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\title{A smooth three-torus diffeomorphism with\\ simple Lebesgue spectrum}
\author{}
\date{}
\begin{document}
\maketitle
\begin{abstract}
We construct a $C^\infty$ diffeomorphism of the three-torus that preserves
standard volume and whose Koopman operator on the entire mean-zero space
$L^2_0$ is unitarily equivalent to multiplication by $w$ on $L^2(S^1)$.
Equivalently, $L^2_0$ has an orthonormal basis of the form
$\{f\circ T^n:n\in\Z\}$ for a single real function $f$. This settles the
probability-preserving form of Banach's simple Lebesgue-spectrum problem in
the smooth category. Every lemma is proved in full. The tools are Fourier
series, stationary phase, and elementary probability.
\end{abstract}
\tableofcontents

\section{The claim, and the shape of the proof}
\label{sec:overview}

\subsection{The statement}
Put $\T=\R/\Z$ and $\ee(t)=\exp(2\pi i t)$. On the torus $\T^3$ with
coordinates $x=(\theta,u,a)$ let $\mu$ be normalized Lebesgue measure. For a
bijection $T$ of $\T^3$ preserving $\mu$, the Koopman operator acts by
$U_T g=g\circ T$ and is unitary on $L^2(\mu)$. Write $L^2_0(\mu)$ for the
closed subspace of mean-zero functions, $\int g\,\dd\mu=0$.

\begin{theorem}[Main theorem]\label{thm:main}
There exist a $C^\infty$ volume-preserving diffeomorphism $T\colon\T^3\to\T^3$
and a real $f\in L^2_0(\mu)$ such that the bilateral orbit
$\{f\circ T^n:n\in\Z\}$ is an orthonormal basis of $L^2_0(\mu)$.
Consequently $U_T$ on $L^2_0(\mu)$ is unitarily equivalent to multiplication
by the coordinate $w$ on $L^2(S^1,m)$ ($m$ normalized Haar measure), and $T$
is ergodic.
\end{theorem}

\subsection{What ``simple Lebesgue spectrum'' means}
The spectral content of $T$ about a vector $f$ is the sequence
$n\mapsto\langle U_T^n f,f\rangle$. By Herglotz this is the Fourier
transform of a unique positive measure $\sigma_f$ on $\T$, the
\emph{spectral measure} of $f$ (Section~\ref{sec:vocab}). The orbit
$\{U_T^n f\}$ is orthonormal exactly when
$\langle U_T^n f,U_T^k f\rangle=\delta_{nk}$, that is, exactly when
$\sigma_f$ is Haar measure $m$. Writing $w(z)=\ee(z)$, each basis vector
$U_T^n f$ then behaves like $w^n$ on $L^2(S^1,m)$. If moreover the orbit
spans all of $L^2_0$, the assignment $U_T^n f\mapsto w^n$ extends to a
unitary intertwining $U_T$ with multiplication by $w$. So ``simple'' means a
single orbit, ``Lebesgue'' means $\sigma_f=m$, and because the orbit is a
basis of the whole mean-zero space the spectral multiplicity is exactly one.

\subsection{Why it is not easy}
We need one vector whose translates are simultaneously mutually orthogonal
and total. Under a generic map translated packets leak: a packet at a fixed
angular index spreads into a range of frequencies. The construction is
therefore a tower of tiny perturbations of integrable \emph{twists}, for
which every packet has a completely explicit spectral density. Each
perturbation must at once (a) barely move the current vectors, (b) reshape
their densities to encode a target, and (c) stay $C^\infty$-close to the
previous map. The bookkeeping is long; the ideas are short:
\begin{itemize}
\item \textbf{Twists and packets:} a twist
$S(\theta,u,a)=(\theta+\phi(u),u,a)$ shears only $\theta$, at speed
$\phi(u)$; packets $A(u,a)\ee(m\theta)$ then have explicit spectral
densities and are visibly orthogonal (Section~\ref{sec:vocab}).
\item \textbf{Passages:} a passage is a volume-preserving coordinate change
to a nearby twist that swaps two coordinates and adds a huge integer to the
speed, separating angular indices so that stationary phase computes new
densities precisely (Section~\ref{sec:passages}).
\item \textbf{Signals:} a positive Fourier series plus a finite prime sieve
lets a passage stamp a prescribed sinusoidal bias onto each density, keyed
to the packet's normalized angular index (Section~\ref{sec:signals}).
\item \textbf{Blocks:} many weak signals are grouped so that the total
$\int\sqrt p$ stays almost flat, keeping the vector near unit norm, while a
predictor of the color label steadily sharpens (Section~\ref{sec:blocks}).
\item \textbf{Completion:} test functions become tagged packets; the
predictor realizes the tags as a function of $T$; whitening restores density
$1$; a diagonal limit makes the orbit complete (Section~\ref{sec:completion}).
\item \textbf{Consequences:} simple Lebesgue spectrum gives mixing, zero
entropy, and vanishing Lyapunov exponents (Section~\ref{sec:consequences}).
\end{itemize}

Every passage is chosen \emph{after} the current finite data (chart,
cutoffs, jitter, error bounds) is fixed; the passage integer is chosen last
and may be arbitrarily large. We never assume a uniform bound on the number
of packets or on the derivatives of the successive charts.


\section{Vocabulary and basic facts}
\label{sec:vocab}

\subsection{The torus and Fourier series}
On $\T^3$ write $x=(\theta,u,a)$ and, for $n=(n_1,n_2,n_3)\in\Z^3$, set
$e_n(x)=\ee(n_1\theta+n_2 u+n_3 a)$. These form an orthonormal basis of
$L^2(\mu)$. A function $A(u,a)\ee(m\theta)$ involves only the angular
frequency $m$ in $\theta$; we call $m$ its \emph{angular index}.

\subsection{Measure-preserving maps and the Koopman operator}
A measurable bijection $T$ of $\T^3$ preserving $\mu$ has a measurable
inverse modulo null sets, and $U_Tg=g\circ T$ is unitary on $L^2(\mu)$.
For an invertible measure-preserving $T$, ergodicity is exactly the
statement that the only $U_T$-invariant vectors are constants.

\subsection{Spectral measure and spectral density}
For $g\in L^2(\mu)$ the sequence $n\mapsto\langle U_T^ng,g\rangle$ is
positive definite, so by Herglotz there is a unique positive finite measure
$\sigma_g$ on $\T$ with
\[
  \int_\T \ee(-nz)\,\dd\sigma_g(z)=\langle U_T^ng,g\rangle,
  \qquad n\in\Z .
\]
(We take the inner product antilinear in the second slot; using $\ee(nz)$
instead of $\ee(-nz)$ only reverses orientation.) We call $\sigma_g$ the
\emph{spectral measure} of $g$. When $\sigma_g=p_g\dd z$ is absolutely
continuous we call $p_g$ the \emph{spectral density}, writing
$\langle U_T^ng,g\rangle=\int_\T\ee(nz)p_g(z)\dd z$. The norm
$\|\cdot\|_1$ of a density is its $L^1(\T)$ norm.

\subsection{Simple Lebesgue spectrum}
By the two preceding paragraphs, the orbit $\{U_T^ng\}$ is orthonormal iff
$\sigma_g=m$ (normalized Haar measure); it is an orthonormal basis of
$L^2_0(\mu)$ iff in addition its closed span is all of $L^2_0(\mu)$. Thus
Theorem~\ref{thm:main} asks for a real $f\in L^2_0(\mu)$ with $\sigma_f=m$
and $\overline{\mathrm{span}}\{U_T^nf:n\in\Z\}=L^2_0(\mu)$.

