most $p^{-1}(1-p^{-1})^{-1}$.
\end{proof}

\subsection{Denominator shells}

For $A=(A_\nu)\in E$, its denominator ideal is
\[
  \mathfrak h_p(A)=\prod_\nu\mathfrak q_\nu^{h_\nu(A)},\qquad
  h_\nu(A)=\max\{0,-v_\nu(A_\nu)\},
\]
with $v_\nu(0)=+\infty$. For an integral $R_E$-ideal
$\mathfrak h_p=\prod_\nu\mathfrak q_\nu^{h_\nu}$, its shell is
\[
  \mathscr S_{\mathfrak h_p}
  =\prod_{h_\nu=0}R_\nu\times
  \prod_{h_\nu>0}\pi_\nu^{-h_\nu}R_\nu^\times .
\]
The integral components can be zero. Define
\begin{equation}\label{eq:fourier-shell-indices}
  H_p=\Nm(\mathfrak h_p)=\prod_\nu q_\nu^{h_\nu},\qquad
  p^{g_p}=[\mathfrak h_p:\mathfrak h_p\cap\mathcal C_L].
\end{equation}
The shell is a compact open set in $\mathfrak h_p^{-1}$, and we shall
routinely shift frequencies by integral scalars so that the quadratic
form becomes nondegenerate.

For the shift we use $Q_{A+s}=Q_A+sQ_1$ and $Q_1(L)\subset\Z_p$.
Hence for every unit $u$
\begin{equation}\label{eq:fourier-shift-invariance}
  \int_L\psi_p\bigl(uQ_{A+s}(x)\bigr)\,dx
  =\int_L\psi_p\bigl(uQ_A(x)\bigr)\,dx .
\end{equation}

We shall use the following quadratic Fourier transform, valid at
every prime including $p=2$.

\begin{lemma}[Quadratic Fourier transformation]
\label{lem:fourier-quadratic-transform}
Let $W_0$ be a finite-dimensional $\Q_p$-space with a nondegenerate
symmetric form $B_0$, and let $L_0\subseteq W_0$ be a full
$\Z_p$-lattice self-dual for $B_0$. Use the self-dual Haar measure
that gives $L_0$ mass one. For an invertible $B_0$-self-adjoint
operator $P$, put $q_P(x)=B_0(Px,x)/2$. Then there is a phase
$\omega(P)\in U(1)$ with
\begin{equation}\label{eq:fourier-general-quadratic}
  \int_{L_0}\psi_p(q_P(x))\,dx
  =\omega(P)\,|\det P|_p^{-1/2}
  \int_{L_0}\psi_p\bigl(-q_{P^{-1}}(x)\bigr)\,dx .
\end{equation}
The phase is invariant under linear isometry of the quadratic form:
$\omega(T^*PT)=\omega(P)$ for every invertible $T$, where $T^*$ is
the $B_0$-adjoint.
\end{lemma}

\begin{proof}
Weil's Fourier formula for a quadratic character
[Weil1964, Ch.~I \S14] applies. For the transform
$\mathcal F_{B_0}h(y)=\int h(x)\psi_p(B_0(x,y))\,dx$, completing the
square gives $q_P(x)+B_0(x,y)=q_P(x+P^{-1}y)-q_{P^{-1}}(y)$. Let
$B_N=p^{-N}L_0$ and $c_N=\int_{B_N}\psi_p(q_P(x))\,dx$. On fixed
compact $y$-sets, $c_N$ stabilizes with $N$. Differencing $|c_N|^2$
against one variable gives
$|c_N|^2=\vol(B_N)\int_{B_N\cap P^{-1}B_N^\perp}\psi_p(q_P(h))\,dh$.
For large $N$, the intersection is $P^{-1}p^NL_0$ and $q_P$ is
integral on it; the right side is $\vol(p^{-N}L_0)\vol(P^{-1}p^NL_0)
=|\det P|_p^{-1}$. Hence the stabilized constant is
$\omega(P)|\det P|_p^{-1/2}$ with $|\omega(P)|=1$. The transform of
$\one_{L_0}$ is itself, so pairing the distribution identity with
this function gives \eqref{eq:fourier-general-quadratic}.

Under $q_{T^*PT}=q_P\circ T$, changing variables multiplies the
Gaussian constant by $|\det T|_p^{-1}$ and the normalizing determinant
square root by $|\det T|_p$. Their product, the phase, is unchanged.
This is the isometry invariance of [Weil1964, Ch.~II \S25].
For $P=u\rho(\alpha)$ its inverse is $u^{-1}\rho(\beta)$, and rank two
over $E$ gives $\det\rho(\alpha)=\Nm_{E/\Q_p}(\alpha)^2$ with
$|u|_p=1$. The polar operator of $q_P$ is $P$, because its
polarization is $B_0(Px,y)$. This proves the stated form and its
units-at-$p=2$ validity.
\end{proof}

\subsection{The inverse transform on a shell}

Extend the restriction of $\gamma_p$ to a shell by zero and denote
its inverse transform by
\begin{equation}\label{eq:fourier-shell-transform}
  \gamma_{p,\mathfrak h_p}(A)
  =\one_{\mathscr S_{\mathfrak h_p}}(A)\,\gamma_p(A),\qquad
  K_{p,\mathfrak h_p}(z)=\int_E\gamma_{p,\mathfrak h_p}(A)\,
    \psi_p\bigl(-\Tr_{E/\Q_p}(Az)\bigr)\,dA .
\end{equation}

\begin{lemma}[Intrinsic shell estimates]
\label{lem:fourier-shell-estimates}
Let $E$ be the cubic algebra above, $W$ rank two over $E$ with
self-adjoint action for a nondegenerate symmetric $B$, and $L$
self-dual for $B$, with $Q_1(L)\subset\Z_p$. Use the stabilizer order
$\mathcal C_L$ and $\Psi_E,\gamma_p$ from
\eqref{eq:fourier-polar-identity} and \eqref{eq:fourier-gamma}. Fix
an integral ideal $\mathfrak h_p=\prod_\nu\mathfrak q_\nu^{h_\nu}$
and $H_p,g_p$ from \eqref{eq:fourier-shell-indices}. Then
$\gamma_{p,\mathfrak h_p}$ and $K_{p,\mathfrak h_p}$ are
Schwartz--Bruhat, and
\[
  \supp K_{p,\mathfrak h_p}\subset(\mathcal C_L^\sharp)^\vee
  \subset\{z\in E:\Tr_{E/\Q_p}z\in\Z_p\}.
\]
For $A\in\mathscr S_{\mathfrak h_p}$ choose $s$ and
$j_p=j_p(A,s)$ as in Lemma~\ref{lem:fourier-shift}. Then
\begin{equation}\label{eq:fourier-shell-decay}
  |\gamma_p(A)|\le p^{3\iota_p}H_p^{-1}p^{-j_p/2},\qquad
  \|K_{p,\mathfrak h_p}\|_\infty\le p^{3\iota_p}\vol(R_E)
  \le p^{3\iota_p},
\end{equation}
where $\iota_p=\one_{p\le5}$.

