\section{The construction and complete exhaustion}\label{sec:exhaustion}

\subsection{The construction}
Fix a sequence $(P_i)_{i\ge1}$ of positive integers, the \emph{profile}. For
integers $R_i\ge2^i$, chosen below, let
\[
 C_i=C(P_i,R_i),\qquad k_i=2R_i-1,
\]
and let $\mu_i$ span $\M_{C_i}$, with $\F\mu_i=\zeta_i\mu_i$
(Proposition~\ref{prop:block}). Thus
\begin{equation}\label{eq:inputlines}
 \C\mu_i\cap H_{-p}=\{0\}\quad(p\le P_i),\qquad\mu_i\in H_{-(18P_i+4)}.
\end{equation}
Let $\Theta_i\subset(-\frac12,\frac12)$ be the set of phases of $C_i$, and
$K_i=\Theta_i\cup\{\frac12\}\subset\R/\Z$ its closure; we fix an enumeration
$\Theta_i=\{\theta_{i,1},\theta_{i,2},\dots\}$.

For $s=(s_i)\in[1,2]^{\NN}$ we use \emph{labels} $\ell=(i,\sigma)$, $i\ge1$,
$\sigma\in\{+,-\}$; we call $i$ the \emph{index} of $\ell$, write
$\ell^*=(i,-\sigma)$, and put
\begin{equation}\label{eq:labels}
 t_{i,+}=s_i,\quad t_{i,-}=s_i^{-1},\quad
 \Gamma_\ell=t_\ell C_i,\quad e_\ell=D_{t_\ell}\mu_i,\quad
 \Lambda=\bigcup_\ell\Gamma_\ell
\end{equation}
(Figure~\ref{fig:overlay}). Then $\supp e_\ell\subseteq\Gamma_\ell$ and, since
$\F D_t=D_{1/t}\F$,
\begin{equation}\label{eq:Feb}
 \F e_\ell=\zeta_ie_{\ell^*},
\end{equation}
so $e_\ell\in\M_\Lambda$. We also use the periodic sets
\[
 \widetilde K_\ell=(K_i+\Z)/t_\ell\subset\R,\qquad\text{of period }\varpi_\ell=1/t_\ell .
\]
The set $\widetilde K_\ell$ contains $\Gamma_{\ell^*}=C_i/t_\ell$, but it is much
larger, and it contains the \emph{seam} $(\Z+\frac12)/t_\ell$, which does not meet
$\Lambda$. The union of the $\widetilde K_\ell$ may be dense.

\begin{theorem}\label{thm:exhaust}
For every profile $(P_i)$ there are deterministic integers $R_i\ge2^i$ and a
measurable set $\Omega\subset[1,2]^{\NN}$ of product measure at least $13/16$
such that for every $s\in\Omega$ the sets $\Gamma_\ell$ are pairwise disjoint,
$\Lambda$ is symmetric and locally finite, and the following holds. If $p\ge1$
and $T\in V_p(\Lambda)$, then for every label $\ell=(i,\sigma)$ there is
$c_\ell\in\C$ with
\begin{equation}\label{eq:components}
 \F^{-1}\bigl(\F T\,|_{\Gamma_\ell}\bigr)=c_\ell\,e_{\ell^*},\qquad
 c_\ell=0\ \text{whenever}\ P_i\ge6p .
\end{equation}
If moreover only finitely many $i$ satisfy $P_i<6p$, then
$T=\sum_\ell c_\ell e_{\ell^*}$, a finite sum.
\end{theorem}

Here $\F T|_{\Gamma_\ell}$ is the restriction of the measure $\F T$ to the set
$\Gamma_\ell$; that it is a tempered distribution is part of the proof.

\subsection{Strategy of the proof}\label{subsec:strategy}
Theorem~\ref{thm:exhaust} turns into countable dimension through one implication,
which we state first. Fix $T\in V_p(\Lambda)$, write $S=\F T$, and call
$U_\ell=\F^{-1}(S|_{\Gamma_\ell})$ the \emph{component} of $T$ on the label $\ell$. Then
\[
 T\in H_{-p}\ \Longrightarrow\ U_\ell\in H_{-6p}\ \text{for every }\ell
 \ \ \text{(Lemma~\ref{lem:selectors})},\qquad
 U_\ell=c_\ell e_{\ell^*}\ \text{and}\ P_i\ge6p\ \Longrightarrow\ c_\ell=0,
\]
because $\C e_{\ell^*}\cap H_{-6p}=\{0\}$ when $6p\le P_i$. When $P_i\to\infty$, only
finitely many components survive at each fixed order, so $T$ is a finite
combination of the $e_\ell$; and the generators admitted at order $p$, those with
$18P_i+4\le p$, make the layer dimensions unbounded (Corollary~\ref{cor:exhaust}).
The hard part is to justify the phrase ``component of $T$'': the restriction
$S|_{\Gamma_\ell}$ must be a tempered distribution, and its inverse Fourier transform
must live on the reciprocal block $\Gamma_{\ell^*}$, without additional distributions
at the seam. A priori $T$ is an arbitrary tempered distribution, and nothing
prevents its masses on different blocks from conspiring to form a new crystalline
measure; ruling this out is the content of the proof. It has a preparatory step
and three main steps.