\subsection{Twists and good arcs}
An \emph{intermediate map}, or \emph{twist}, is
\begin{equation}\label{eq:twist}
  S(\theta,u,a)=(\theta+\phi(u),\,u,\,a),
\end{equation}
where $\phi\in C^\infty(\T;\R)$. Being a shear, $S$ preserves $\mu$, and its
inverse is obtained by replacing $\phi$ with $-\phi$. We say $\phi$ has a
\emph{good arc} $I\subset\T$ if $\phi'>0$ on $I$; derivatives on an arc are
taken in an increasing real lift. The good arc is where stationary phase is
nondegenerate, and its complement can always be made arbitrarily small.
Since $\phi(u)$ is constant along an orbit,
$S^n(\theta,u,a)=(\theta+n\phi(u),u,a)$, so
\begin{equation}\label{eq:twist-koopman}
  (U_S^ng)(\theta,u,a)=g(\theta+n\phi(u),u,a).
\end{equation}

\subsection{Packets and their spectral measures}
A \emph{packet} is a function
\[
  F(\theta,u,a)=A(u,a)\,\ee(m\theta),\qquad m\in\Z\setminus\{0\},
\]
with $A$ smooth and compactly supported in the good arc, meaning the
$u$-support of $A$ lies in a compact subinterval of $I$. By
\eqref{eq:twist-koopman},
\[
  U_S^nF=\ee(nm\phi(u))\,A(u,a)\,\ee(m\theta),
  \qquad
  \langle U_S^nF,F\rangle=\int_{\T^2}\ee(nm\phi(u))|A(u,a)|^2\dd u\,\dd a .
\]
Hence the spectral measure is the pushforward
\begin{equation}\label{eq:packet-spectral-measure}
  \sigma_F=(m\phi\bmod1)_*\bigl(|A(u,a)|^2\dd u\,\dd a\bigr),
\end{equation}
which is absolutely continuous because the support of $A$ sits in the good
arc. If $|A(u,a)|=w(u)$ does not depend on $a$, the density is
\begin{equation}\label{eq:packet-density}
  p_F(z)=\sum_{u:\,m\phi(u)\equiv z}\frac{w(u)^2}{|m\phi'(u)|},
  \qquad z\in\T,
\end{equation}
a smooth function on the good arc (away from its endpoints).

\subsection{Colors, orthogonality, cyclic subspaces}
Group the packets of a finite list into finitely many \emph{colors}, and let
$F_r$ be the sum of the packets of color $r$. A finite list is
\emph{disjoint} if any two packets with the same angular index have
disjoint coefficient supports. For such a list, packets with different
angular indices are orthogonal because the $\theta$-integral of a
nontrivial character vanishes; packets with equal angular index and
disjoint supports are orthogonal after any two time-translates because one
coefficient function vanishes pointwise. Thus
\begin{equation}\label{eq:packet-orthogonality}
  \langle U_S^nF_r,\,U_S^kF_l\rangle=0
  \qquad(r\ne l,\ n,k\in\Z),
\end{equation}
so distinct colors generate mutually orthogonal cyclic subspaces. We will
repeatedly use that disjoint angular-index supports make different colors'
cyclic subspaces orthogonal.

\subsection{The density-norm inequality}
Suppose $F,G$ for the same map have spectral densities $p_F,p_G$. Then
\begin{equation}\label{eq:density-norm-bound}
  \|p_F-p_G\|_1\le(\|F\|_2+\|G\|_2)\,\|F-G\|_2 .
\end{equation}
\emph{Proof.} Test $\sigma_F-\sigma_G$ against any bounded measurable $h$
with $\|h\|_\infty\le1$. By the functional calculus,
$\langle h(U)F,F\rangle=\int_\T h\,\dd\sigma_F$, and $\|h(U)\|\le1$, so
\[
  \int_\T h\,(p_F-p_G)
  =\langle h(U)F,F-G\rangle+\langle h(U)(F-G),G\rangle
\]
has modulus at most $(\|F\|_2+\|G\|_2)\|F-G\|_2$. Supremum over such $h$.
$\square$

\subsection{Spectral multipliers}
For measurable $v$ on $\T$, the operator $v(U_S)$ acts on the packet
$A\ee(m\theta)$ by multiplying $A$ by the scalar $v(m\phi(u)\bmod1)$:
indeed $U_S^n$ multiplies $A$ by $\ee(nm\phi)$, a trigonometric polynomial
$v$ multiplies by $v(m\phi)$, and $L^2$-limits extend this. Whenever
$v(m\phi(\cdot))A\in L^2$ the result is a well-defined vector. This is the
only functional calculus we use, and it turns spectral surgery (whitening)
into a pointwise operation on the coefficients.


\section{Smooth passages and spectral densities}
\label{sec:passages}

A \emph{volume-preserving coordinate system}, or \emph{chart}, is a smooth
volume-preserving identification of the physical torus with $\T^3$; all
charts below preserve normalized volume. An intermediate map has the form
\begin{equation}
  S(\theta,u,a)=(\theta+\phi(u),\,u,\,a),
  \qquad (\theta,u,a)\in\T^3,
\end{equation}
with $\phi\in C^\infty(\T;\R)$ strictly increasing ($\phi'>0$) on a good arc
$I$. Recall the packet $A(u,a)\ee(m\theta)$, $m\in\Z\setminus\{0\}$, with
$A$ smooth and compactly supported in $I\times\T$, its spectral measure
\eqref{eq:packet-spectral-measure}, and the orthogonality
\eqref{eq:packet-orthogonality}. Everything in this section is proved in a
fixed chart; when the chart later changes we always compare in the fixed
physical $L^2(\T^3)$.

The passage is the elementary move that changes the twist. It swaps the
roles of two coordinates and enlarges the speed by a huge integer, so that
stationary phase computes the new spectral densities on a much finer scale.

\begin{lemma}[Passage]\label{lem:passage}
Fix the twist, its physical chart, and a finite disjoint packet list. Choose
a positive integer $c$ so large that the compact ranges
$m(c+\phi(\T))$, for the distinct old angular indices $m$, are pairwise
disjoint and avoid zero. Set $V=c+\phi$. Fix $J\in C^\infty(\T;\R)$ with
$J'>0$ on an open arc $I'$ and fix $0\le\psi\le1$ smooth, compactly
supported in the interior of $I'$. For integers $N\to\infty$ choose
$\Delta_N$ with
\begin{equation}\label{eq:passage-rounding}
  |\Delta_N|\le N^{-1},\qquad
  k_N:=N\Bigl(\int_\T V(u)\,\dd u+\Delta_N\Bigr)\in\Z,
  \qquad H_N'=N(V+\Delta_N),
\end{equation}
and put $\sigma_N(a)=N^{-1}+N^{-2}J(a)$. Then
\begin{equation}\label{eq:passage-map}
  T_N(\theta,u,a)=
  \bigl(\theta+H_N(u+\sigma_N(a))-H_N(u),\,
  u+\sigma_N(a),\,a\bigr)
\end{equation}
is a volume-preserving smooth torus diffeomorphism, and $T_N$ and $T_N^{-1}$
converge smoothly to $S$ and $S^{-1}$ in the fixed physical chart. In the
new chart
\begin{equation}\label{eq:passage-chart}
  (\theta',u',a')=C_N(\theta,u,a)=(u,\,a,\,\theta-H_N(u)),
\end{equation}
$T_N$ is the twist of speed $\sigma_N$ and good arc $I'$.