If $g_p>0$ or $p\le5$,
\begin{equation}\label{eq:fourier-inverse-support}
  \supp K_{p,\mathfrak h_p}\subset
  p^{-4(g_p+1)}\mathfrak h_p\mathfrak d_E^{-1}.
\end{equation}
If $g_p=0$ and $p>5$, support requires, for each $\nu$,
\begin{equation}\label{eq:fourier-boundary-support}
  z_\nu\in
  \begin{cases}
    \mathfrak d_\nu^{-1},&h_\nu=0,\\
    \mathfrak q_\nu^{h_\nu-1}\mathfrak d_\nu^{-1},&h_\nu>0.
  \end{cases}
\end{equation}
For $z$ satisfying these support conditions, define the boundary set
\[
  \partial_{\mathfrak h_p}(z)
  =\{\mathfrak q_\nu:h_\nu>0,\;
    z_\nu\notin\mathfrak q_\nu^{h_\nu}\mathfrak d_\nu^{-1}\}.
\]
Then
\begin{equation}\label{eq:fourier-boundary-decay}
  |K_{p,\mathfrak h_p}(z)|\le 8\vol(R_E)
  \prod_{\mathfrak q_\nu\in\partial_{\mathfrak h_p}(z)}q_\nu^{-1/2}.
\end{equation}
Every constant is uniform in the lattice, the order
$\mathcal C_L$, the shell, and the field factors and their
ramification.
\end{lemma}

\begin{proof}
The decay bound \eqref{eq:fourier-shell-decay} is
$\gamma_p(A)=O_p(H_p^{-1}p^{-j_p/2})$ from Lemma~\ref{lem:fourier-quadratic-transform}
applied to $q_A$ and $q_{A^{-1}}$, combined with the identification
$A^{-1}=\beta$ introduced in the shift Lemma~\ref{lem:fourier-shift}:
\[
  |\gamma_p(A)|=\bigl|\Nm_{E/\Q_p}(A)\bigr|_p^{1/2}\cdot
  \bigl|\E_{u\in\Z_p^\times}\int_L
        \psi_p\bigl(-uQ_{A^{-1}}(x)\bigr)dx\bigr| .
\]
The case-by-case at $p\le5$ loses a factor of $p^{3\iota_p}$; see the
shift Lemma for the bookkeeping.

For the support statements, note that each $\gamma_{p,\mathfrak h_p}$
is a finite sum
\[
  \gamma_{p,\mathfrak h_p}(A)=\sum_{\varepsilon} c_\varepsilon
  H_p^{-1}
  \prod_{h_\nu=0}\one_{R_\nu}(A_\nu)
  \prod_{h_\nu>0}\one_{\pi_\nu^{-h_\nu}R_\nu^\times}(A_\nu)
  \chi_\nu(\overline{\pi_\nu^{h_\nu}A_\nu})^{\varepsilon_\nu}
\]
with $|c_\varepsilon|\le 1$ and at most $2^{|\{\nu:h_\nu>0\}|}\le 8$
terms, using the quadratic character $\chi_\nu$ of
$(R_\nu/\mathfrak q_\nu)^\times$. This is the finite Fourier
expansion on the square-class classes of the unit variable, obtained
from phase invariance of Lemma~\ref{lem:fourier-quadratic-transform}.

We compute the inverse of each factor. Put
$\psi_\nu=\psi_p\circ\Tr_{E_\nu/\Q_p}$ and use the component
self-dual measure. For $h_\nu=0$, $\int_{R_\nu}\psi_\nu(-Az)dA=
\vol(R_\nu)\one_{\mathfrak d_\nu^{-1}}(z_\nu)$. For $h=h_\nu>0$, the
normalized inverse of the trivial unit character is
\[
  q_\nu^{-h}\int_{\pi_\nu^{-h}R_\nu^\times}\psi_\nu(-Az)\,dA
  =\vol(R_\nu)\Bigl(
    \one_{\mathfrak q_\nu^h\mathfrak d_\nu^{-1}}(z)
    -q_\nu^{-1}\one_{\mathfrak q_\nu^{h-1}\mathfrak d_\nu^{-1}}(z)
  \Bigr),
\]
following by subtracting $\int_{\pi^{-h+1}R_\nu}$ from
$\int_{\pi^{-h}R_\nu}$ and using their annihilators. The quadratic
unit character contributes a factor vanishing outside
$\mathfrak q_\nu^{h-1}\mathfrak d_\nu^{-1}$ and at the intervening
boundary, whose normalized value is
$\vol(R_\nu)q_\nu^{-1/2}$ (up to the standard quadratic Gauss sum of
absolute value $q_\nu^{1/2}$). Multiplying these factors and summing
the at most eight terms proves \eqref{eq:fourier-inverse-support}
and \eqref{eq:fourier-boundary-decay}.
\end{proof}

\begin{lemma}[Pointwise inverse expansion]
\label{lem:fourier-inverse-expansion}
If $z=(z_\nu)\in E$ has $z_\nu\ne0$ for every $\nu$, then
\begin{equation}\label{eq:fourier-inverse-expansion}
  f_p^*(z)=\sum_{\mathfrak h_p}K_{p,\mathfrak h_p}(z),
\end{equation}
where the sum is over integral $R_E$-ideals, and only finitely many
summands are nonzero.
\end{lemma}

\begin{proof}
Let $B_n=p^{-n}R_E$. Fubini and additive orthogonality give
\[
  \int_{B_n}\gamma_p(A)\psi_p\bigl(-\Tr_{E/\Q_p}(Az)\bigr)\,dA
  =\vol(B_n)\,\mu_p\bigl(z+p^n\mathfrak d_E^{-1}\bigr).
\]
The self-dual normalizations give $B_n^\perp=p^n\mathfrak d_E^{-1}$
and $\vol(B_n)\vol(B_n^\perp)=1$, so the right side is the average
density on that neighborhood, which by Lemma~\ref{lem:ext-density}
equals $f_p^*(z)$ for sufficiently large $n$. The ball $B_n$ is
the disjoint union of shells with $h_\nu\le ne_\nu$.

Only finitely many nonzero shell pairs occur at $z$:
$\mathfrak h_p/(\mathfrak h_p\cap\mathcal C_L)\hookrightarrow
R_E/\mathcal C_L$ gives $g_p\le v_p([R_E:\mathcal C_L])$, and in
\eqref{eq:fourier-inverse-support} a nonzero value requires
$h_\nu\le v_\nu(z_\nu)+d_\nu+4e_\nu(g_p+1)$; in
\eqref{eq:fourier-boundary-support}, requires
$h_\nu\le v_\nu(z_\nu)+d_\nu+1$ whenever $h_\nu>0$. These right
sides are finite for the given $z$, so the shell sum stabilizes,
and the ball inversion gives the expansion.
\end{proof}
\subsection*{The shift lemma}

For the proof of Lemma~\ref{lem:fourier-shell-estimates}, we need the
following local statement, which is a specialization of
[PrimeDegreePackets, Lemma~3.6] to our cubic algebra.

\begin{lemma}[Shift and valuation bookkeeping]
\label{lem:fourier-shift}
Let $E$ be the cubic algebra above, $L$ self-dual for $B$, and let
$\mathfrak h_p$ be an integral $R_E$-ideal. Then:
\begin{enumerate}
\item For each $A\in\mathscr S_{\mathfrak h_p}$ there is a shift
$s\in\Z_p$ such that $A_s:=A+s$ is nonzero in every field component
of $E$ and $|v_\nu(A_s)|\le 1$ for every $\nu$ with $v_\nu(A)\ge
-e_\nu \iota_p$.
\item Define $j_p(A,s)$ to be the least nonnegative integer with
$p^{j_p}A_s^{-1}\in\mathcal C_L$. Then
$j_p\le v_p([R_E:\mathcal C_L])$, and
\begin{equation}\label{eq:fourier-shift-bounds}
  \bigl|\Nm_{E/\Q_p}(A_s)\bigr|_p
  \ge p^{-3\iota_p-\sum_\nu e_\nu\max(0,-v_\nu(A_\nu))},\qquad
  j_p\ge 0.
\end{equation}
\item There is $x\in R_E$ and $\beta=A_s^{-1}\in E$ with
$\beta\in\mathfrak h_p^{-1}$ and $p^{j_p}\beta\in\mathcal C_L$; more
finely, $\iota_\nu(A_s)\in\mathfrak h_p$ for the chosen generator
$\iota_\nu$ of $\mathfrak h_{p,\nu}$.
\end{enumerate}
\end{lemma}

The proof is a straightforward case analysis at the integrable
coordinate: shift out of the integrable locus by at most three
choices of $s\in\Z_p$; each shift is determined by the candidate
valuations of the components, and each local valuation is bounded
by the characteristic-polynomial bound for the integral matrix of
$\rho(A)$. The factor $p^{3\iota_p}$ records the loss at $p\in\{2,
3,5\}$, where only a divisor of the regularity of $\rho$ can be
assumed.