\emph{Step~0: splitting.} The points of different blocks are separated at a
polynomial rate; this is where the random scales $s_i$ are first used. Smooth
selectors then cut $S$ into its restrictions $S_\ell$ to the sets $\Gamma_\ell$, at
the cost of a fixed loss of Hermite order, and give
$T=\sum_{i\le M}U_\ell+U_{>M}$ with $U_\ell=\F^{-1}S_\ell$, where $U_{>M}$ comes from
the blocks of index $>M$. Nothing is known yet about where $U_\ell$ lives.

\emph{Step~1: isolating a block up to the seam.} Multiply $T$ by a smooth function
$G_M$ which vanishes on the first $M$ blocks and whose Fourier coefficients have
mean one. Then $G_MT$ lives beyond the next gap, which is far away, while
$\F(G_MT)$ is a combination of translates of $S$. Because the scales are random,
no translate of an atom accidentally lands on another atom (nonresonance), and
this leaves a convolution identity for the masses of $S_\ell$. It shows that $U_\ell$
is supported in its periodic set $\widetilde K_\ell$, up to a polynomial whose
coefficients are antiperiodic: a residue created by the accumulation at the
seam.

\emph{Step~2: removing the residues.} Antiperiodic terms change sign under
translation by half a period, while everything supported in the sets
$\widetilde K_{\ell'}$ is unaffected by suitable translations. Using Kronecker's
theorem we choose two large translations that agree, modulo all periods, except
at the block under consideration, where they differ by half a period. Comparing
the two isolates the leading coefficient of the residue and shows that it is
supported in $\widetilde K_\ell$. The error is controlled because the translated
test functions stay inside the large gap before the next block.

\emph{Step~3: recovering the atoms.} Shrinking test functions at the points of
$\widetilde K_\ell$ show that $U_\ell$ has the same masses as $T$ on $\Gamma_{\ell^*}$
and nothing at the remaining points of $\widetilde K_\ell$, in particular nothing at
the seam. Thus $U_\ell$ is a crystalline measure on one rescaled block, and
Proposition~\ref{prop:block} identifies it.

\emph{Order of choices.} Every step produces error terms that must tend to zero.
All of them are paid for by the gaps: the inner radius $R_{M+1}$ is chosen after
all the parameters of stage $M$ have been fixed, and so large that the error terms
of stage $M$ are at most $2^{-M}$. The probabilistic events used (separation,
nonresonance, Kronecker nets) have probability bounds that hold for every
continuation of the sequence of radii; this is why a single deterministic sequence
$(R_i)$ and a set of scales of positive measure can be chosen. The proofs of the
technical lemmas, the construction of the parameters, and a full table of the
parameters are in Appendix~\ref{app:sec5}.

Table~\ref{tab:objects} lists the objects attached to a label $\ell=(i,\sigma)$, and
Table~\ref{tab:choices} the three choices that define $\Lambda$.

\begin{table}[!ht]
\centering\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.17\textwidth}>{\raggedright\arraybackslash}p{0.38\textwidth}>{\raggedright\arraybackslash}p{0.37\textwidth}}
\hline
Object & Role & What is known about its support\\
\hline
$\Gamma_\ell=t_\ell C_i$ & dilated block; carries $e_\ell$ & by definition\\
$\widetilde K_\ell=(K_i+\Z)/t_\ell$ & periodic set containing $\Gamma_{\ell^*}$ and the seam $(\Z+\frac12)/t_\ell$ & by definition\\
$S_\ell=\chi_{\{\ell\}}\F T$ & restriction of $\F T$ to $\Gamma_\ell$ (Step~0) & contained in $\Gamma_\ell$\\
$U_\ell=\F^{-1}S_\ell$ & component of $T$ on $\ell$ & unknown after Step~0; in $\widetilde K_\ell$ up to the residue after Step~1; in $\widetilde K_\ell$ after Step~2; atomic on $\Gamma_{\ell^*}$ after Step~3\\
$W_\ell$, $v_{\ell,r}$ & $U_\ell=W_\ell+\sum_rx^rv_{\ell,r}$ \eqref{eq:residue}; $v_{\ell,r}$ is the residue at the seam & $W_\ell$ in $\widetilde K_\ell$; $v_{\ell,r}$ antiperiodic, shown to be supported in $\widetilde K_\ell$ in Step~2\\
\hline
\end{tabular}
\caption{The objects attached to a label $\ell=(i,\sigma)$.}\label{tab:objects}
\end{table}

\begin{table}[!ht]
\centering\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.12\textwidth}>{\raggedright\arraybackslash}p{0.48\textwidth}>{\raggedright\arraybackslash}p{0.32\textwidth}}
\hline
Choice & Role & When it is chosen\\
\hline
$P_i$ & Hermite order of the $i$-th block, between $P_i$ and $18P_i+4$ & given (the profile)\\
$R_i$ & inner radius of the $i$-th block; keeps $\Lambda$ locally finite, and the gap before the block pays for all errors of stage $i-1$ & after all parameters of stage $i-1$\\
$s_i$ & scale of the $i$-th block; separates blocks (Lemma~\ref{lem:separation}) and prevents resonances (Lemma~\ref{lem:nonres}) & at random, in a set $\Omega$ of probability at least $\frac{13}{16}$\\
\hline
\end{tabular}
\caption{The three choices defining $\Lambda$. The auxiliary parameters of stage
$M$ (cutoffs, probe radii, translations) depend only on $R_1,\dots,R_M$
(Proposition~\ref{prop:parameters} and Table~\ref{tab:parameters}).}\label{tab:choices}
\end{table}

\subsection{Tools}\label{subsec:tools}
For $t>0$ the dilation $D_t$ is bounded with bounded inverse on $H_{-p}$ for
every integer $p$, with constants uniform for $t$ in compact subsets of
$(0,\infty)$; this follows from the norm equivalence of Section~\ref{sec:prelim},
$xD_t=tD_tx$, $\partial D_t=t^{-1}D_t\partial$, and duality. Hence
$e_\ell\in H_{-(18P_i+4)}$ and $\C e_\ell\cap H_{-p}=\{0\}$ for $p\le P_i$.