Here are the packet and density properties. For an old packet indexed by
$\alpha$, with angular index $m_\alpha$ and amplitude $A_\alpha$, set
\begin{equation}\label{eq:stationary-image}
  E_\alpha=\{(m_\alpha V(u),a):(u,a)\in\supp A_\alpha\}.
\end{equation}
Choose pairwise disjoint, arbitrarily small fixed neighborhoods of the
compact sets $E_\alpha$, and smooth cutoffs $\chi_\alpha(s,a)$ supported in
these neighborhoods and equal to one near $E_\alpha$. Retain the finitely
many new packets
\begin{align}
  G_{\alpha,j}(\theta',u',a')
    &=\ee(j\theta')\,\ee(m_\alpha a')\,
      \psi(u')\,\chi_\alpha(j/N,u')\,C_{\alpha,j}(u'),
      \label{eq:retained-packets}\\
  C_{\alpha,j}(a)
    &=\int_\T A_\alpha(u,a)\,\ee(m_\alpha H_N(u)-ju)\,\dd u .
      \label{eq:exact-passage-coefficient}
\end{align}
All retained angular indices are nonzero for large $N$. For each $\alpha$,
\begin{equation}\label{eq:transport-norm}
  \Bigl\|\sum_j G_{\alpha,j}\circ C_N
       -\psi(a)A_\alpha(u,a)\ee(m_\alpha\theta)\Bigr\|_2=o(1).
\end{equation}
The retained packets are disjoint, and their moduli are independent of the
new spare coordinate $a'$; every constant attached to an old packet is
preserved by this linear transport and these cutoffs.

For a single old packet write, with zero extension off its branch,
\begin{equation}\label{eq:W-definition}
  W(s,a)=\frac{|A(u,a)|^2}{|m\phi'(u)|},
  \qquad s=mV(u).
\end{equation}
Let $K(s,y)$ be the density of the pushforward of $W(s,a)\psi(a)^2\dd a$
under $a\mapsto sJ(a)$. It is continuous with compact support in $(s,y)$,
and its $s$-support stays away from zero. For any fixed positive integer
$P$ and residue $h\pmod P$, the unfolded temporal density of the retained
packets with $j\equiv h\pmod P$ differs in $L^1(\R)$ by $o(1)$ from
\begin{equation}\label{eq:fine-density-unfolded}
  Q_{N,h}(t)=\sum_{k\in\Z}K(t,\,Nt-h-kP),
\end{equation}
and consequently their density on $\T$ differs in $L^1(\T)$ by $o(1)$ from
$\sum_{l\in\Z}Q_{N,h}(z+l)$. These assertions hold after summing over any of
the finitely many colors. All $o(1)$ terms refer to the old data
$c,J,\psi,P$ and support neighborhoods, all fixed before $N\to\infty$; in
particular $N$ may be required to exceed any prescribed bound.
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lem:passage}]
\emph{Step 1: Geometry and smooth convergence.} Choose $k_N$ by rounding
$N\int_\T V$ to an integer; then \eqref{eq:passage-rounding} gives
$H_N'=N(V+\Delta_N)$ and $H_N(u+1)-H_N(u)=k_N$, so $u\mapsto H_N(u)\bmod1$
is globally defined and $C_N$ is a smooth torus diffeomorphism of Jacobian
one, with inverse
$(\theta',u',a')\mapsto(a'+H_N(\theta'),\theta',u')$. Direct substitution,
using that $H_N$ is affine along the $u$-flow lines, yields
\[
  C_NT_NC_N^{-1}(\theta',u',a')=(\theta'+\sigma_N(u'),\,u',\,a').
\]
For smooth convergence, the integral identity
\[
  H_N(u+\sigma_N(a))-H_N(u)
  =N\!\int_0^{\sigma_N(a)}\!(V(u+t)+\Delta_N)\,\dd t
  =V(u)+O_{C^k}(N^{-1})
\]
holds for each fixed $k$, with constants depending on the fixed $c,J,\phi$.
Since $c$ is an integer, addition of $V$ and of $\phi$ agree on $\T$, so the
right side is the speed $\phi(u)$ modulo integers and $T_N\to S$ smoothly.
The inverse
\[
  (\theta,u,a)\longmapsto
  (\theta-H_N(u)+H_N(u-\sigma_N(a)),\,u-\sigma_N(a),\,a)
\]
gives the same estimate, so $T_N^{-1}\to S^{-1}$ smoothly. Composing with a
fixed physical chart and its inverse preserves convergence in every fixed
smooth norm. No bound uniform over future charts is asserted.

\emph{Step 2: Fourier coefficients by stationary phase.} Work on a compact
subarc of $I$ containing all old coefficient supports in its interior. For
$s=j/N$ the phase in \eqref{eq:exact-passage-coefficient}, divided by $N$,
has critical equation
\begin{equation}
  m(V(u)+\Delta_N)=s,
\end{equation}
with second derivative $m\phi'(u)$; it has at most one critical point on
this arc. With $u_*=u_*(s-m\Delta_N)$ on that branch put
\[
  L_{m,j}(a)=
  \frac{A(u_*,a)}{\sqrt{N|m\phi'(u_*)|}}\,
  \ee\Bigl(mH_N(u_*)-ju_*+\frac{\operatorname{sgn}(m)}{8}\Bigr),
\]
and set it to zero where the amplitude at the critical point vanishes or
there is no point on the branch. We claim that, uniformly for $j/N$ in any
fixed compact interval and $a\in\T$,
\begin{equation}\label{eq:stationary-uniform}
  C_{m,j}(a)-L_{m,j}(a)=o(N^{-1/2}).
\end{equation}
The phase normalization is $m H_N(u)-ju$ with $H_N'=N(V+\Delta_N)$, so
$(mH_N(u)-ju)/N=m(V(u)+\Delta_N)-su$ has critical value at $u_*$ with
second derivative $m\phi'(u_*)$. A smooth change of variable, depending
smoothly on the critical point, sends the divided phase to its critical
value plus $\operatorname{sgn}(m)y^2/2$, with Jacobian
$|m\phi'(u_*)|^{-1/2}$ at zero. After $v=\sqrt N\,y$ the integral over
$|v|\le D$ converges uniformly to the constant amplitude times the
quadratic exponential integral, the local tails are $O(N^{-1/2}/D)$ since
the rescaled amplitudes have uniformly bounded supremum and total variation,
and the part away from the critical point is $O(N^{-1})$ by integration by
parts. Finally
\[
  \lim_{D\to\infty}\int_{-D}^{D}\ee(\pm v^2/2)\,\dd v
  =\exp(\pm i\pi/4),
\]
obtained by inserting a Gaussian factor, evaluating the Gaussian integral,
and removing the factor with the same tail bound. Taking first $N\to\infty$
then $D\to\infty$ proves \eqref{eq:stationary-uniform}. When the only
possible critical point lies outside the enlarged support interval, the
integral is uniformly nonstationary, which also covers zero amplitudes and
the ends of the branch.