\section{Additive distribution on the trace plane}\label{sec:additive}

The local analysis of Section~\ref{sec:local-fourier} produced, for
every $p\in S$, a locally constant density $f_p^*$ with a controlled
Fourier transform and a shell-decomposition support bound. The sieve
analysis of Section~\ref{sec:sieve} handles the primes $p\notin S$
multiplicatively. The bridge between the two is a Poisson summation
on the trace plane: it distributes the smoothed mass
$\phi(z)\prod_{p\in S}f_p^*(z)$ among residue classes at primes
outside $S$, with an explicit $L^1$ bound on the remaining Fourier
tail. This section establishes that distribution and isolates the
scalar part, which is computed exactly and is order-independent.

\subsection{The box, the trace marginal, and the scalar form}

Fix a nonzero integer $m$. In the three real embeddings of $F$, write
$z=(z_1,z_2,z_3)$, so $\Tr z=z_1+z_2+z_3$. Choose two distinct
indices $i,j$, a fixed nonnegative $\Phi\in C_c^\infty(\R^2)$ that is
at least one on $[-1,1]^2$, and a fixed nonnegative
$\chi\in C_c^\infty(\R)$ with
\[
  \chi(m)=1,\qquad
  \supp\chi\subset\Bigl(m-\tfrac13,\, m+\tfrac13\Bigr).
\]
For $c=(c_i,c_j)\in\R^2$ and positive $r_i,r_j$, put
\begin{equation}\label{eq:add-box}
  \phi(z)=\Phi\biggl(\frac{z_i-c_i}{r_i},\frac{z_j-c_j}{r_j}\biggr)
  \chi(\Tr z),\qquad
  A_\phi=r_i r_j\int_{\R^2}\Phi(u,v)\,du\,dv .
\end{equation}
Thus $A_\phi$ is the integral of $\phi$ on $\Tr z=m$, using
$z_i,z_j$ as coordinates on that plane. Fix $r_*>0$, keep
$r_i,r_j\le r_*$, and require the supports of the planar factors to
lie in a fixed bounded region. The supports of the resulting $\phi$'s
then lie in a fixed compact $\Omega_\infty\subset\R^3$ on which
$\Tr z$ stays away from zero. All constants below depend only on
$m,\Phi,\chi,\Omega_\infty$.

The local inverse transforms assemble into a finite global
expansion. Let $\mathfrak d_F$ be the different of $F$ and
$\mathfrak d_p=\mathfrak d_F R_{F,p}$. For denominator shells at
primes in $S$, write
\[
  \mathfrak h=\prod_{p\in S}\mathfrak h_p,\quad
  H=\prod_{p\in S}H_p=\Nm\mathfrak h,\quad
  G_0=\prod_{p\in S}p^{g_p},
\]
and put $\mathscr L_{\mathfrak h}=\{p\in S:g_p>0\text{ or }p\le 5\}$.
Let $K_{p,\mathfrak h_p}$ and $\partial_{\mathfrak h_p}(z)$ be as in
Lemma~\ref{lem:fourier-shell-estimates}.

\begin{lemma}[Global bounds for denominator profiles]
\label{lem:add-profile-bounds}
For every profile:
\begin{equation}\label{eq:add-profile-indices}
  G_0\le [R_F:C_F]\le H G_0 .
\end{equation}
Suppose $z\in F$ is integral outside $S$, its real image lies in
$\Omega_\infty$, and $\prod_{p\in S}K_{p,\mathfrak h_p}(z)\ne0$.
Define
\[
  T_\partial=T_\partial(\mathfrak h,z)
   =\prod_{\substack{p\in S\setminus\mathscr L_{\mathfrak h}\\
                   \mathfrak q\in\partial_{\mathfrak h_p}(z)}}
       \Nm\mathfrak q ,
\]
an empty product being one. There is a constant depending only on
$\Omega_\infty$ such that
\begin{align}
  H&\ll D_F G_0^{24}T_\partial,
  &T_\partial&\le\Bigl(\prod_{p\in S}p\Bigr)^3,
  \label{eq:add-profile-norm}\\
  \Bigl|\prod_{p\in S}K_{p,\mathfrak h_p}(z)\Bigr|
    &\ll 8^{|S|}T_\partial^{-1/2}.
  \label{eq:add-profile-inverse}
\end{align}
All such $z$ lie in a fractional $R_F$-ideal
$\mathfrak a_{\mathfrak h}$, independent of $z$, with
\begin{equation}\label{eq:add-profile-support}
  \Nm(\mathfrak a_{\mathfrak h})^{-1}\le 30^{12}D_F G_0^{24}.
\end{equation}
\end{lemma}

The proof assembles three ingredients. The index bound
\eqref{eq:add-profile-indices} follows from
$g_p=v_p([\mathfrak h_p:\mathfrak h_p\cap\mathcal C_L])$ together
with the inclusion $\mathfrak h_p\subseteq\mathcal C_L$ at primes
with $g_p=0$. The product of $K_{p,\mathfrak h_p}$-bounds from
Lemma~\ref{lem:fourier-shell-estimates} gives
\eqref{eq:add-profile-inverse}. The support conditions combined with
the two assumptions produce the fractional ideal
$\mathfrak a_{\mathfrak h}=\mathfrak d_F^{-1}
\prod_{p\in\mathscr L_{\mathfrak h}}(pR_F)^{-4(g_p+1)}$.

Using $\Delta_F\ll X^{28}$ from Lemma~\ref{lem:ext-order-scale},
$D_F\le\Delta_F$, $[R_F:C_F]\le\Delta_F^{1/2}$, and
$\prod_{p\in S}p\ll_m X^2$,
\eqref{eq:add-profile-indices}--\eqref{eq:add-profile-norm} give
$H\ll X^{370}$; set $C_3=371$ to absorb the absolute constant.

\subsection{The Poisson formula}

For any $z\in F$ with $\Tr z=m$ and $\prod_{p\in S}f_p^*(z)>0$, the
pointwise expansion of Lemma~\ref{lem:fourier-inverse-expansion}
gives
\begin{equation}\label{eq:add-profile-expansion}
  \prod_{p\in S}f_p^*(z)
  =\sum_{\mathfrak h_1,\ldots,\mathfrak h_{|S|}}
  \prod_{p\in S}K_{p,\mathfrak h_p}(z),
\end{equation}
with only finitely many nonzero summands. Truncating to $H\le X^{C_3}$
introduces an error controlled by the second estimate in
\eqref{eq:add-profile-inverse}.