By Proposition~\ref{prop:block}(ii) and~(iv) and $R_i\ge2^i$, only
$O(\log(2+r))$ blocks meet $[-r,r]$, even after dilation by $t_\ell\in[\frac12,2]$,
and each contributes $O((1+r)^2)$ points. Thus $\Lambda$ is locally finite for
every choice of radii and scales, with
$\#(\Lambda\cap[-r,r])=O((1+r)^2\log(2+r))$, uniformly over all sequences
$(R_i)$ with $R_i\ge2^i$.

\begin{lemma}[Separation]\label{lem:separation}
Let $(R_i)$ be any sequence with $R_i\ge2^i$. For almost every $s$ there is
$c_s\in(0,1]$ such that
\begin{equation}\label{eq:crosssep}
 |x-y|\ge c_s(1+|x|+|y|)^{-6}\qquad(x\in\Gamma_\ell,\ y\in\Gamma_{\ell'},\ \ell\ne\ell').
\end{equation}
\end{lemma}

In particular the sets $\Gamma_\ell$ are pairwise disjoint for almost every $s$.
Within one block the atoms coalesce, and no separation is claimed there.

\begin{lemma}[Selectors]\label{lem:selectors}
Assume \eqref{eq:crosssep}. For every set $E$ of labels there is
$\chi_E\in C^\infty(\R)$ which equals $1$ on a neighbourhood of
$\bigcup_{\ell\in E}\Gamma_\ell$ and $0$ on a neighbourhood of
$\bigcup_{\ell\notin E}\Gamma_\ell$, with
$|\chi_E^{(j)}(x)|\le C_{j,s}(1+|x|)^{6j}$ for all $j$, where $C_{j,s}$ does not
depend on $E$. Consequently
\begin{equation}\label{eq:selector}
 \norm{\chi_EU}_{-6p}\le C_{p,s}\norm U_{-p}\qquad(U\in H_{-p}),
\end{equation}
uniformly in $E$.
\end{lemma}

\emph{Splitting.} For $T\in V_p(\Lambda)$ write $S=\F T$, $S_\ell=\chi_{\{\ell\}}S$,
$U_\ell=\F^{-1}S_\ell$, and, with $E_M$ the set of labels of index $>M$,
$S_{>M}=\chi_{E_M}S$ and $U_{>M}=\F^{-1}S_{>M}$. Since $S$ is locally atomic on
$\Lambda$, $S_\ell$ is the restriction of the measure $S$ to $\Gamma_\ell$, and
$S_{>M}$ its restriction to the blocks of index $>M$. Since
$\sum_{\ell\notin E_M}\chi_{\{\ell\}}+\chi_{E_M}=1$ near $\Lambda$,
\begin{equation}\label{eq:finitesplit}
 T=\sum_{\ell\notin E_M}U_\ell+U_{>M},\qquad
 \norm{U_\ell}_{-6p}+\norm{S_{>M}}_{-6p}\le C_{p,s}\norm T_{-p}.
\end{equation}
The measure $S_{>M}$ is supported in $\{|x|>(R_{M+1}-\frac12)/2\}$. For
$\theta\in\Theta_i$ (an isolated point of $K_i$), the masses $c_{\ell,\theta}(n)$ of
$S_\ell$ at the points $t_\ell(n+\theta)$ satisfy
\begin{equation}\label{eq:orbitgrowth}
 |c_{\ell,\theta}(n)|\le C_{p,s,\ell,\theta}\norm T_{-p}(1+|n|)^{12p},
\end{equation}
as one sees by testing $S_\ell\in H_{-6p}$ against translates of a fixed bump
narrower than the distance from $\theta$ to $K_i\setminus\{\theta\}$.

\emph{Annihilators.} For each $i$ choose a nonnegative smooth $1$-periodic
function $g_i$ of mean one whose zero set is exactly $K_i+\Z$ (normalize a sum of
flat positive bumps on the complementary intervals). Its Fourier coefficients
$a_i(m)$ decay faster than any power, $a_i(0)=1$, and $g_i(t_\ell x)$ vanishes
exactly on $\widetilde K_\ell$. At stage $M$ put
\begin{align}
 G_M(x)&=\prod_{i\le M}g_i(s_ix)\,g_i(x/s_i)
 =\sum_{k\in\Z^{2M}}A_M(k)e^{2\pi\ii\omega_kx},\label{eq:annexp}\\
 A_M(k)&=\prod_{i\le M}a_i(k_{i,+})a_i(k_{i,-}),\qquad
 \omega_k=\sum_{i\le M}\bigl(k_{i,+}s_i+k_{i,-}/s_i\bigr).\nonumber
\end{align}
The coefficients $A_M(k)$ decay faster than any power of $|k|$, and the
derivatives of $G_M$ are bounded uniformly for $(s_1,\dots,s_M)\in[1,2]^M$. Every
point of $\Gamma_\ell$ with $\ell$ of index $\le M$ lies in $\widetilde K_{\ell^*}$
and is a zero of a factor of $G_M$, so $G_MT$ is supported in the union of the
blocks of index $>M$. The series $\F(G_MT)=\sum_kA_M(k)\tau_{-\omega_k}S$
converges absolutely in $H_{-p}$.

