\section{Rigidity of lattice-supported modular distributions and the
localization obstruction}
\label{sec:rigidity}

Expanding the spectrum of an admissible partition function to finite
order about points of circle lattices produces a distribution on
momentum space with discrete support. This section determines the
distributions of this kind that are invariant under $\mathbb S$ and
$\mathcal C$, and derives a quantitative consequence. The results are
of three kinds.
\begin{enumerate}[(a)]
\item A tempered distribution that is supported on a finite union of
 circle lattices and invariant under $\mathbb S$ and $\mathcal C$ is a
 finite sum of polynomials in the boost generator $\mathcal K$ applied
 to circle combs (Theorems~\ref{thm:grid-rigidity},
 \ref{thm:circle-union} and~\ref{thm:circle-jets}). In particular its
 local expansion at the origin is a multiple of $\delta_0$, and in the
 even sector it is a combination of radius derivatives of the circle
 Fock measures $\rhoF_r$.
\item The statement about the origin has a finite quantitative form:
 finitely many Gaussian-mixture test functions bound every derivative
 coefficient at the origin of a $\mathcal C$-invariant distribution by
 its $\mathbb S$-defect (Theorem~\ref{thm:origin-certificate}), with at
 most a finite-power loss when one radius moves
 (Proposition~\ref{prop:moving-certificate}).
\item The shifted vacuum of an admissible partition function
 contributes $-\frac a4(\partial_P^2+\partial_Q^2)\delta_0$ at the
 origin. Consequently the low spectrum cannot be localized, with a
 power-law error in a logarithmic window, on a fixed finite union of
 circle lattices together with one moving pair
 (Theorems~\ref{thm:fixed-localization}
 and~\ref{thm:moving-localization}).
\end{enumerate}

\subsection{Distributions with discrete support}
\label{subsec:rig-conventions}

In this section $T$ denotes a distribution on $\RR^2$. Multi-indices
refer to the momentum coordinates,
$\partial^\alpha=\partial_P^{\alpha_1}\partial_Q^{\alpha_2}$, and
$\mathcal A_q=\{\alpha\in\NN^2:|\alpha|\leq q\}$.

\paragraph{Local coefficients.}
Let $S\subset\RR^2$ be closed and discrete. A distribution $T$ with
$\supp T\subset S$ has a unique expansion
\begin{equation}\label{eq:rig-local-form}
 T=\sum_{x\in S}\sum_{\alpha}c_\alpha(x)\,\partial^\alpha\delta_x ,
\end{equation}
in which, for each $x$, only finitely many $c_\alpha(x)$ are nonzero;
conversely every such array of complex numbers defines a distribution,
the sum being locally finite. We say that $T$ has \emph{local order at
most $q$} if $c_\alpha(x)=0$ whenever $|\alpha|>q$. The numbers
$c_\alpha(0)$ with $|\alpha|\geq1$ are the \emph{positive-order origin
coefficients} of $T$. A tempered distribution has finite order, so a
tempered distribution with support in $S$ has local order at most $q$
for some $q<\infty$. A function \emph{vanishes to order at least
$q+1$} at a point if all its derivatives of order at most $q$ vanish
there.

\begin{lemma}\label{lem:rig-local-pairing}
Let $T$ be a tempered distribution with support in a closed discrete
set $S$ and of local order at most $q$, with coefficients as in
\eqref{eq:rig-local-form}.
\begin{enumerate}[(i)]
\item Let $F_0\subset S$ be finite and let $\psi$ be a Schwartz
 function that vanishes to order at least $q+1$ at every point of
 $S\setminus F_0$. Then
 \[
 \langle T,\psi\rangle=\sum_{x\in F_0}\sum_{|\alpha|\leq q}
 (-1)^{|\alpha|}c_\alpha(x)\,\partial^\alpha\psi(x).
 \]
\item Let $\omega$ be a smooth function, all of whose derivatives are
 polynomially bounded, that vanishes to order at least $q+1$ at every
 point of $S$. Then $\omega T=0$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i) Let $\chi\in C_c^\infty(\RR^2)$ equal one near the origin and put
$\chi_\varrho(X)=\chi(X/\varrho)$. Then $\chi_\varrho\psi\to\psi$ in the
Schwartz space as $\varrho\to\infty$, so
$\langle T,\psi\rangle=\lim_\varrho\langle T,\chi_\varrho\psi\rangle$.
The support of $\chi_\varrho\psi$ meets finitely many points of $S$, and
by \eqref{eq:rig-local-form}
\[
 \langle T,\chi_\varrho\psi\rangle=\sum_{x\in S}\sum_{|\alpha|\leq q}
 (-1)^{|\alpha|}c_\alpha(x)\,\partial^\alpha(\chi_\varrho\psi)(x).
\]
By the Leibniz rule the derivatives of order at most $q$ of
$\chi_\varrho\psi$ vanish at every $x\in S\setminus F_0$, and for large
$\varrho$ they agree with those of $\psi$ at the points of $F_0$.