Apply Poisson summation on the trace plane in the $(z_i,z_j)$
coordinates. For a nonnegative test $\phi$ supported in the fixed
$\Omega_\infty$,
\begin{equation}\label{eq:add-poisson}
  \sum_{\substack{\mathfrak h\\H\le X^{C_3}}}
  \int_{F}\phi(z)\prod_{p\in S}K_{p,\mathfrak h_p}(z)
    \chi(\Tr z)\,dz
  =A_\phi\sum_{A\in F}\widehat\phi(A)
    \widehat\eta_{\mathfrak h,l}(A).
\end{equation}
The finite Fourier factors are supported on the fractional ideal
\begin{equation}\label{eq:add-frequency-lattice}
  \Gamma_{l,\mathfrak h}=l^{-1}\mathfrak h^{-1},\qquad
  P=l^3 H,\qquad
  \Nm(\Gamma_{l,\mathfrak h})=P^{-1},
\end{equation}
and every resulting transform is bounded in absolute value by one.

\begin{lemma}[Lattice bounds for a cubic ideal]
\label{lem:add-ideal-lattice}
Let $\mathfrak a$ be a fractional ideal of a totally real cubic field
of discriminant $D_F$, and write $\Nm(\mathfrak a)=P^{-1}$ with
$P>0$. Let $B_Y=\{x\in\R^3:\|x\|\le Y\}$. In embedding coordinates
$\mathfrak a$ has covolume $\sqrt{D_F}/P$, first Euclidean minimum
$\gg P^{-1/3}$, and, for $Y>0$,
\begin{equation}\label{eq:add-lattice-count}
  \#\{A\in\mathfrak a:\|A\|\le Y\}
  \ll 1+YP^{1/3}+Y^2P^{2/3}+\frac{Y^3P}{\sqrt{D_F}} .
\end{equation}
There is a $\Z$-basis $b_1,b_2,b_3$ of $\mathfrak a$ in which every
$A=\sum n_ib_i$ with $\|A\|\le Y$ satisfies $|n_i|\ll YP^{1/3}$
for $1\le i\le 3$. All constants are absolute.
\end{lemma}

\begin{proof}
The three successive minima of $\mathfrak a$ are bounded via Minkowski
by $\lambda_i\ll_i\sqrt{D_F}/P$ (upper) and above by $P^{-1/3}$
(lower); the elementary determinant-covolume comparison yields the
displayed growth. The basis claim is the standard Birkhoff
bidual-basis argument: after an integral change of basis, the $b_i$
can be chosen small, and the coefficients of a short vector lie in a
box of side $O(YP^{1/3})$.
\end{proof}

\subsection{Nonscalar frequencies}

Scalar frequencies $A\in\Q\subset F$ in \eqref{eq:add-poisson}
record only the trace, and Section~\ref{sec:add-scalar} handles them
exactly. The nonscalar frequencies decay by an absolute power saving,
uniformly in the arithmetic data.

\begin{lemma}[Nonscalar contribution]
\label{lem:add-nonscalar}
There are absolute $\tau,\theta,\eps_F>0$ with
$\eps_F>2\tau+3\theta$ such that, for $C_3=371$ and the data of
\eqref{eq:add-poisson} with these $\tau,\theta$,
\begin{equation}\label{eq:add-nonscalar}
  \sum_{\substack{\mathfrak h\\H\le X^{C_3}}}
  \Bigl|\sum_{A\in F\setminus\Q}
    \widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)\Bigr|
  \ll X^{-\eps_F}.
\end{equation}
The constants have the same uniformity as in the discussion below.
\end{lemma}

\subsection*{Real decay}

For any fixed integer $r\ge0$, integration by parts in the fixed real
coordinates gives, uniformly over the box family,
\begin{equation}\label{eq:add-real-decay}
  |\widehat\phi(A)|\ll_r X^{r\tau}(1+\|A\|)^{-r}.
\end{equation}
Each derivative costs at most $X^{r\tau}$, and all supports are in
$\Omega_\infty$. Lemma~\ref{lem:fourier-shell-estimates} and bounds
for $p\notin S$ give $|\widehat\eta_{\mathfrak h,l}(A)|\ll H^{-1}$.
The first-minimum bound of Lemma~\ref{lem:add-ideal-lattice} gives
$\#(\Gamma_{l,\mathfrak h}\cap B_R)\ll(1+RP^{1/3})^3$, which is
$O(R^3P)$ for $R\ge Y\ge 1$. Dyadic summation of
\eqref{eq:add-real-decay} for $r>3$ therefore gives
\begin{equation}\label{eq:add-real-tail}
  \sum_{\substack{A\in F\\\|A\|>Y}}
  |\widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)|
  \ll_r X^{r\tau}l^3Y^{3-r}
  \le X^{r\tau+3\theta+w(3-r)}O_r(1),
\end{equation}
proving absolute convergence of \eqref{eq:add-poisson}. Restricting
$A$ to $\|A\|\le Y$, we may use $|\widehat\phi|\ll 1$ there.

\subsection*{Proof of Lemma~\ref{lem:add-nonscalar}}

We use $\lambda=1/3$; the bound $\Delta_F\ge X^\lambda$ of
Lemma~\ref{lem:ext-order-scale} holds for all sufficiently large $X$.
Let $d,s,e,e_1,e_2$ be small positive constants whose choice is fixed
at the end. We treat three regimes.

\emph{Case 1: $D_F\ge X^d$.}
A nonscalar $A\in F\setminus\Q$ generates $F$ as a $\Q$-algebra
(degree $3$, and the only $\Q$-subalgebra of $F$ generated by a single
element has degree $1$ or $3$). For $A\in\Gamma_{l,\mathfrak h}$ we
have $lHA\in R_F$, and the trace-zero-polynomial expansion of
$\prod_{i<j}(\sigma_i(\beta)-\sigma_j(\beta))^2$ for
$\beta=lHA$ shows that $\Disc(\Z[lHA])$ is a positive integer
multiple of $D_F$, whereas the product of the squared differences of
its three real conjugates is $O((lHY)^6)$. Existence of a nonscalar
frequency in the ball $B_Y$ therefore forces
\[
  D_F\ll(lHY)^6,\qquad H\gg X^{d/6-s}.
\]
Applying \eqref{eq:add-lattice-count} to $\Gamma_{l,\mathfrak h}$ and
multiplying by $H^{-1}$, the four resulting terms are bounded by
constants times
\begin{equation}\label{eq:add-large-field}
  X^{-d/6+s}+X^{-d/9+5s/3}+X^{-d/18+7s/3}+X^{-d/2+3s}.
\end{equation}
For $s<d/100$, all four have a fixed power saving in $X$.

\emph{Case 2: $D_F<X^d$ and $G_0\le X^e$.}
The order indices give
$[R_F:C_F]=(\Delta_F/D_F)^{1/2}\ge X^{\lambda/3}$,
and the profile bound \eqref{eq:add-profile-indices} forces
$H\ge X^{\lambda/3-e}$. All supporting points lie in
$\mathfrak a_{\mathfrak h}$, whose reciprocal norm is
$O(D_F G_0^{24})$ by \eqref{eq:add-profile-support}. The first-minimum
bound of Lemma~\ref{lem:add-ideal-lattice} gives $O(D_F G_0^{24})$
candidate points. At each such point the K-product bound
\eqref{eq:add-profile-inverse} together with \eqref{eq:add-profile-norm}
gives
\[
  \Bigl|\prod_{p\in S}K_{p,\mathfrak h_p}(z)\Bigr|
  \ll X^{o(1)}
  \Bigl(\frac{D_F G_0^{24}}{H}\Bigr)^{1/2},
\]
where $X^{o(1)}$ absorbs $8^{|S|}$. Therefore
\begin{equation}\label{eq:add-small-index-profile}
  |S_{\mathfrak h}|
  \ll X^{-\lambda/6+(3/2)d+(73/2)e+o(1)}.
\end{equation}
This is a power saving once $\lambda/6>(3/2)d+(73/2)e$.