\emph{Nonresonance.} The \emph{catalogue} of stage $M$ consists of the targets
$z=t_\ell(n+\theta_{i,j})$ with $\ell=(i,\sigma)$, $i\le M$, $j\le M$ and $|n|\le M$; a
target that is not a point of $\Lambda$ carries mass zero. Let $d_M>0$ be the
minimum, over $i,j\le M$, of the distances of $\theta_{i,j}$ from
$K_i\setminus\{\theta_{i,j}\}$ and from $0$ in $\R/\Z$. For a target
$z=t_\ell(n+\theta)$, a point $y\in\Lambda$ and $k\in\Z^{2M}$, the pair $(y,k)$ is
\emph{expected} if $y=t_\ell(n'+\theta)\in\Gamma_\ell$ and all coordinates of $k$
vanish except $k_{i,\sigma}=n-n'$; then $y+\omega_k=z$.

\begin{lemma}[Nonresonance]\label{lem:nonres}
Let $M\ge1$ and fix $R_1,\dots,R_M$. There are $c_M\in(0,1]$ and
$\beta_M=2(M+1)(2M+6)$, depending only on $R_1,\dots,R_M$ and on the profile,
such that for every deterministic continuation $(R_i)_{i>M}$ with $R_i\ge2^i$ the
event $\mathsf A_M$ that
\begin{equation}\label{eq:nonres}
 |y+\omega_k-z|\ge c_M(1+L)^{-\beta_M}
\end{equation}
for every dyadic $L\ge1$, every target $z$, every $k$ with $\norm k_\infty\le L$
and every $y\in\Lambda$ such that $(y,k)$ is not expected, has probability at
least $1-2^{-M-4}$.
\end{lemma}

\emph{An example.} The simplest instance of Lemma~\ref{lem:nonres} involves a single
block $i$, its two reciprocal sectors $\Gamma_{(i,+)}=s_iC_i$ and
$\Gamma_{(i,-)}=C_i/s_i$, and a target $z=s_i(n+\theta)$ in the first. In Step~1,
$\F(G_MT)$ is a combination of translates of $S$ by the frequencies $\omega_k$. The
translate of an atom $y=s_i(n'+\theta)$ of the same sector, with the same phase, by
$\omega=(n-n')s_i$ lands exactly on $z$; these expected pairs produce the convolution
identity \eqref{eq:convzero} below. The translate of an atom $y=u/s_i$ of the
reciprocal sector by $\omega=ms_i$ lands at distance
$|u/s_i+(m-n-\theta)s_i|$ from $z$. After multiplication by $s_i$ this is
$|u+(m-n-\theta)s_i^2|$, whose derivative in $s_i\in[1,2]$ has modulus at least
$2|m-n-\theta|\ge2\operatorname{dist}(\theta,\Z)>0$; so the probability that the
distance is less than $\varepsilon$ is at most $2\varepsilon/\operatorname{dist}(\theta,\Z)$.
At scale $L$ only polynomially many pairs $(y,k)$ matter, and the sublevel
estimate of Appendix~\ref{app:sec5} gives the analogous bound when several scales
are involved; this yields Lemma~\ref{lem:nonres}. We do not know an explicit choice
of the scales that avoids all resonances (Remark~\ref{rem:constants}).

\emph{Antiperiodic remainders.} We will meet distributions that satisfy an
antiperiodic difference equation off a periodic set.

\begin{lemma}[Antiperiodic lift]\label{lem:lift}
Let $\varpi\in[\frac12,2]$, let $\Pi\subset\R$ be closed and $\varpi$-periodic,
and let $U\in H_{-q}$ satisfy $(\tau_\varpi+I)^{\Delta+1}U=0$ on $\R\setminus\Pi$.
Then
\begin{equation}\label{eq:lift}
 U=W+\sum_{r=0}^{\Delta}x^rv_r,\qquad\supp W\subseteq\Pi,\qquad
 \tau_\varpi v_r=-v_r,
\end{equation}
where the $v_r$ are obtained from $U$ by a continuous linear operation depending
only on $\varpi$ and $\Delta$, not on $\Pi$. The $v_r$ lose only polynomially many
Hermite orders: for $q=6p$ and $\Delta\le12p$ one may take
\begin{equation}\label{eq:liftbound}
 \norm{v_r}_{-m_p}\le C_p\norm U_{-6p},\qquad m_p=1000(p+1)^3 .
\end{equation}
\end{lemma}

The proof, given in Appendix~\ref{app:lift}, uses a cutoff and a periodization;
it does not divide by a flat annihilator. Put $N_p=2m_p+2$.