There are $O(N)$ indices in a fixed bounded $j/N$ range. Outside a
sufficiently large such range, repeated integration by parts gives for any
fixed positive integer $q$
\[
  |C_{m,j}(a)|\le C_q|j|^{-q}
  \quad\hbox{uniformly in }a,N,
\]
because the phase derivative has size comparable to $|j|$ and its higher
derivatives are $O(N)$ with $N=O(|j|)$. Hence
\eqref{eq:stationary-uniform} implies
\[
  \sum_j\int_\T|C_{m,j}-L_{m,j}|^2\dd a=o(1).
\]
The $L^2$ norm of the leading array is a Riemann sum, so Cauchy--Schwarz
gives
\begin{equation}\label{eq:summed-stationary-mass}
  \sum_j\int_\T
  \Bigl||C_{m,j}(a)|^2-\frac1N W(j/N-m\Delta_N,a)\Bigr|\dd a=o(1).
\end{equation}

\emph{Step 3: Cutoffs and transport.} The map $(u,a)\mapsto(mV(u),a)$ is
injective on $I\times\T$ and sends disjoint compact supports with the same
$m$ to disjoint compact sets; our choice of $c$ separates the images for
different $m$. Hence the cutoffs $\chi_\alpha$ exist. For large $N$ they
equal one on the support of the corresponding leading array, since its
$s$-coordinate is shifted only by $m_\alpha\Delta_N$. Because each cutoff is
one on the shifted leading-array support, the deleted exact coefficient mass
is bounded by the summed squared error $C_{m,j}-L_{m,j}$, which tends to
zero by Step 2 and Parseval. The same argument holds after multiplying by
$\psi$, proving \eqref{eq:transport-norm} and
\eqref{eq:summed-stationary-mass} with the factor $\psi^2$ on the right. The
cutoffs make the separation exact rather than asymptotic: different old $m$
never contribute to the same retained $j$, and at a shared $j$ the remaining
supports in $u'=a$ are disjoint. The factor $\ee(m_\alpha a')$ in
\eqref{eq:retained-packets} shows the retained modulus is independent of the
new spare coordinate $a'$, also after combining packets at one index.
Linearity gives the assertion about attached constants.

\emph{Step 4: Fine temporal density.} We first remove the rounding from
any fine-scale sampling. Smoothness and compact support of the fixed $W$
give
\begin{equation}\label{eq:rounding-mass-removal}
  \frac1N\sum_j\int_\T
  |W(j/N-m\Delta_N,a)-W(j/N,a)|\dd a=o(1):
\end{equation}
a uniform bound on $\partial_sW$ and $O(N)$ nonzero summands bound each sum
by $O(|m|/N)$. The same holds with $\psi^2$ and for any residue subset.
Pushforward contracts total variation in $L^1$, so
\eqref{eq:summed-stationary-mass} and \eqref{eq:rounding-mass-removal} let
us replace the retained squared coefficient measures by
$N^{-1}W(j/N,a)\psi(a)^2\dd a$, at total error $o(1)$, before pushing to
temporal frequency. In particular the actual lattice $j/N$ is never
translated by $m\Delta_N$.

On the retained arc the change of variable $y=sJ(a)$ gives
\begin{equation}\label{eq:K-jacobian}
  K(s,y)=\frac{W(s,a)\psi(a)^2}{|s|J'(a)},
  \qquad a=J|_{I'}^{-1}(y/s),
\end{equation}
with zero extension; compact support of $\psi$ inside $I'$ makes this
extension continuous. The real temporal frequency of a new angular index
$j$ is
\begin{equation}\label{eq:output-frequency}
  t=j\sigma_N(a)=j/N+(j/N)J(a)/N .
\end{equation}
The density of the pushforward of $N^{-1}W(j/N,a)\psi(a)^2\dd a$ under this
map is exactly $K(j/N,Nt-j)$, the factor $N$ in the density change of
variable cancelling the coefficient mass $N^{-1}$.

Choose $Y$ with $K(s,y)=0$ for $|y|>Y$. For fixed $t$ at most $2Y+2$
integers contribute to either $\sum_jK(j/N,Nt-j)$ or
$\sum_jK(t,Nt-j)$, and on every contributing term $|j/N-t|\le Y/N$. Uniform
continuity of $K$, and a fixed bounded interval containing the $t$-supports
for large $N$, show that replacing $j/N$ by $t$ costs $o(1)$ in $L^1(\R)$.
The same proof works after restriction to $j\equiv h\pmod P$. This is
\eqref{eq:fine-density-unfolded}. Folding modulo one contracts the $L^1$
norm and finishes the proof.
\end{proof}

We record the normalization in \eqref{eq:fine-density-unfolded}. The
periodization $\sum_kK(t,y-kP)$ is a density relative to length on the
circle of length $P$, so its mean value is
\begin{equation}\label{eq:residue-mean}
  \frac1P\int_\T W(t,a)\psi(a)^2\dd a .
\end{equation}
For $P=1$, folding $t=z+l$ leaves the fast phase $Nz$ unchanged since $N$
is an integer. For general $P$ the folded expression has phase
$N(z+l)-h$ modulo $P$; no divisibility condition on $N$ is imposed.

\begin{lemma}[Fast-variable averaging]\label{lem:fast-average}
Let $P>0$ and let $G(t,y)$ be continuous, compactly supported in $t$, and
$P$-periodic in $y$. Then, uniformly in real offsets $\beta_N$,
\begin{equation}\label{eq:fast-average}
  \int_\R G(t,Nt+\beta_N)\dd t
  \longrightarrow\frac1P\int_\R\int_0^P G(t,y)\dd y\,\dd t .
\end{equation}
In particular, let $w\ge0$ be continuous and compactly supported in a
compact set $K\subset\R\setminus\{0\}$. For continuous integrable densities
$\rho,\widetilde\rho$ define $\mathcal P_{s,P}\rho(y)=\sum_{k\in\Z}
|s|^{-1}\rho((y+kP)/s)$ and suppose these periodizations are jointly
continuous for $s\in K$. Then
\begin{align}\label{eq:law-diagonal-comparison}
  &\lim_{N\to\infty}\int_\R w(s)\,
  |\mathcal P_{s,P}\rho(Ns+\beta_N)
   -\mathcal P_{s,P}\widetilde\rho(Ns+\beta_N)|\dd s \notag\\
  &\qquad=\frac1P\int_\R w(s)\int_0^P
   |\mathcal P_{s,P}\rho(y)-\mathcal P_{s,P}\widetilde\rho(y)|\dd y\,\dd s
   \le\frac{\|w\|_1}{P}\|\rho-\widetilde\rho\|_1 .
\end{align}
Joint continuity holds for example if the densities are continuous with
bounds $C/(1+y^2)$, or are continuous with compact support. Neither density
need have mass one.
\end{lemma}

\begin{proof}
Partition the $t$-axis into intervals of length $P/N$ on each of which
$Nt+\beta_N$ runs through one period. On each interval replace the slow
argument $t$ of $G$ by an endpoint; uniform continuity on the fixed compact
support bounds the integrated error by a constant times the modulus of
continuity at $P/N$, with an additional $O(N^{-1})$ boundary error. The
resulting averages are Riemann sums for the right side of
\eqref{eq:fast-average}, uniformly in the offset. Applying this to the
continuous absolute difference in \eqref{eq:law-diagonal-comparison}
strengthens it to the displayed equality. The integral of the difference
over one period is
\[
  \sum_k\int_0^P|s|^{-1}
  |\rho((y+kP)/s)-\widetilde\rho((y+kP)/s)|\dd y
  =\|\rho-\widetilde\rho\|_1 ,
\]
proving the final bound. On a compact set of nonzero $s$, quadratic tail
bounds give a uniform summable bound $C'/(1+k^2)$ on the periodizing
summands for $0\le y\le P$, so uniform convergence gives joint continuity;
compactly supported densities have only finitely many nonzero summands.
\end{proof}