(ii) For a Schwartz function $f$, the function $\omega f$ is a Schwartz
function vanishing to order at least $q+1$ on $S$, so
$\langle\omega T,f\rangle=\langle T,\omega f\rangle=0$ by (i) with
$F_0=\emptyset$.
\end{proof}

\paragraph{Translations, characters, reflections.}
For $w\in\RR^2$ let $(\tau_wf)(X)=f(X+w)$ and
$\chi_w(X)=e^{2\pi iB(w,X)}$; on distributions,
$\langle\tau_wT,f\rangle=\langle T,\tau_{-w}f\rangle$, so that
$\tau_w\delta_x=\delta_{x-w}$. From \eqref{eq:S-transform},
$\mathbb S(\chi_wf)=\tau_w\mathbb Sf$ and
$\mathbb S(\tau_{-w}f)=\chi_w\mathbb Sf$ for Schwartz functions, and by
duality $\mathbb S(\chi_wT)=\tau_w\mathbb ST$ for tempered $T$.
Hence, for vectors $w_i$ and integers $m_i\geq0$,
\begin{equation}\label{eq:rig-intertwine}
 \prod_i(\tau_{w_i}-I)^{m_i}
 =\mathbb S\circ\Bigl(\prod_i(\chi_{w_i}-1)^{m_i}\Bigr)\circ\mathbb S^{-1}
\end{equation}
on tempered distributions, the middle factor acting by multiplication.
Let $\check T$ denote the image of $T$ under $X\mapsto-X$. Since
$\mathbb S^2f=\check f$, the distribution $\mathbb S^{-1}T$ is the
reflection of $\mathbb ST$; it has the same local orders as
$\mathbb ST$, and its support is the reflection of that of
$\mathbb ST$. If $T$ is even, then $\mathbb S^{-1}T=\mathbb ST$, and
$\langle T,f\rangle=\langle T,f^{\rm ev}\rangle$ for every test
function $f$.

\paragraph{Lattices, grids and combs.}
For a full-rank lattice $\Gamma\subset\RR^2$ the dual lattice with
respect to the Lorentzian pairing is
\[
 \Gamma^\vee=\{w\in\RR^2:B(w,\lambda)\in\ZZ\text{ for all }
 \lambda\in\Gamma\},\qquad(\Gamma^\vee)^\vee=\Gamma .
\]
If $w\in\Gamma^\vee$ then $\chi_w-1$ vanishes on $\Gamma$. We call
$\Gamma$ a \emph{grid} if $\Gamma^\vee\subset\Gamma$. The quotient
$\Gamma/\Gamma^\vee$ is then finite, and for
$\gamma\in\Gamma/\Gamma^\vee$ we define the \emph{coset comb} and the
space of comb measures
\[
 e_\gamma=\sum_{\lambda\in\gamma+\Gamma^\vee}\delta_\lambda,\qquad
 \mathcal E(\Gamma)=\operatorname{span}
 \{e_\gamma:\gamma\in\Gamma/\Gamma^\vee\}.
\]
Two families of grids occur. The \emph{rational momentum grid}
$(2k)^{-1}\ZZ^2$, in the coordinates $(P,Q)$ and for an integer
$k\geq1$, has dual $k\ZZ^2$. The \emph{hyperbolic grid}
\[
 d^{-1}\Lambda_r=\Bigl\{(u,v)=\Bigl(\frac{mr}d,\frac n{dr}\Bigr):
 m,n\in\ZZ\Bigr\},\qquad d\geq1\text{ an integer},\ r>0,
\]
has dual $d\Lambda_r$, which consists of the points with
$m,n\in d^2\ZZ$; its spin function is $s=mn/d^2$. For $d=1$ the grid
is the self-dual lattice $\Lambda_r$ and its only comb is $e_r$.