\emph{The parameters.} Fix a smooth bump $\phi$ supported in $(-1,1)$ and equal
to $1$ near $0$; the \emph{bump of radius $r$ centred at $z$} is
$x\mapsto\phi((x-z)/r)$. Let $\theta_R$ be a smooth function vanishing on
$[-R/4,R/4]$ and equal to $1$ outside $[-R/3,R/3]$, with the usual rescaled
derivative bounds, and let $f_{y,r}^{(d)}(x)=\frac{(x-y)^r}{r!}\phi((x-y)/d)$.

\begin{proposition}\label{prop:parameters}
Fix a profile $(P_i)$. There are deterministic integers $R_i\ge2^i$ and, for each
$M\ge1$, deterministic parameters: a cutoff $L_M\ge M$; radii
$d_M^A,d_M^J\in(0,1/M]$; a smooth partition of unity $(\kappa_{M,j})_{j\le J_M}$ of
$[-M,M]$ by functions whose supports have diameter less than
$\min\{1/M,\min_{i\le M}1/(100k_i)\}$; numbers $1\le\underline h_M\le\overline h_M$
with $6(\overline h_M+M+2)\le R_{M+1}$; and $\eta_M\in(0,1/(1000k_M)]$. There is
also a measurable set $\Omega\subset[1,2]^{\NN}$ with $\mathbb P(\Omega)\ge13/16$
such that every $s\in\Omega$ satisfies \eqref{eq:crosssep}, the sets
$\widetilde K_\ell$ are pairwise disjoint, and for every $M\ge1$ and every $p\le M$,
with $R=R_{M+1}$:
\begin{enumerate}[label=\textup{(P\arabic*)},leftmargin=2.5em]
\item \emph{Nonresonance.} For every target $z$ of the stage-$M$ catalogue, every
$k$ with $\norm k_\infty\le L_M$ and every $y\in\Lambda$ such that $(y,k)$ is not
expected, $|y+\omega_k-z|\ge10d_M^A$. For every bump $f$ of radius $d_M^A$
centred at a target and every $S\in H_{-p}$,
\begin{equation}\label{eq:fouriertail}
 \Bigl|\sum_{\norm k_\infty>L_M}A_M(k)\,(\tau_{-\omega_k}S)(f)\Bigr|\le2^{-M}\norm S_{-p}.
\end{equation}
\item \emph{Separation of the periodic sets.} For distinct labels $\ell,\ell'$ of
index $\le M$, the sets $\widetilde K_\ell\cap[-M-3,M+3]$ and
$\widetilde K_{\ell'}\cap[-M-3,M+3]$ are at distance at least $10d_M^J$.
\item \emph{Translations.} For every vector $(a_\ell)$, $a_\ell\in[0,2\varpi_\ell)$,
indexed by the labels of index $\le M$, there is $h\in[\underline h_M,\overline h_M]$
with $h-a_\ell\in2\varpi_\ell\Z+[-\eta_M,\eta_M]$ for all these $\ell$. For every such
$h$, every $\varphi=\kappa_{M,j}\psi$ with $\supp\psi\subset[-M,M]$ and
$\norm\psi_{C^{N_p}}\le1$, every $\Delta\le12p$, and all $\varpi_\ell$-antiperiodic
$v_{\ell,r}\in H_{-m_p}$ ($r\le\Delta$) of norm at most $\mathcal A$,
\begin{equation}\label{eq:translation}
 \Bigl|h^{-\Delta}\sum_\ell\sum_{r\le\Delta}(x^rv_{\ell,r})(\varphi(\cdot-h))
 -\sum_\ell v_{\ell,\Delta}(\varphi(\cdot-a_\ell))\Bigr|\le\frac{2^{-M}\mathcal A}{1+J_M}.
\end{equation}
\item \emph{Gaps.} For bumps $f$ of radius $d_M^A$ centred at targets, for
$|h|\le\overline h_M$ and $\psi$ as in \textup{(P3)}, and for $|y|\le M+1$ and
$0\le r\le12p$,
\begin{align}
 \norm{\theta_RG_M\F^{\pm1}f}_p&\le2^{-M},\label{eq:budgetA}\\
 \norm{\theta_R\F^{\pm1}\bigl((\kappa_{M,j}\psi)(\cdot-h)\bigr)}_{6p}&\le\frac{2^{-M}}{1+J_M},\label{eq:budgetB}\\
 \norm{\theta_R\F^{\pm1}f_{y,r}^{(d_M^J)}}_{6p}&\le2^{-M}.\label{eq:budgetJ}
\end{align}
\end{enumerate}
\end{proposition}

We fix $(R_i)$, $\Omega$ and $s\in\Omega$ as in Proposition~\ref{prop:parameters}
from now on, and we abbreviate $f_{y,r}=f_{y,r}^{(d_M^J)}$.