In the special case $|A(u,a)|=w(u)$ does not depend on $a$,
\eqref{eq:W-definition} has the form $W(s,a)=w(s)$. If
$\widetilde\rho\dd y=J_*(\psi^2\dd a)$, then
$K(s,y)=w(s)|s|^{-1}\widetilde\rho(y/s)$, so replacing the realized jitter
law by a regular target law $\rho$ in the fine-density prescription costs,
after $N$ is large, at most
$\|w\|_1\|\rho-\widetilde\rho\|_1+o(1)$ for the full density. Summing over
the fixed packet list and folding modulo one. This is an averaged estimate
along the fast variable, not a pointwise estimate from total variation, and
it is applied only after \eqref{eq:rounding-mass-removal}, so the family
being averaged is fixed.

\begin{lemma}[Realization of jitter laws]\label{lem:law-approx}
For any strictly positive smooth probability density $\rho$ on $\R$ and any
$\eps>0$ there are a smooth real circle function $J$, an open arc $I'$ with
$|\T\setminus I'|<\eps$ and $J'>0$ on $I'$, and $0\le\psi\le1$ smooth and
compactly supported in $I'$, such that
\[
  J_*(\psi^2\dd a)=\widetilde\rho(y)\dd y,\qquad
  \widetilde\rho\in C_c^\infty(\R),\qquad
  \|\rho-\widetilde\rho\|_1<\eps .
\]
Here the cutoff law is a subprobability measure, not renormalized; also
$\int_\T(1-\psi^2)\dd a<\eps$.
\end{lemma}

\begin{proof}
Let $F(y)=\int_{-\infty}^y\rho(v)\dd v$. Strict positivity makes $F$ a
smooth diffeomorphism from $\R$ to $(0,1)$. Choose
$0<\delta<\min(1/8,\eps/4)$ and a smooth function $\xi$ compactly supported
in $(\delta/2,1-\delta/2)$ and equal to one on $[\delta,1-\delta]$. In the
circle coordinate $0<a<1$ set $J(a)=\xi(a)F^{-1}(a)$ and extend it by zero
near the identified endpoints. It is a smooth bounded circle function, and
on $I'=(\delta,1-\delta)$ it has derivative $1/\rho(F^{-1}(a))>0$. Choose
$\psi\in C_c^\infty(I')$ with $0\le\psi\le1$ and $\psi=1$ on
$[2\delta,1-2\delta]$. Change of variables gives
\[
  \widetilde\rho(y)=\rho(y)\psi(F(y))^2\le\rho(y),\qquad
  \|\rho-\widetilde\rho\|_1=1-\int_\T\psi^2\dd a\le4\delta<\eps.
\]
Its support is compact because $\psi$ is compactly supported inside
$(0,1)$. All claims follow.
\end{proof}

\begin{lemma}[Weighted neutral refresh]\label{lem:neutral}
Given a twist, its fixed physical chart, a finite disjoint smooth packet list
with color densities $p_r$, a positive integer $P$, and a positive tolerance,
one can perform a passage with the following simultaneous properties. Its
physical map and inverse lie in any prescribed smooth neighborhood of the
old ones. Its new good-arc complement is arbitrarily small. Every color
vector is transported within the prescribed $L^2$ tolerance, retaining
constant tags and disjointness. The new packet moduli are independent of the
new spare coordinate. The output densities $p'_r$ and the densities
$p'_{r,h}$ contributed by output indices $j\equiv h\pmod P$ satisfy
\begin{equation}\label{eq:neutral-densities}
  \|p'_r-p_r\|_1<\eps,\qquad
  \|p'_{r,h}-p_r/P\|_1<\eps\quad(0\le h<P),
\end{equation}
where $\eps>0$ is arbitrary. The integer $c$ may first be required to exceed
any fixed bound, and the retained supports may use any sufficiently small
fixed enlargements of \eqref{eq:stationary-image}. If colors were disjoint
in their old angular indices alone, their output angular-index sets are
still disjoint. The final passage integer $N$ may exceed any prescribed
bound.
\end{lemma}

\begin{proof}
Fix the old finite data and choose $c$ as in Lemma~\ref{lem:passage}, also
satisfying its prescribed lower bound. Identify all but a small interval of
the $a$-circle with an interval in $\R$. Choose an open arc $I'$ whose
complement is as small as desired, and $0\le\psi\le1$ compactly supported
there and equal to one except on a set of sufficiently small length.
Boundedness of the finitely many old amplitudes lets us make both their
$L^2$ cutoff errors and the removed squared masses arbitrarily small; fix
this cutoff before proceeding.

For a positive parameter $L$, take a smooth periodic function $J_L$ equal to
$La+\beta$ on $I'$ and extended smoothly across the complementary interval
(using a smooth cutoff equal to one on the closure of $I'$ in a slightly
larger real interval); in particular $J_L'=L$ on $I'$. Put
$f_s(a)=W(s,a)\psi(a)^2$, extended by zero in this real coordinate. The
periodization of its pushforward by $sJ_L$ is
\begin{equation}\label{eq:neutral-riemann-sum}
  \sum_{k\in\Z}\frac1{|s|L}
  f_s\Bigl(\frac{(y+kP)/s-\beta}{L}\Bigr).
\end{equation}
This is $1/P$ times a Riemann sum for $\int f_s(a)\dd a$ of mesh
$P/(|s|L)$, with an arbitrary lattice offset and possibly reversed
orientation. Its error is bounded by
\begin{equation}\label{eq:neutral-riemann-error}
  \frac{\|\partial_af_s\|_{L^1(\R)}}{|s|L}:
\end{equation}
compare each lattice value times the mesh with the integral on its cell
using the fundamental theorem of calculus and sum the absolute derivative.
Since $s$ ranges over a fixed compact set away from zero, the bound tends to
zero uniformly in $s,y$, and remains valid for arbitrary fixed dependence
of the amplitude on $a$.

Choose $L$ large enough that these uniform errors are as small as needed
for the finite packet list. Now $J_L$ and all its norms are fixed. Choose
the support cutoffs in Lemma~\ref{lem:passage}, then take $N$ sufficiently
large. Its fine-density rule with \eqref{eq:neutral-riemann-sum}
approximates each unfolded residue density by
$P^{-1}\int W(s,a)\psi(a)^2\dd a$ in $L^1$. Folding modulo one gives $p_r/P$
up to the small cutoff loss; the integer $mc$ in $mV$ has no effect on this
folding. The estimate is uniform in the periodic phase, so the folding
offsets $Nl$ cause no difficulty. Summing the residue estimates gives the
total density estimate, after initially making each error smaller by the
fixed factor $P$ if necessary. Norm transport, exact support properties, and
smooth proximity follow from the same passage lemma. All constraints on $N$
are lower bounds imposed after the chart, amplitudes, $c$, cutoff, $L$, and
support neighborhoods are fixed.
\end{proof}

\begin{remark}\label{rem:finite-family-passages}
A common passage can handle finitely many packet lists, requiring
disjointness only within each list. Choose $c$ using the union of their old
indices, choose the neutral cutoff and slope to meet all their finitely many
mass and density bounds, and choose $N$ above all resulting lower bounds.
Support cutoffs are chosen separately for each list, disjoint within that
list. The norm and density conclusions then hold for every list; no
orthogonality between different lists is asserted.
\end{remark}

All these operations preserve real data. Include together the packets
$(m,A)$ and $(-m,\overline A)$, attach conjugate tags, and choose
$\chi_{-m}(s,a)=\chi_m(-s,a)$. The exact coefficients obey
$C_{-m,-j}=\overline{C_{m,j}}$, and all other cutoffs are real. Thus
conjugate packets and conjugate colors can be retained at every passage.