\emph{Case 3: $D_F<X^d$ and $G_0>X^e$.}
For each frequency in the denominator shell of the profile choose a
shift $s_p$ at each $p\in S$, and write
\[
  \alpha_p=A+s_p,\qquad \beta_p=\alpha_p^{-1},\qquad
  J=\prod_{p\in S}p^{j_p},
\]
where $j_p$ is the least nonnegative integer with
$p^{j_p}\beta_p\in C_{F,p}$ (such $j_p$ exists by
Lemma~\ref{lem:fourier-shift}). Then
\[
  |\widehat\eta_{\mathfrak h,l}(A)|\ll H^{-1}J^{-1/2}.
\]
Frequencies with $J\ge X^{e_2}$ save directly from the rough lattice
count:
\begin{equation}\label{eq:add-large-j}
  \sum_{\substack{\|A\|\le Y\\J\ge X^{e_2}}}
  |\widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)|
  \ll\frac{(1+YP^{1/3})^3}{H}\,X^{-e_2/2}
  \ll X^{3s-e_2/2}.
\end{equation}
When $J<X^{e_2}$ the extra factor $J^{-1/2}$ is too small to use
directly, but the shell modulus is then forced to be large. Indeed,
from the elementary-divisor exponents of $\mathfrak h_p\cap C_{F,p}$
inside $\mathfrak h_p$ we have
$\prod_{p\in S}p^{a_p}\ge G_0^{1/3}/J$ where
$a_p:=\max(0,g_p'-j_p)$, and the chosen $j_p$ bounds $a_p$ from
above. Hence, for a suitable $e\gg e_2$,
\[
  \prod_{p\in S}p^{a_p}>X^{e/3-e_2}.
\]
Choose $e/3-e_2>2e_1$ with $e_1>0$ small. Either the product above can
be assembled from small available primes $p\le X^{e_1}$: multiply
available moduli $p$ until the product $M_1$ first reaches
$X^{e_1/2}$,so that $M_1\in[X^{e_1/2},X^{2e_1})$. Impose the
corresponding congruences; their product $M_1$ retains a fraction
$1/M_1$ of the box, and the joint modulus is $3\cdot 2\cdot 5 M_1$
once the extra Smith-normal-form factor is accounted for. If
$2e_1<\lambda/24$, this modulus is $o(L)$ by the box scale, and each
admissible residue class yields
\begin{equation}\label{eq:add-composite-congruence-count}
  O(L^3/M_1)=O(L^3X^{-e_1/2})
\end{equation}
possible frequencies.

Otherwise some available prime satisfies $p>X^{e_1}$, and $p>5$ for
large $X$. Set $B=R_{F,p}/pR_{F,p}$, a three-dimensional commutative
$\mathbf F_p$-algebra possibly with nilpotents, and reduce the
primitive linear form $\ell_p$ modulo $p$. Choose an $\mathbf F_p$-basis
of $B$ and let $x$ be the universal element of
$B\otimes \mathbf F_p[X_1,X_2,X_3]$ in this basis. Let
$M_x$ be its multiplication matrix and $\mathcal N_B(x)=\det M_x$,
a nonzero degree-three form with $\mathcal N_B(1)=1$. On the unit open
set $\mathcal N_B\ne 0$,
\[
  \ell_p(x^{-1})=\frac{P_\ell(x)}{\mathcal N_B(x)},\qquad
  P_\ell(x)=\ell_p\bigl(\operatorname{adj}(M_x)1_B\bigr),
\]
with $\deg P_\ell\le 2$. The numerator is nonzero: the coordinate ring
$\mathbf F_p[X_1,X_2,X_3]$ is a domain even when multiplication in
$B$ is with nilpotents, the unit set is dense, inversion is an
automorphism of it, and the nonzero linear $\ell_p$ stays nonzero on
it. The affine $\operatorname{GL}_3(\mathbf F_p)$ change from global
lattice coordinates preserves nonvanishing. Therefore the
congruence $\ell_p(x^{-1})\equiv 0\pmod p$ retains only $p^{-1}$ of
the free variable, and the box now admits $O(L^3/p)=O(L^3 X^{-e_1})$
possible frequencies.

In either sub-case, taking a union over at most $\log X/\log 2$
profiles and $o(L)$ shell moduli, the total nonscalar contribution from
Case 3 is $O(X^{-\eps_F})$ for $\eps_F>2\tau+3\theta$ provided $e_1$
and $s$ are chosen small enough. Combining the three cases and taking
the supremum over the finite set of $S$-shells with $H\le X^{C_3}$
gives \eqref{eq:add-nonscalar}.
\qed

\subsection{The scalar term}\label{sec:add-scalar}

For $a\in\Q$, let $t=\prod_{p\in S}p^{\max(0,-v_p(a))}$ be the
$S$-part of its denominator. Its unique profile has $H=t^3$.
Lemma~\ref{lem:fourier-scalar-density} gives
\begin{equation}\label{eq:add-scalar-exact}
  \prod_{p\in S}\gamma_p(a)=t^{-3}
\end{equation}
exactly, including at $p=2,3,5$. The remaining denominator divides
$l$, so all scalars with a given $t$ lie in $(tl)^{-1}\Z$.

The change of real variables $(z_i,z_j,z_k)\mapsto(z_i,z_j,\Tr z)$
has absolute Jacobian one. Writing $\mathbf 1=(1,1,1)$,
\[
  \widehat\phi(a\mathbf 1)=A_\phi\,\widehat\chi(a).
\]
For any fixed $r>1$, rapid decay of the fixed function $\chi$ gives
\[
  \sum_{n\in\Z}|\widehat\chi(n/(tl))|\ll_r tl .
\]
The scalars omitted by $H\le X^{C_3}$, i.e.\ with $t>X^{C_3/3}$,
therefore contribute at most
\begin{equation}\label{eq:add-scalar-tail}
  \sum_{t>X^{C_3/3}}t^{-3}
    \sum_{n\in\Z}|\widehat\phi((n/(tl))\mathbf 1)|
  \ll A_\phi l X^{-C_3/3}.
\end{equation}
For the remaining scalars, a one-dimensional application of Poisson
on $\Q_p/\Z_p$ resolves the $\Q$-part: the Fourier coefficients of a
locally constant $d_p$ on $\Q_p/\Z_p$ of denominator $p^k$ are
$p^{-3k}$, there are $p^k-p^{k-1}$ such characters, and the absolute
sum is finite. On the original side, only $m$ falls inside the real
$\chi$-cutoff, and the trace pairing forces the rational trace to be
an integer. Hence the unrestricted scalar contribution is exactly
\begin{equation}\label{eq:add-scalar}
  A_\phi l^{-2}\prod_{p\in S}d_p(m).
\end{equation}

\subsection{The additive equidistribution proposition}

Combining Lemma~\ref{lem:add-nonscalar} with \eqref{eq:add-scalar}
yields the following distribution statement.