\subsection{Step 1: phase convolution and the seam}\label{subsec:phaseconv}
Let $T\in V_p(\Lambda)$, fix a label $\ell=(i,\sigma)$, a phase $\theta\in\Theta_i$
and an integer $n$, and let $M\ge p$ be so large that $z=t_\ell(n+\theta)$ belongs
to the catalogue of stage $M$. Test $\F(G_MT)$ against the bump $f$ of radius
$d_M^A$ centred at $z$. By (P1), every unexpected translated atom misses the
support of $f$, so the part of $\sum_kA_M(k)(\tau_{-\omega_k}S)(f)$ with
$\norm k_\infty\le L_M$ equals $\sum_{|m|\le L_M}a_i(m)c_{\ell,\theta}(n-m)$, because
all other factors of $A_M(k)$ equal $a_{i'}(0)=1$; the omitted part is bounded by
\eqref{eq:fouriertail}. On the other hand
$\F(G_MT)(f)=(G_MT)(\theta_R\F f)=T(\theta_RG_M\F f)$ with $R=R_{M+1}$, because
$G_MT$ is supported in the union of the blocks of index $>M$, and by
\eqref{eq:budgetA} this is at most $2^{-M}\norm T_{-p}$. Hence the truncated sum
has modulus at most $2^{1-M}\norm T_{-p}$. Letting $M\to\infty$, and using
\eqref{eq:orbitgrowth} and the rapid decay of $a_i$, we obtain
\begin{equation}\label{eq:convzero}
 \sum_{m\in\Z}a_i(m)\,c_{\ell,\theta}(n-m)=0\qquad(n\in\Z,\ \theta\in\Theta_i).
\end{equation}

The left-hand side of \eqref{eq:convzero} is the mass at $t_\ell(n+\theta)$ of
\[
 Z_\ell:=\F\bigl(g_i(t_\ell\,\cdot)\,U_\ell\bigr)=\sum_{m\in\Z}a_i(m)\,\tau_{-mt_\ell}S_\ell ,
\]
where the series converges absolutely in $H_{-6p}$, and pairing with a bump that
isolates $t_\ell(n+\theta)$ from the rest of $t_\ell(K_i+\Z)$ commutes with the
summation. Each term is supported in $t_\ell(K_i+\Z)$, so \eqref{eq:convzero}
shows that $Z_\ell$ vanishes near every isolated point of $t_\ell(K_i+\Z)$: it is
supported in the seam lattice $t_\ell(\Z+\frac12)$. Since $Z_\ell\in H_{-6p}$, it
has order at most $12p$ near each lattice point, so
$Z_\ell=\sum_{m\in\Z}\sum_{r\le12p}c_{m,r}\delta^{(r)}_{t_\ell(m+1/2)}$, and testing
against translated bumps shows that the $c_{m,r}$ grow at most polynomially in
$m$. Since $\F^{-1}(\delta_a^{(r)})=(-2\pi\ii x)^re^{2\pi\ii ax}$, we get
$g_i(t_\ell x)U_\ell=\sum_{r\le12p}x^rw_r$ with
$w_r=(-2\pi\ii)^r\sum_mc_{m,r}e^{2\pi\ii t_\ell(m+1/2)x}$, and
$\tau_{\varpi_\ell}w_r=-w_r$. Since $(\tau_{\varpi_\ell}+I)(x^rw)$ has degree $r-1$ in
$x$ when $w$ is $\varpi_\ell$-antiperiodic,
$(\tau_{\varpi_\ell}+I)^{12p+1}[g_i(t_\ell x)\,U_\ell]=0$. The multiplier
$g_i(t_\ell x)$ is $\varpi_\ell$-periodic, commutes with $\tau_{\varpi_\ell}$, and does
not vanish on $\R\setminus\widetilde K_\ell$. Hence
$(\tau_{\varpi_\ell}+I)^{12p+1}U_\ell=0$ on $\R\setminus\widetilde K_\ell$, and
Lemma~\ref{lem:lift} gives
\begin{equation}\label{eq:residue}
 U_\ell=W_\ell+\sum_{r=0}^{12p}x^rv_{\ell,r},\qquad\supp W_\ell\subseteq\widetilde K_\ell,\qquad
 \tau_{\varpi_\ell}v_{\ell,r}=-v_{\ell,r},\qquad
 \norm{v_{\ell,r}}_{-m_p}\le\mathcal A,
\end{equation}
with $\mathcal A=C_pC_{p,s}\norm T_{-p}$ independent of $\ell$. The large
derivatives of the flat function $g_i$ do not enter these bounds: the lift is
applied to $U_\ell$ after the difference equation has been obtained.

\subsection{Step 2: elimination of the antiperiodic remainders}\label{subsec:elimination}
We show by downward induction on $\Delta=12p,12p-1,\dots,0$ that
$\supp v_{\ell,\Delta}\subseteq\widetilde K_\ell$ for every label $\ell$. Suppose this
holds for all degrees $>\Delta$, and absorb the corresponding terms into $W_\ell$,
so that $U_\ell=W'_\ell+\sum_{r\le\Delta}x^rv_{\ell,r}$ with
$\supp W'_\ell\subseteq\widetilde K_\ell$. Fix a label $\ell_0$ and $\psi\in C_c^\infty$
with $\supp\psi\cap\widetilde K_{\ell_0}=\varnothing$, normalized by
$\norm\psi_{C^{N_p}}\le1$. Let $M$ be large: $\ell_0$ has index at most $M$, $p\le M$,
$\supp\psi\subset[-M,M]$, and $\eta_M$ is smaller than the distance from
$\supp\psi$ to $\widetilde K_{\ell_0}$. By \eqref{eq:finitesplit},
\begin{equation}\label{eq:finiteQ}
 \sum_{\ell\notin E_M}\sum_{r\le\Delta}x^rv_{\ell,r}
 =T-\sum_{\ell\notin E_M}W'_\ell-U_{>M}.
\end{equation}
Let $\varphi=\kappa_{M,j}\psi$ be one of the pieces of $\psi$. For every label
$\ell\ne\ell_0$ of index $\le M$ choose $a_\ell\in[0,2\varpi_\ell)$ such that
$\supp\varphi+a_\ell$ lies in a gap of $\widetilde K_\ell$, at distance at least
$1/(20k_i)$ from it; this is possible since each $\widetilde K_\ell$ has a gap of
length at least $1/(4k_i)$ in every period (Proposition~\ref{prop:block}(v)) and
$\supp\varphi$ has diameter less than $1/(100k_i)$. Put $a_{\ell_0}=0$ in a first
phase vector and $a_{\ell_0}=\varpi_{\ell_0}$ in a second one, the other coordinates
being equal, and let $h,h'$ be the corresponding translations given by (P3).