\section{Small Fourier signals}
\label{sec:signals}

The aim of this section is to modulate a packet's spectral density according
to its angular index. The size of the modulation will depend on a fixed band
of \emph{normalized} indices, not on the scale of those indices. The integer
added to the old speed may depend on the actual finite set of indices.

Fix numbers $0<b<D_0$ and a function
\begin{equation}\label{eq:signal-d}
  d\in C_c^\infty(\R\setminus\{0\};\C),\qquad
  d(-x)=\overline{d(x)} .
\end{equation}
A \emph{scale} is a positive real $B$, and the normalized index of a packet
at angular index $m$ is $x=m/B$.

\begin{lemma}[Transmission of an index signal]\label{lem:signal}
For the fixed data $b,D_0,d$ there are an integer $P\ge2$ and a number
$\gamma_0>0$ with the following property. Any fixed $0<\gamma\le\gamma_0$ may
be used, independently of $B$.

Let $S$ be an intermediate map and let
$F_r=\sum_{m\in\mathcal M_r}A_{r,m}(u,a)\ee(m\theta)$ be finite smooth packet
data in its good arc. Suppose the finite sets
$\mathcal M_r\subset\Z\setminus\{0\}$ are mutually disjoint, that
$|A_{r,m}(u,a)|$ is independent of $a$, and that $b\le|m|/B\le D_0$. Write
$p_{r,m}$ for the spectral density of the indicated packet. Given any
$\eps,\delta_x,\delta_z>0$ and prescribed positive bounds for the physical
$L^2$ transport errors, a passage produces finite output packet sums
$F_{r,m}^+$, approximating the corresponding input packets in physical
$L^2$, whose spectral densities satisfy
\begin{equation}\label{eq:signal-conclusion}
  \sum_{r,m}\Bigl\|p_{r,m}^+
  -p_{r,m}(z)\bigl[1+2\gamma\1_{\gcd(m,P)=1}
    \Rea\{d(m/B)\ee(-Nz)\}\bigr]\Bigr\|_{L^1(\T)}<\eps .
\end{equation}
The new scale is $B^+=NcB$, and the new speed is $\sigma_N$ of
Lemma~\ref{lem:passage}. For every point $(u',a')$ in the coefficient
support of a retained output packet of index $j$ descending from $(r,m)$
there is $(u,a)\in\supp A_{r,m}$ with
\begin{equation}\label{eq:signal-support-closeness}
  \Bigl|\frac{j}{B^+}-\frac mB\Bigr|<\delta_x,\qquad
  \dist_\T\bigl(j\sigma_N(u'),m\phi(u)\bigr)<\delta_z .
\end{equation}
Its modulus is independent of the new spare coordinate. Distinct old angular
indices have disjoint output angular-index supports. Conjugate symmetry is
preserved when present.

The passage and its inverse can lie in any prescribed smooth neighborhood of
$S$ and $S^{-1}$ in physical coordinates, and the new good arc can have
arbitrarily small complement. The choices may be made in the order $c$, a
bounded smooth jitter and cutoffs, and finally the integer $N$. After the
preceding choices every sufficiently large integer $N$ works; in particular
an arbitrary lower bound on $N$ can be imposed.
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lem:signal}]
\emph{A positive law.} We first build a positive law independent of $B$.
Use the positive-exponent Fourier convention
$\widehat q(v)=\int_\R q(R)\ee(vR)\dd R$. Choose $0<\alpha<b/2$, put
$a_0=(2\alpha)^{-1}$, and define
\begin{equation}\label{eq:signal-baseline}
  s_\alpha(R)=\alpha\Bigl(\frac{\sin(\pi\alpha R)}{\pi\alpha R}\Bigr)^2,
  \qquad
  q_0(R)=\frac1{2a_0}\int_{-a_0}^{a_0}s_\alpha(R-t)\dd t,
\end{equation}
with the removable value at the origin understood. Fourier inversion for an
interval, followed by convolution, gives
$\widehat{s_\alpha}(v)=(1-|v|/\alpha)_+$, so $q_0$ is a smooth probability
density with Fourier support in $[-\alpha,\alpha]$. It is strictly positive
because the interval of translations cannot sit inside the discrete zero set
of $s_\alpha$. For a quantitative tail, the averaging interval has length
$1/\alpha$ and contains exactly one period of $\sin^2(\pi\alpha(R-t))$ in
$t$, so for $|R|\ge2a_0+1$,
\[
  \frac1{2\pi^2\alpha(|R|+a_0)^2}\le q_0(R)
  \le\frac1{\pi^2\alpha(|R|-a_0)^2}.
\]
Positivity and continuity on the remaining compact interval give
$c_0,C_0>0$ with
\begin{equation}\label{eq:signal-quadratic-tails}
  \frac{c_0}{1+R^2}\le q_0(R)\le\frac{C_0}{1+R^2}
  \qquad(R\in\R).
\end{equation}

Choose $D>b$ with $\supp d\subset[-D,D]$, then an integer
$L>\max(2,D/b)$, and let $P$ be the product of all primes at most $L$ (the
strict inequality allows a small dilation of Fourier arguments). Define
\[
  g(R)=\int_\R d(v)\ee(-vR)\dd v,\qquad
  C_P(h)=\sum_{k\in(\Z/P\Z)^\times}\ee\Bigl(-\frac{kh}{P}\Bigr)
  \quad(0\le h<P).
\]
The function $g$ is Schwartz, real by \eqref{eq:signal-d}, and
$\int g=d(0)=0$. The number $C_P(h)$ is real because the units are
negation-invariant. Set
\begin{equation}\label{eq:signal-joint-law}
  Q_h(R)=\frac1P\bigl[q_0(R)+\gamma C_P(h)g(R)\bigr].
\end{equation}
Let $G=\sup_R(1+R^2)|g(R)|$ and $C_*=\max_h|C_P(h)|$ and choose
$\gamma_0>0$ so small that $\gamma_0C_*G\le c_0/2$ and
$2\gamma_0\|d\|_\infty\le1/2$ (a zero factor imposes no restriction). Then
for every permitted $\gamma$,
\begin{equation}\label{eq:signal-positive-law}
  \frac{q_0(R)}{2P}\le Q_h(R)\le\frac{3q_0(R)}{2P},\qquad
  \int_\R Q_h(R)\dd R=\frac1P .
\end{equation}
Thus $Q_h(R)\dd R$ is a probability law on $\R\times\{0,\ldots,P-1\}$.
Finite Fourier inversion, summing $C_P(h)$ against the character at $k$ and
leaving precisely the indicator that $k$ is a unit, gives, for all $v$ and
$k$,
\begin{equation}\label{eq:signal-joint-fourier}
  \sum_{h=0}^{P-1}\int_\R Q_h(R)\ee\Bigl(vR+\frac{kh}{P}\Bigr)\dd R
  =\widehat q_0(v)\1_{k\equiv0\ (P)}
   +\gamma d(v)\1_{\gcd(k,P)=1}.
\end{equation}
Everything so far is independent of $B$, the map, and its packet list.