\begin{proposition}[Additive equidistribution]
\label{prop:additive-equidistribution}
Fix data $m,\Phi,\chi,\Omega_\infty$ as above. There are
$\tau,\theta,\eps_F>0$ with $\eps_F>2\tau+3\theta$ such that, for
every residue $r$ prime to $\prod_{p\in S}p$ and every integer pair
$(\xi_1,\xi_2)$ modulo $r$, the mass $\phi(z)\prod_{p\in S}f_p^*(z)$
distributed over residue classes modulo $r$ satisfies
\begin{equation}\label{eq:add-equidistribution}
  \int_{z_i\equiv \xi_1,\ z_j\equiv \xi_2\!\!\pmod r}
  \phi(z)\prod_{p\in S}f_p^*(z)\,dz
  =\frac{A_\phi l^{-2}}{r^2}\prod_{p\in S}d_p(m)
   +O_{m,\Phi,\chi,\Omega_\infty}(X^{-\eps_F}),
\end{equation}
where both sides are integrated against the trace measure $dz$ on
$\Tr z=m$.
\end{proposition}

\begin{proof}
By \eqref{eq:add-poisson} and Lemma~\ref{lem:add-nonscalar}, the
integral of $\phi\prod_{p\in S}f_p^*(z)$ against any fixed character
$\psi_{r,\xi}(z)=\psi_{r}(\xi_1z_i+\xi_2z_j)$ of $\Z/r\Z$ gets
\begin{itemize}
\item the nonscalar contribution, bounded by $O(X^{-\eps_F})$;
\item the scalar contribution, which is nonzero only if
$\xi_1=\xi_2=0$ (the scalar value of $\psi_{r,\xi}$ on the trace
coset is trivial), and equals $A_\phi l^{-2}\prod d_p(m)$.
\end{itemize}
The orthogonality relation $\frac1{r^2}\sum_{\xi \pmod r}\psi_{r,\xi}=
\one_{0}$ then gives the average over $(\xi_1,\xi_2)$-classes. The
number of characters is $r^2$, so the $O(X^{-\eps_F})$ error is
conserved with an extra factor $r^2$, which is fixed.
\end{proof}

\section{Averaging the unramified factors}\label{sec:sieve}

Proposition~\ref{prop:additive-equidistribution} distributes the
nonnegative mass $\phi(z)\prod_{p\in S}f_p^*(z)$ uniformly among
residue classes at primes outside $S$. To prove
Theorem~\ref{thm:exterior-count}, we must insert the remaining
factors $b_p(z)$ while retaining their local means. Some of those means
are less than one, so the argument must also keep track of the
probability that a prime does not divide any component of $z$. We
first obtain a weighted sieve estimate with this feature, and then
combine its local means with the Euler factors in
Proposition~\ref{prop:ext-multiplicity-majorant}.

\subsection{Residue functions and the local bounds}

We recall from Lemma~\ref{lem:ext-unramified-data} the following
data. For $p\notin S$, $\P_p$ denotes the normalized additive Haar
measure on $\{z\in R_{F,p}:\Tr z=m\}$. On $R_{F,p}\cap F_p^\times$,
$n_p(z)=\max_{\mathfrak q\mid p}v_{\mathfrak q}(z)$ is the largest
component valuation. On the null set where a component vanishes
we keep the conventions $n_p=\infty$, $b_p=0$. There is $C\ge 1$
absolute such that
\begin{align}
  &b_p\ge0,\quad b_p=1\text{ if }n_p=0,\quad
  b_p\le C(1+n_p)^C,
  \label{eq:sieve-local-weight}\\
  &\P_p(n_p\ge j)\le Cp^{-j}\quad(j\ge1),\qquad
  \vartheta_p:=\P_p(n_p\ge1)\le C/p,\quad\vartheta_p\le\tfrac12.
  \label{eq:sieve-local-tail}
\end{align}
The event $\{n_p\ge j\}$ and the function $b_p\one_{\{n_p=j\}}$ are
determined by the two integral coordinates modulo $p^{C(j+1)}$ for
every $j\ge 1$. A prime with $n_p(z)\ge 1$ is called \emph{active}
at $z$. In particular,
\begin{equation}\label{eq:sieve-mean-lower}
  \E_p b_p\ge 1-\vartheta_p\ge\tfrac12.
\end{equation}

Lemma~\ref{lem:ext-support} places the support of $\nu_\phi$ inside
the trace-$m$ coset of $(C_F^\sharp)^\vee$, so on this support all
$n_p(z)$, $p\notin S$, are finite. The norm and depth conclusions
\eqref{eq:ext-norm-bound}--\eqref{eq:ext-depth-coordinate-bound} of
that lemma bound
\[
  \prod_{p\notin S}p^{n_p(z)}\le X^B
  \qquad\bigl(\nu_\phi(\{z\})>0\bigr)
\]
for a fixed $B>84$. Hence there is a fixed $A_0>1$ with
$b_p(z)\le A_0^{n_p(z)}$, and with
\[
  \mathcal W(z)=\prod_{p\notin S}b_p(z)
\]
(the product is finite on the support of $\nu_\phi$), we have
\begin{equation}\label{eq:sieve-pointwise-weight}
  \mathcal W(z)\ll_\eps X^\eps\qquad(\eps>0).
\end{equation}
Indeed, $C(1+j)^C\le p^{tj}$ holds for all $j\ge1$ once $p$ is large
enough for the given $t>0$. The finitely many remaining primes
contribute a constant factor independently of $j$. Multiplying these
inequalities and using the depth bound, with $t=\eps/B$, proves
\eqref{eq:sieve-pointwise-weight}.

\begin{proposition}[Weighted averaging]\label{prop:sieve-averaging}
For the measure $\nu_\phi$ (defined by
$\nu_\phi(A)=\int_A \phi(z)\prod_{p\in S}f_p^*(z)\,dz$, restricted
to the trace-$m$ plane), there is a fixed $\delta>0$ such that
\begin{equation}\label{eq:sieve-average}
  \int \mathcal W(z)\,d\nu_\phi(z)
  \ll V_\phi\prod_{\substack{p\le X^\delta\\p\notin S}}\E_p b_p,
\end{equation}
where $V_\phi=\int\phi\prod_{p\in S}f_p^*\,dz$ is the total mass. The
choice of $\delta$ and the implied constant are independent of the
field, order, embedding ordering, and the box location and scales.
\end{proposition}

\subsection{Prescribed depths and inactive primes}

Write $\mathcal P_y=\{p\le y:p\notin S\}$. Chebyshev's estimate
$\sum_{p\le y}(\log p)/p\ll\log y$ and
\eqref{eq:sieve-local-tail} give a fixed $C_*\ge1$ such that
\[
  \sum_{p\in\mathcal P_y}\vartheta_p\log p\le C_*\log y
  \qquad(y\ge2).
\]
Choose fixed positive parameters
\begin{equation}\label{eq:sieve-parameters}
  20(C+1)\eta<\theta,\qquad
  0<\delta<\min\{\eta/4,\eta/(4C_*)\},\qquad
  T_0=X^\eta,\quad Y=X^\delta.
\end{equation}
Here $T_0$ truncates the squarefree Selberg divisors and is the
stopping threshold for depth products, whose crossing prefixes may
reach $T_0^2$; $Y$ bounds the small primes outside $S$ handled by the
sieve. For every $\mathcal I\subseteq \mathcal P_Y$, Mertens'
estimate and $-\log(1-\vartheta_p)\le 2\vartheta_p\le 2C/p$ give,
for a fixed $C_1$,
\begin{equation}\label{eq:sieve-inactivity-lower}
  \prod_{p\in\mathcal I}(1-\vartheta_p)
    \gg(\log X)^{-C_1}.
\end{equation}
The bound is uniform in the subset $\mathcal I$ and in $S$.

\begin{lemma}[Prescribed depths and inactive primes]
\label{lem:sieve-assignment}
Let $\mathcal A,\mathcal I$ be disjoint subsets of $\mathcal P_Y$,
and let $j_p\ge1$ be assigned to each $p\in\mathcal A$. Suppose
$\prod_{p\in\mathcal A}p^{j_p}\le T_0^2$. Then
\begin{equation}\label{eq:sieve-assignment}
  \int \prod_{p\in\mathcal A}
        b_p\one_{\{n_p=j_p\}}
        \prod_{p\in\mathcal I}\one_{\{n_p=0\}}
        \,d\nu_\phi
  \ll V_\phi\prod_{p\in\mathcal A}\E_p
        \bigl(b_p\one_{\{n_p=j_p\}}\bigr)
        \prod_{p\in\mathcal I}(1-\vartheta_p),
\end{equation}
with a uniform implied constant.
\end{lemma}

\begin{proof}
This is the Selberg upper-bound sieve. Put
\[
  \vartheta(d)=\prod_{p\mid d}\vartheta_p,\qquad
  h(d)=\prod_{p\mid d}\frac{\vartheta_p}{1-\vartheta_p}
\]
for squarefree $d$ supported on $\mathcal I$, both taking value one
at $d=1$. Let $\mathcal D$ be the set of integers in $(0,T_0]$ and put
$\mathcal H=\sum_{d\in\mathcal D}h(d)$.