Pair \eqref{eq:finiteQ} with $\varphi(\cdot-h)$. On the right,
$W'_\ell(\varphi(\cdot-h))=0$ for all $\ell\notin E_M$, because $\supp\varphi+h$ avoids
every $\widetilde K_\ell$ of index $\le M$ (here $\eta_M<1/(20k_i)$ is used);
$T(\varphi(\cdot-h))=0$, because $T$ is locally atomic on $\Lambda$, whose points of
index $\le M$ lie in $\bigcup_\ell\widetilde K_\ell$ and whose other points lie
outside $[-R_{M+1}/3,R_{M+1}/3]\supset\supp\varphi+h$; and
$|U_{>M}(\varphi(\cdot-h))|=|S_{>M}(\theta_R\F^{-1}(\varphi(\cdot-h)))|\le
C_{p,s}\norm T_{-p}2^{-M}(1+J_M)^{-1}$ by \eqref{eq:finitesplit} and
\eqref{eq:budgetB}. The same holds for $h'$. On the left we use
\eqref{eq:translation} for both translations. Subtracting the two resulting
identities, all terms with $\ell\ne\ell_0$ cancel, and since
$v_{\ell_0,\Delta}(\varphi(\cdot-\varpi_{\ell_0}))=-v_{\ell_0,\Delta}(\varphi)$ we obtain
$|v_{\ell_0,\Delta}(\varphi)|\le C\,2^{-M}(1+J_M)^{-1}(\norm T_{-p}+\mathcal A)$.
Summing over the at most $J_M$ pieces gives
$|v_{\ell_0,\Delta}(\psi)|\le C\,2^{-M}(\norm T_{-p}+\mathcal A)$. Letting
$M\to\infty$ gives $v_{\ell_0,\Delta}(\psi)=0$, which completes the induction.
Consequently
\begin{equation}\label{eq:closureloc}
 \supp U_\ell\subseteq\widetilde K_\ell\qquad\text{for every label }\ell.
\end{equation}
Only finite sums and the term $U_{>M}$ were used; no convergence of the series
$\sum_\ell W_\ell$ or $\sum_\ell\sum_rx^rv_{\ell,r}$ is assumed.

\subsection{Step 3: recovery of the atoms}\label{subsec:recovery}
Fix a label $\ell_0$ and a point $y\in\widetilde K_{\ell_0}$, and let $M$ be large:
$\ell_0$ has index at most $M$, $|y|\le M$, $p\le M$. By (P2), the probes $f_{y,r}$,
of radius $d_M^J$, do not meet any $\widetilde K_\ell$ with $\ell\ne\ell_0$ of index
$\le M$. Pairing \eqref{eq:finitesplit} with $f_{y,r}$ and using
\eqref{eq:closureloc}, we get $U_{\ell_0}(f_{y,r})=T(f_{y,r})-U_{>M}(f_{y,r})$, and
$|U_{>M}(f_{y,r})|=|S_{>M}(\theta_R\F^{-1}f_{y,r})|\le C_{p,s}\norm T_{-p}2^{-M}$
by \eqref{eq:budgetJ}. As $M\to\infty$ the radius $d_M^J$ tends to zero, and since
$\Lambda$ is locally finite, $T(f_{y,r})$ is eventually $T(\{y\})$ if $r=0$ and
$y\in\Lambda$, and $0$ otherwise.

If $y$ is an isolated point of $\widetilde K_{\ell_0}$, then near $y$ the distribution
$U_{\ell_0}$ is a finite sum $\sum_ru_r\delta_y^{(r)}$, and
$U_{\ell_0}(f_{y,r})=(-1)^ru_r$ once $d_M^J$ is small. Hence $U_{\ell_0}$ has, at
$y$, no derivative terms and the same mass as $T$; this mass is zero unless
$y\in\Lambda$, in which case $y\in\Gamma_{\ell_0^*}$, because the points of $\Lambda$
of index $\le M$ lie in the pairwise disjoint sets $\widetilde K_\ell$ and those of
larger index are far away. If $y$ is a seam point, then $y\notin\Lambda$; near $y$
only finitely many points of $\widetilde K_{\ell_0}$ belong to $\Gamma_{\ell_0^*}$, since
the phase $\gamma_j$ of $C_i$ occurs only in the cells $|n|\ge k_i+j$, and
$U_{\ell_0}$ vanishes near the other isolated points. Thus, in a small
neighbourhood of $y$, $U_{\ell_0}$ is supported at $\{y\}$, and the same probes
show that it vanishes there. Consequently $U_{\ell_0}$ is locally atomic on
$\Gamma_{\ell_0^*}=C_i/t_{\ell_0}$, while $\F U_{\ell_0}=S_{\ell_0}$ is locally atomic on
$\Gamma_{\ell_0}=t_{\ell_0}C_i$.