\emph{Moments and the finite sieve.} For fixed $B$ and integer $c>0$
consider $J_*=(R/B+h/P)/c$ under the law \eqref{eq:signal-joint-law}; its
density is
\[
  \rho_{B,c}(y)=cB\sum_{h=0}^{P-1}Q_h\bigl(B(cy-h/P)\bigr),
\]
a strictly positive smooth probability density whose
\eqref{eq:signal-positive-law} upper and lower bounds give quadratic tails
with constants depending on $B,c$. Write $x=m/B$ and $s=mct$. For its
positive integer Fourier moments the exact formula is
\begin{equation}\label{eq:signal-exact-moment}
  M_\ell(t)=\frac1P\sum_{h=0}^{P-1}\ee\Bigl(\frac{\ell mt h}{P}\Bigr)
  \bigl[\widehat q_0(\ell xt)+\gamma C_P(h)d(\ell xt)\bigr],
  \quad\ell\ge1.
\end{equation}
At $t=1$ formula \eqref{eq:signal-joint-fourier} applies; its baseline term
vanishes since $|\ell x|\ge b>\alpha$. If $\ell\ge2$ and $d(\ell x)\ne0$
then $\ell\le D/b<L$, so some prime divisor of $\ell$ divides $P$ and
$\gcd(\ell m,P)>1$. Hence
\begin{equation}\label{eq:signal-ideal-moments}
  M_1(1)=\gamma d(x)\1_{\gcd(m,P)=1},\qquad
  M_\ell(1)=0\quad(\ell\ge2).
\end{equation}

For finite $c$ the discrete phases in \eqref{eq:signal-exact-moment} need
not be residue characters, so we estimate their errors before using the
sieve. Choose $\tau\in(1/2,1)$ with $D/(\tau b)<L$ and $c$ large enough that
$t_N(u)=1+(\phi(u)+\Delta_N)/c\ge\tau$, $|\Delta_N|\le1/N$, on the old
supports, uniformly for $N\ge1$. The baseline term of
\eqref{eq:signal-exact-moment} then vanishes for \emph{every} $\ell\ge1$
since $|\ell xt_N|\ge\tau b>\alpha$, and the perturbation vanishes
identically for $\ell>D/(\tau b)$; thus nonintegral discrete phases cannot
introduce an unbounded collection of harmonics. For the remaining finite set
put $C_1=P^{-1}\sum_h|C_P(h)|$; the bound
$|\ee(v)-\ee(w)|\le2\pi|v-w|$ gives
\begin{equation}\label{eq:signal-finite-c}
  |M_\ell(t)-M_\ell(1)|
  \le\gamma C_1\ell
  \bigl[D_0\|d'\|_\infty+2\pi|m|\|d\|_\infty\bigr]|t-1|.
\end{equation}
So the sum of all moment errors can be made arbitrarily small by choosing
$c$ after the actual finite set of old indices; $c$ is allowed to grow with
$\max|m|$, and $\gamma$ need not decrease. The same choice can make the
ranges $m(c+\phi(\T))$ for distinct old indices disjoint and off zero.

To pass from moments to densities, write
\[
  \mathcal P_s\rho(y)=\sum_{k\in\Z}\frac1{|s|}\rho\Bigl(\frac{y+k}{s}\Bigr),
  \qquad y\in\T .
\]
Once $c$ is chosen, every relevant $s$-range is a fixed compact set away
from zero, and no bound uniform over future passages is needed. For $s$ in
a fixed compact subset of $\R\setminus\{0\}$ the quadratic upper tail of
$\rho_{B,c}$ makes $\mathcal P_s\rho_{B,c}$ jointly continuous in $(s,y)$
with uniformly summable tails; its Fourier coefficients in the
positive-exponent convention are the moments just computed. The compact
bandwidth makes its Fourier series a finite sum on the relevant branches.
Fourier uniqueness together with \eqref{eq:signal-ideal-moments} and
\eqref{eq:signal-finite-c} therefore gives, uniformly on those branches and
in $y$,
\begin{equation}\label{eq:signal-periodic-density}
  \mathcal P_s\rho_{B,c}(y)
  =1+2\gamma\1_{\gcd(m,P)=1}\Rea\{d(m/B)\ee(-y)\}
   +o_{c\to\infty}(1),
\end{equation}
the error being for the fixed old list and scale $B$. The negative exponent
comes from our convention: $\int\mathcal P_s\rho(y)\ee(\ell y)\dd y=M_\ell$.

\emph{Realization and support.} Lemma~\ref{lem:law-approx} provides a
bounded smooth circle function $J$, strictly increasing on a good arc, and a
smooth cutoff $\psi$ compactly supported in that arc, such that
$\widetilde\rho=J_*(\psi(a)^2\dd a)$ is arbitrarily close to
$\rho_{B,c}$ in total variation (its missing mass included in the error),
with the good arc having as small a complement as requested. Since
$|A_{r,m}|$ is independent of $a$, its stationary weight is a function of
$s$ alone,
\[
  w_{r,m}(s)=\frac{|A_{r,m}(u,a)|^2}{|m\phi'(u)|},\qquad m(c+\phi(u))=s,
\]
extended by zero. For this nonnegative compactly supported weight,
Lemma~\ref{lem:fast-average} and contraction of total variation under
pushforward and periodization give
\begin{align}\label{eq:signal-law-comparison}
  &\lim_{N\to\infty}\int_\R w_{r,m}(s)
  |\mathcal P_s\widetilde\rho(Ns)-\mathcal P_s\rho_{B,c}(Ns)|\dd s \notag\\
  &\qquad=\int_\R w_{r,m}(s)\int_\T
  |\mathcal P_s\widetilde\rho(y)-\mathcal P_s\rho_{B,c}(y)|\dd y\,\dd s
  \le\|w_{r,m}\|_1\|\widetilde\rho-\rho_{B,c}\|_{L^1(\R)}.
\end{align}
The realized law has a continuous compactly supported density; the ideal law
has the uniform tail bound above. These give the continuity and domination
needed for fast averaging; the independence of the weight from $a$ is
essential here.

Lemma~\ref{lem:passage} identifies the output density, up to an arbitrarily
small $L^1$ error, as the folding to $\T$ of
$w_{r,m}(s)\mathcal P_s\widetilde\rho(Ns)$. In using it the rounding
correction is removed at the coefficient-mass level: replacing
$w_{r,m}(s)$ by $w_{r,m}(s-m\Delta_N)$ costs $o(1)$ in the summed masses
and does not translate the fast argument $Ns$; alternatively
\eqref{eq:signal-finite-c} holds uniformly with $t=t_N$. Since $cm$ and $N$
are integers,
\[
  p_{r,m}(z)=\sum_{k\in\Z}w_{r,m}(z+k),\qquad
  \ee(-N(z+k))=\ee(-Nz).
\]
Now \eqref{eq:signal-periodic-density}, \eqref{eq:signal-law-comparison},
and the passage density formula prove \eqref{eq:signal-conclusion}. Choose
$c$ first to make its error small, then the law approximation, then $N$;
the finite packet list allows all errors to be summed.