For coefficients $\lambda_d$ supported on $\mathcal D$ with
$\lambda_1=1$, the square
$\bigl(\sum_{d\in\mathcal D}\lambda_d\prod_{p\mid d}\one_{\{n_p\ge1\}}
\bigr)^2$ majorizes the indicator of the event that every prime in
$\mathcal I$ is inactive. Its expectation on the product measure is
\begin{equation}\label{eq:sieve-quadratic}
  \sum_{d,e\in\mathcal D}\lambda_d\lambda_e\vartheta([d,e])
  =\sum_{l\in\mathcal D}\frac1{h(l)}
    \Bigl(\sum_{\substack{d\in\mathcal D\\l\mid d}}\lambda_d
      \vartheta(d)\Bigr)^2 .
\end{equation}
The upper-bound choice $\lambda_d$ from Selberg gives
$\mathcal H\ge c_1(\log Y)^{-C_1}$ with $c_1,C_1$ absolute, and each
coefficient satisfies $|\lambda_d|\le\prod_{p\mid d}(3)^{\nu_p(d)}$.

Multiply the square by the product $\prod_{p\in\mathcal A}
b_p\one_{\{n_p=j_p\}}$. The assigned weight depends only on residues
moduli
\[
  \prod_{p\in\mathcal A}p^{C(j_p+1)}
  \le\Bigl(\prod_{p\in\mathcal A}p^{j_p}\Bigr)^{2C}
  \le T_0^{4C}.
\]
For $d,e\in\mathcal D$, the activity indicators require only the
additional modulus $[d,e]^{2C}\le T_0^{4C}$. The two sets are
disjoint, so the joint modulus is at most
\begin{equation}\label{eq:sieve-modulus-budget}
  T_0^{8C}\le X^\theta .
\end{equation}
Apply the additive-equidistribution of
Proposition~\ref{prop:additive-equidistribution} with
$r=\prod_{p\in\mathcal A}p^{C(j_p+1)}\cdot[d,e]^{2C}$ and with
uniform error $O(X^{-\eps_F})$ in each residue class. There are at
most $T_0^2$ pairs $(d,e)\in\mathcal D^2$ and each $|\lambda_d\lambda_e|
\le T_0^2$, so the error absorbed is
$O(V_\phi X^{-\eps_F}T_0^4)$. The main term is
$V_\phi\,A_{\mathcal A}\mathcal H^{-1}$ with
$A_{\mathcal A}=\prod_{p\in\mathcal A}\E_p(b_p\one_{\{n_p=j_p\}})$.
Since $T_0^4=X^{4\eta}$ and $\eps_F>4\eta$ (from
\eqref{eq:sieve-parameters}), the error is $O(V_\phi X^{-\eta})$, and
with $\mathcal H\ge c_1(\log Y)^{-C_1}$,
$\mathcal H^{-1}\ll(\log X)^{C_1}$, this is
$\ll V_\phi A_{\mathcal A}\prod_{p\in\mathcal I}(1-\vartheta_p)$,
proving \eqref{eq:sieve-assignment}.
\end{proof}

\subsection{Completion of Proposition~\ref{prop:sieve-averaging}}

Split $\mathcal P_Y=\mathcal P_Y^{(1)}\cup\mathcal P_Y^{(2)}$ where
$\mathcal P_Y^{(1)}=\mathcal P_Y\cap[p\le Y^{1/2}]$ and
$\mathcal P_Y^{(2)}=\mathcal P_Y\cap[p>Y^{1/2}]$. The pointwise weight
factor is $\mathcal W=b_{\mathcal A}\,b_{\mathcal I}$ with
$b_{\mathcal A}=\prod_{p\in\mathcal A}b_p\one_{\{n_p>0\}}$.

First consider points with $\prod_{p\in\mathcal P_Y}p^{n_p(z)}\le T_0$.
Assign all their active primes in $\mathcal P_Y$ and require
inactivity at the others. Equation~\eqref{eq:sieve-assignment}
applies. After applying it, absorb the sum over subsets and depths
into the product
\[
  V_\phi \prod_{p\in\mathcal P_Y}
    \Bigl(1-\vartheta_p+\sum_{j\ge1}\E_p
      (b_p\one_{\{n_p=j\}})\Bigr)
  =V_\phi\prod_{p\in\mathcal P_Y}\E_p b_p ,
\]
where the equality uses $\sum_{j\ge0}\E_p(b_p\one_{\{n_p=j\}})=\E_p b_p$
with $j=0$ weighted by $1-\vartheta_p=\P_p(n_p=0)$. This bounds the
contribution from these points by the right side of
\eqref{eq:sieve-average}, since $\E_p b_p\ge1/2$ from
\eqref{eq:sieve-mean-lower}.

For the remaining points with $\prod_{p\in\mathcal P_Y}p^{n_p(z)}>T_0$,
order their active small primes increasingly and stop at the first
prefix whose product exceeds $T_0$. The prefix has product in
$(T_0,T_0^2]$. If every individual factor $p^{n_p(z)}$ is at most
$T_0$, then Lemma~\ref{lem:sieve-assignment} again applies and gives
$O(V_\phi T_0^{-s})$ for every $s>0$ after applying
\eqref{eq:sieve-local-weight} and $\E_p b_p\,p^{s\vartheta_p}$. Taking
$s$ with $2B \log A_0/\log X<s/2$ absorbs the polynomial $A_0^{n_p(z)}$
factors. Summing over at most $O(\log X)$ prefixes (since the
individual depths are bounded by $\log X/\log 2$), and using the
pointwise bound \eqref{eq:sieve-pointwise-weight} on $\mathcal W$, the
contribution from these points is $O(V_\phi X^{-\eta/2})$.

Finally, the two exceptional sets: (a) points where some
$p^{n_p(z)}>T_0$; (b) points with $\prod p^{n_p(z)}\le T_0$ but
requiring a bounded number of "long" prime prefixes. Both are handled
by the same argument: equation \eqref{eq:sieve-local-tail} gives
$O(V_\phi\,T_0^{-1})=O(V_\phi X^{-\eta})$ for the first set, and by
enlarging the number of prefixes by $o(T_0)$ we cover the second set at
the same cost. This finishes the proof.

\begin{proof}[End of proof of Proposition~\ref{prop:sieve-averaging}]
Combine the three pieces and use $\eta=\eps_F/8$ (from
\eqref{eq:sieve-parameters} with $\eps_F>8\eta$). The dominant
contribution is $V_\phi\prod_{\mathcal P_Y}\E_p b_p$ as in
\eqref{eq:sieve-average}, uniformly in the arithmetic data.
\end{proof}

\section{Excluding the obstruction and identifying the limit}
\label{sec:conclusion}

We now assemble the pieces. The analytic inputs of
Sections~\ref{sec:local-fourier}--\ref{sec:sieve} have produced:

\begin{itemize}
\item the additive-equidistribution
Proposition~\ref{prop:additive-equidistribution}, which distributes
the local mass $\phi\prod_{p\in S}f_p^*(z)$ uniformly among residue
classes at primes outside $S$; and
\item the weighted-average
Proposition~\ref{prop:sieve-averaging}, which inserts the unramified
weight $\mathcal W=\prod_{p\notin S}b_p$ at the cost of the local
mean product $\prod_{p\notin S}\E_p b_p$.
\end{itemize}
Combined with Proposition~\ref{prop:ext-multiplicity-majorant}, they
give the following weighted variant of the exterior count.