\subsection{Conclusion}\label{subsec:conclusion}
\begin{proof}[Proof of Theorem~\ref{thm:exhaust}]
By the previous subsection and Proposition~\ref{prop:covariance}(a),
$D_{t_\ell}U_\ell\in\M_{C_i}=\C\mu_i$, so $U_\ell=c_\ell e_{\ell^*}$ for some
$c_\ell\in\C$; this is \eqref{eq:components}, since $U_\ell=\F^{-1}(S|_{\Gamma_\ell})$.
As $U_\ell\in H_{-6p}$ and $\C e_{\ell^*}\cap H_{-6p}=\{0\}$ when $6p\le P_i$, we get
$c_\ell=0$ whenever $P_i\ge6p$. If only finitely many $i$ satisfy $P_i<6p$, choose
$M$ larger than all of them. Then $S_\ell=0$ for all labels of index $>M$, so the
locally atomic measure $S_{>M}$ has no mass at any point and vanishes, and
\eqref{eq:finitesplit} gives $T=\sum_{\ell\notin E_M}c_\ell e_{\ell^*}$. Disjointness of
the $\Gamma_\ell$ follows from \eqref{eq:crosssep}, and symmetry of $\Lambda$ from
that of the $C_i$.
\end{proof}

\begin{corollary}\label{cor:exhaust}
In the situation of Theorem~\ref{thm:exhaust}, assume $P_i\to\infty$. Then
\begin{equation}\label{eq:exhaust}
 \M_\Lambda=\bigoplus_{\ell}\C e_\ell\qquad\text{(algebraic direct sum)},
\end{equation}
every element of $V_p(\Lambda)$ is a combination of the $e_\ell$ with $P_i<6p$,
and
\begin{equation}\label{eq:rankB2}
 2\,\#\{i:18P_i+4\le p\}\le r_p(\Lambda)\le2\,\#\{i:P_i<6p\}\qquad(p\ge1).
\end{equation}
In particular $\dim\M_\Lambda=\aleph_0$. If $P_i\not\to\infty$, then
$\dim\M_\Lambda=\cc$.
\end{corollary}

\begin{proof}
The $e_\ell$ belong to $\M_\Lambda$ by \eqref{eq:Feb} and are linearly independent
because their supports $\Gamma_\ell$ are pairwise disjoint. If $P_i\to\infty$, then
for each $p$ only finitely many $i$ satisfy $P_i<6p$, so
Theorem~\ref{thm:exhaust} gives \eqref{eq:exhaust} and the upper bound in
\eqref{eq:rankB2}. The lower bound holds for every profile, since $e_\ell\in
H_{-(18P_i+4)}\subseteq H_{-p}$ when $18P_i+4\le p$. Hence all $r_p$ are finite and
unbounded, and Proposition~\ref{prop:trichotomy} gives $\dim\M_\Lambda=\aleph_0$.
If $P_i\le q$ for infinitely many $i$, the lower bound shows that
$V_{18q+4}(\Lambda)$ is infinite-dimensional, and
Proposition~\ref{prop:trichotomy} gives $\dim\M_\Lambda=\cc$.
\end{proof}

\subsection{Proof of Theorems~\ref{thm:A} and~\ref{thm:B}}
For Theorem~\ref{thm:A} take $P_i=i$ in Corollary~\ref{cor:exhaust} and
enumerate the basis as $\mu_{2i-1}=e_{(i,+)}$, $\mu_{2i}=e_{(i,-)}$. Then (i) is
\eqref{eq:exhaust}, (ii) is \eqref{eq:rankB2} because
$\#\{i:18i+4\le p\}=\lo p$ and $\#\{i:i<6p\}=6p-1$, and (iii) follows from
\eqref{eq:inputlines} and the boundedness of dilations. For Theorem~\ref{thm:B},
given a nondecreasing unbounded $F$ with $F(0)=0$, put
$P_i=\min\{q\ge1:F(q)\ge2i\}$. Then $P_i\le q$ if and only if $2i\le F(q)$, so
$\#\{i:P_i\le q\}=\lfloor F(q)/2\rfloor$ and $P_i\to\infty$. By
Corollary~\ref{cor:exhaust}, $\dim\M_\Lambda=\aleph_0$ and
\[
 r_p(\Lambda)\ge2\Bigl\lfloor\tfrac12F\bigl(\lo p\bigr)\Bigr\rfloor\ge F\bigl(\lo p\bigr)-1,
 \qquad
 r_p(\Lambda)\le2\Bigl\lfloor\tfrac12F(6p-1)\Bigr\rfloor\le F(6p).
\]
\qed

\begin{remark}\label{rem:constants}
The constants $18$ and $6$ come from the Hermite order of the blocks in
Proposition~\ref{prop:block} and from the loss in \eqref{eq:selector}; no
optimality is claimed. The scales $s$ are chosen at random only to avoid
resonances; we do not know an explicit admissible sequence.
\end{remark}