For completeness we give the simultaneous support estimates, used also for
neutral passages. Put $M=\max_{r,m}|m|$, with $M=0$ for an empty list. By
the support cutoff of Lemma~\ref{lem:passage}, at every retained point of an
output packet at index $j$ there is $u$ in its old support with
\begin{equation}\label{eq:signal-retained-support}
  j/N=m(c+\phi(u)+\Delta_N)+\eta,\qquad |\eta|\le\eta_0,
\end{equation}
where the image enlargement $\eta_0$ can be as small as desired. If
$x'=j/(NcB)$ and $z'=j/N+(j/N)J(a)/N$ is the new temporal frequency, then
\begin{align}
  |x'-m/B|
  &\le\frac{D_0(\|\phi\|_\infty+1)}c+\frac{\eta_0}{cB},
  \label{eq:signal-index-drift}\\
  \dist_\T(z',m\phi(u))
  &\le\frac MN+\eta_0
   +\frac{[M(c+\|\phi\|_\infty+1)+\eta_0]\|J\|_\infty}{N}.
  \label{eq:signal-temporal-drift}
\end{align}
Choose $c$ for the first bound and index separation, then $\eta_0$ and the
bounded $J$, finally $N$ for the second bound; a large $\|J\|_\infty$ only
increases the eventual requirement on $N$. These choices also give the
physical smooth closeness and the required $L^2$ approximations by
Lemma~\ref{lem:passage}. Every new coefficient descending from old index $m$
has the form $\ee(ma_{\rm new})$ times a scalar function of the new speed
coordinate; because different old indices have disjoint retained new index
ranges, their spare-coordinate phases are never added at one index, so the
resulting moduli are independent of the new spare coordinate. All cutoffs
can be chosen in conjugate pairs, so conjugation is retained.
\end{proof}

We next remove the coprimality restriction by placing a neutral passage
before transmission: it is the \emph{output} residue classes of the neutral
passage, not the input classes, that get tested.

\begin{lemma}[Refresh and transmission]\label{lem:refresh-transmit}
Fix $b,D_0,d,P,\gamma$ as in Lemma~\ref{lem:signal}, and put
\begin{equation}\label{eq:signal-lambda}
  \lambda=\gamma\frac{|(\Z/P\Z)^\times|}{P}>0 .
\end{equation}
Suppose finite colored packet data have mutually disjoint angular-index sets
and moduli independent of the spare coordinate, and their normalized
indices lie in a fixed compact subset of $\{x:b<|x|<D_0\}$. Let $p_r$ be the
color densities, and let $d_r:\T\to\C$ be continuous with
\begin{equation}\label{eq:signal-color-match}
  |d(m/B)-d_r(m\phi(u)\bmod1)|\le\omega
\end{equation}
on the support of every input packet of color $r$. Then a neutral passage
followed by a transmission gives output colors with
\begin{equation}\label{eq:signal-refreshed}
  \sum_r\Bigl\|p_r^+
  -p_r(z)\bigl[1+2\lambda\Rea\{d_r(z)\ee(-Nz)\}\bigr]\Bigr\|_1
  \le2\gamma\omega\sum_r\|p_r\|_1+\eps,
\end{equation}
where $\eps>0$ is arbitrary and $N$ is the second passage integer. The
physical map and inverse, the physical packet vectors, and the
normalized-index and temporal-frequency displacements can all be controlled
within arbitrary prescribed positive bounds. Output packet moduli remain
independent of the spare coordinate, and colors remain disjoint in angular
indices. The final $N$ can exceed any prescribed bound; conjugate-symmetric
data are treated conjugate-symmetrically.

If different colors start in separated compact normalized-index sets, any
fixed sufficiently small enlargements of those sets may contain their
outputs. The same holds through any fixed finite number of such pairs, with
positive margins around the band $b<|x|<D_0$. If the input supports satisfy
$\dist_\T(m/B,m\phi(u))\le\kappa$, the output supports satisfy the same
inequality with $\kappa$ increased by an arbitrarily small amount.
\end{lemma}

\begin{proof}
Apply Lemma~\ref{lem:neutral} with residue modulus $P$. Denote the refreshed
scale by $\widetilde B$, its packet densities by $\widetilde p_{r,m}$, and
its color-residue densities by
$\widetilde p_{r,h}=\sum_{m\equiv h(P)}\widetilde p_{r,m}$. That lemma makes
\begin{equation}\label{eq:signal-residue-error}
  E_r=\sum_{h=0}^{P-1}\|\widetilde p_{r,h}-p_r/P\|_1
\end{equation}
arbitrarily small, retains disjoint output angular-index sets, and gives
moduli independent of the new spare coordinate. For its support, first
choose its $c$ as large as needed for \eqref{eq:signal-index-drift}, then
its cutoff and broad linear jitter for the residue equalization, and finally
its passage integer for \eqref{eq:signal-temporal-drift}; thus equalizing
residues does not prevent either displacement from being arbitrarily small.

For a continuous function $f$ on its relevant compact domain let
$\nu_f(t)=\sup_{\dist(v,w)\le t}|f(v)-f(w)|$ be its modulus of continuity.
If the refresh changes normalized indices by at most $\delta_x$ and temporal
frequencies by at most $\delta_z$, its retained color supports satisfy
\begin{equation}\label{eq:signal-refreshed-match}
  |d(m/\widetilde B)-d_r(z)|
  \le\rho_r:=\omega+\nu_d(\delta_x)+\nu_{d_r}(\delta_z),
\end{equation}
by linking an output support point to an old one and using
\eqref{eq:signal-color-match}; the extra terms tend to zero. The compact
margin in the normalized band lets us apply Lemma~\ref{lem:signal} with the
same $P,\gamma$ after this refresh, although its actual indices and scale may
be very large.

By that lemma the transmission density, with an arbitrarily small summed
error $T_r$ for color $r$, is
\[
  \widetilde p_r(z)+2\gamma\Rea\Bigl\{\ee(-Nz)
  \sum_{\gcd(m,P)=1}d(m/\widetilde B)\,\widetilde p_{r,m}(z)\Bigr\}.
\]
Replacing the coefficient in this sum by $d_r(z)$ costs at most
$2\gamma\rho_r\|\widetilde p_r\|_1$ in $L^1$. Also
\[
  \|\widetilde p_r-p_r\|_1\le E_r,\qquad
  \Bigl\|\sum_{h\in(\Z/P\Z)^\times}\widetilde p_{r,h}
  -\frac{|(\Z/P\Z)^\times|}{P}p_r\Bigr\|_1\le E_r .
\]
Hence the color's total error against the target in
\eqref{eq:signal-refreshed} is bounded by
\begin{equation}\label{eq:signal-refresh-error}
  T_r+(1+2\gamma\|d_r\|_\infty)E_r
  +2\gamma\rho_r(\|p_r\|_1+E_r).
\end{equation}
Choosing the refresh errors, displacements, and transmission errors small
enough and summing over the finite list proves \eqref{eq:signal-refreshed}
with the exact value of $\lambda$. Only residue balancing after summing over
a color was needed, because \eqref{eq:signal-refreshed-match} holds
pointwise before the sum.

The remaining assertions follow from the two passage lemmas and
\eqref{eq:signal-index-drift}--\eqref{eq:signal-temporal-drift}. The
triangle inequality
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