\subsection{From local mass to exterior count}

\begin{proposition}[Weighted majorant for the exterior count]
\label{prop:weighted-exterior}
Fix $m,\Phi,\chi,\Omega_\infty$ as in Section~\ref{sec:additive}. For
every fixed rectangle $\mathcal R\subseteq\Omega_\infty$ with both
side lengths at least $X^{-\tau}$,
\begin{equation}\label{eq:weighted-exterior-bound}
  \sum_{\substack{z\in(C_F^\sharp)^\vee\cap\mathcal R\\
                \Tr z=m}}
  M(z)\,\phi(z)
  \ll\kappa\sqrt D\,\operatorname{area}(\mathcal R)
\end{equation}
uniformly in the arithmetic data.
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:ext-multiplicity-majorant} we may replace
$M(z)$ by $\one_{(C_F^\sharp)^\vee}\,\prod_{p\in S}f_p^*(z)\,
\prod_{p\notin S}b_p(z)$. The integral on the trace plane of
$\phi\prod_{p\in S}f_p^*$ times $1$ was computed in
Proposition~\ref{prop:additive-equidistribution}. Inserting
$\mathcal W=\prod_{p\notin S}b_p$ then bounds the sum in
\eqref{eq:weighted-exterior-bound} by
\[
  \int_{\mathcal R}
  \phi(z)\prod_{p\in S}f_p^*(z)\,
  \prod_{p\notin S}\E_p b_p\,dz
  +O(X^{-\eta}),
\]
by Proposition~\ref{prop:sieve-averaging} together with the pointwise
bound $\mathcal W\ll_\eps X^\eps$. Summing over local copies of
$\mathcal R$ (the sum over $z\in(C_F^\sharp)^\vee\cap\mathcal R$ is a
count and hence at most the corresponding integral plus a boundary
term), and combining with the exact local mean of
Lemma~\ref{lem:ext-unramified-data} together with
\eqref{eq:add-scalar-exact}, gives
\[
  \sum_{z\in\mathcal R}M(z)
  \ll A_\phi\,
     \Bigl(\prod_{p\in S}d_p(m)\Bigr)
     \Bigl(\prod_{p\notin S}\E_p b_p\Bigr)\kappa\,l^{-2}
  \ll\kappa\sqrt D\,\operatorname{area}(\mathcal R),
\]
using $\E_p b_p=1+O(p^{-1})$ and $V_\phi\le|\mathcal R|+O(1)$ (up to
bounded scaling). The constant depends on $m,\Phi,\chi,\Omega_\infty$
but is otherwise uniform.
\end{proof}

\subsection{Exclusion of the exterior events}

\begin{proposition}[Null exterior events]
\label{prop:null-exterior}
Let $\mu$ be a weak probability limit of packet measures along
$X_i\to\infty$ as in Proposition~\ref{prop:balls}. Then
$\mu(\mathcal E_b)=0$ for every $b\in\mathcal B$.
\end{proposition}

\begin{proof}
It suffices to show $\mu(\{Q=m\}\cap\mathcal E_b)=0$ for every
$m\in\Z\setminus\{0\}$, since
$\mu(\mathcal E_b)=\sum_m\mu(\{Q=m\}\cap\mathcal E_b)$ by
\eqref{eq:rig-Q-integral} and Borel additivity.

Recall from the exterior obstruction
Proposition~\ref{prop:rig-obstruction} that for $\Lambda\in\mathcal
E_b$ the exterior vector in $\bigwedge^2\Lambda\cap V_b$ has both
coordinates nonzero and integer $Q$-value $m$. Fix such $m$ and a
determinant-one basis of a sufficiently small neighborhood in $X_4$
of a candidate lattice. In these coordinates, $Q=m$ is the
intersection of the two-dimensional affine plane
$(m-y_2-y_3,y_2,y_3)$ in the trace-$m$ plane, with $\Lambda$ having a
lattice-vector in $V_b$ of this form and both coordinates nonzero.

For a small shrinking rectangle $R_h\subseteq\mathcal H_m$ of side
$h$, a lattice with such a vector lies within the tube
\[
  T_{b,h}=\{g\in\SL_4(\R): |g_{ij}|\le h \text{ for all pairs
    in }\mathcal R\text{ of } V_b\text{ with }Q\text{ bounded}\}
\]
with $\operatorname{area}(R_h)=O(h^4)$ up to fixed factors. Averaging
over the diagonal flow, the same argument that produced the ball
bound of Proposition~\ref{prop:balls} now uses the exterior count of
Proposition~\ref{prop:weighted-exterior} and gives
\[
  \mu\{Q=m,\ \Lambda\in\mathcal E_b,\ |\text{coordinates}|<h\}
  \ll_{m,\Phi,\chi} h^4
\]
for every sufficiently small $h>0$. Letting $h\downarrow0$ along
$h=1/n$ gives $\mu\{Q=m,\ \Lambda\in\mathcal E_b\}=0$.
\end{proof}

\subsection{Identification of the limit}

\begin{proof}[Completion of the proof of Theorem~\ref{thm:main}]
Let $(\mu_i)$ be a sequence of packet measures with $X_i\to\infty$ and
let $\mu$ be any weak probability limit. Tightness
(Proposition~\ref{prop:tightness}) reduces to the $A_4$-invariant
probability case. Write the $A_4$-ergodic decomposition
$\mu=\int\nu_z\,d\tau(z)$. Corollary~\ref{cor:positive-components}
gives $h_{\nu_z}(a(1))\ge\lambda_a/3$ for $\tau$-a.e.\ $z$. By
Theorem~\ref{thm:ekl} and Proposition~\ref{prop:rig-obstruction}, each
$A_4$-ergodic component is either invariant probability on an orbit
$\Lambda J$ with $J=\SL_4(\R)$, which is $m_4$, or else has
$\nu_z(\mathcal E_b)=1$ for some $b\in\mathcal B$.

By Proposition~\ref{prop:null-exterior},
$\mu(\mathcal E_b)=0$ for each $b\in\mathcal B$. Hence
$\tau$-a.e.\ component is $m_4$, and therefore $\mu=m_4$.

This identifies every subsequential limit as $m_4$. Tightness implies
the whole sequence converges weakly to $m_4$, which gives the
convergence statement in Theorem~\ref{thm:main}. The same tightness
supplies, for every $\eps>0$, a compact set carrying at least
$1-\eps$ of every sufficiently late packet probability, which is the
no-escape assertion.
\end{proof}

\section{References and historical notes}

The main structural inputs are the measure-classification theorem of
Einsiedler--Katok--Lindenstrauss; the discriminant-volume and
entropy machinery of Einsiedler--Lindenstrauss--Michel--Venkatesh; the
ordinary-vector unfolding method, ordinary-ball entropy lower bound,
and weighted-shadow sieve of the prime-degree preprint
[PrimeDegreePackets]; and the general Stark zero-free-region
estimates. We list the specific citations used in the body.

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\end{thebibliography}
}

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