Poisson summation takes the following form. Let
$(\mathsf b_1,\mathsf b_2)$ be a basis of $\Gamma^\vee$ and
$(\mathsf b_1^*,\mathsf b_2^*)$ the basis of $\Gamma$ with
$B(\mathsf b_i,\mathsf b_k^*)=\one_{i=k}$. For a Schwartz function $f$
and $\gamma\in\Gamma$ put
$\phi(x)=f(x_1\mathsf b_1^*+x_2\mathsf b_2^*)\,
e^{2\pi iB(\gamma,x_1\mathsf b_1^*+x_2\mathsf b_2^*)}$, $x\in\RR^2$.
The substitution $X'=x_1\mathsf b_1^*+x_2\mathsf b_2^*$ in
\eqref{eq:S-transform} gives
$\mathbb Sf(\gamma+n_1\mathsf b_1+n_2\mathsf b_2)
=c_\Gamma\int\phi(x)e^{2\pi in\cdot x}\dd x$,
where $c_\Gamma$ is the absolute value of the determinant of
$(\mathsf b_1^*,\mathsf b_2^*)$ in light-cone coordinates, and the
classical Poisson formula
$\sum_{n\in\ZZ^2}\int\phi(x)e^{2\pi in\cdot x}\dd x
=\sum_{n\in\ZZ^2}\phi(n)$ yields
$\langle\mathbb Se_\gamma,f\rangle
=c_\Gamma\sum_{\mu\in\Gamma}e^{2\pi iB(\gamma,\mu)}f(\mu)$. The phase
depends only on the classes of $\gamma$ and $\mu$ modulo $\Gamma^\vee$,
so
\begin{equation}\label{eq:rig-poisson}
 \mathbb Se_\gamma=c_\Gamma\sum_{\delta\in\Gamma/\Gamma^\vee}
 e^{2\pi iB(\gamma,\delta)}\,e_\delta,\qquad
 c_\Gamma=[\Gamma:\Gamma^\vee]^{-1/2}.
\end{equation}
The value of $c_\Gamma$ follows because the matrix of $B$ in light-cone
coordinates has determinant $-1$, so that the covolumes of $\Gamma$ and
$\Gamma^\vee$ are reciprocal. For $\Gamma=d^{-1}\Lambda_r$ one has
$c_\Gamma=d^{-2}$, and \eqref{eq:rig-poisson} with $d=1$ is the first
identity in \eqref{eq:circle-invariance}. In particular
$\mathcal E(\Gamma)$ is mapped into itself by $\mathbb S$, and by
$\mathbb S^{-1}$ because $\check e_\gamma=e_{-\gamma}$.

\paragraph{Polynomial differential operators.}
Let $\mathcal W$ be the algebra of linear differential operators on
$\RR^2$ with polynomial coefficients. It acts on distributions and
preserves temperedness. Every $\mathsf A\in\mathcal W$ has a unique
\emph{normal form} $\mathsf A=\sum_\alpha\partial^\alpha\circ p_\alpha$
with polynomials $p_\alpha$, almost all zero, and for a closed discrete
set $S$
\begin{equation}\label{eq:rig-normal-action}
 \mathsf A\sum_{\lambda\in S}\delta_\lambda
 =\sum_\alpha\sum_{\lambda\in S}p_\alpha(\lambda)\,
 \partial^\alpha\delta_\lambda .
\end{equation}
We use the generators
\begin{equation}\label{eq:rig-generators}
 \mathsf x_1=P,\qquad\mathsf x_2=Q,\qquad
 \mathsf y_1=\frac{\partial_P}{4\pi i},\qquad
 \mathsf y_2=-\frac{\partial_Q}{4\pi i},
\end{equation}
whose only nonzero commutators are
\begin{equation}\label{eq:rig-commutators}
 [\mathsf y_1,\mathsf x_1]=\frac1{4\pi i},\qquad
 [\mathsf y_2,\mathsf x_2]=-\frac1{4\pi i}.
\end{equation}
For $\mathsf A\in\mathcal W$ and an integer $\nu$ put
$\sigma(\mathsf A)=\mathbb S\mathsf A\mathbb S^{-1}$ and
$\kappa_\nu(\mathsf A)=\mathcal C^\nu\mathsf A\mathcal C^{-\nu}$.
Differentiating \eqref{eq:S-transform} under the integral sign and
integrating by parts gives, for Schwartz functions,
$\mathbb S(Pf)=(4\pi i)^{-1}\partial_P\mathbb Sf$,
$\mathbb S(\partial_Pf)=-4\pi iP\,\mathbb Sf$,
$\mathbb S(Qf)=-(4\pi i)^{-1}\partial_Q\mathbb Sf$ and
$\mathbb S(\partial_Qf)=4\pi iQ\,\mathbb Sf$; these identities extend to
tempered distributions by duality. Together with
$\mathcal C^\nu\partial_P\mathcal C^{-\nu}=\partial_P-4\pi i\nu P$ and
$\mathcal C^\nu\partial_Q\mathcal C^{-\nu}=\partial_Q+4\pi i\nu Q$ they
show that $\sigma$ and $\kappa_\nu$ are automorphisms of $\mathcal W$,
given on the generators by
\begin{equation}\label{eq:rig-sigma-kappa}
 \sigma:(\mathsf x_j,\mathsf y_j)\mapsto(\mathsf y_j,-\mathsf x_j),
 \qquad
 \kappa_\nu:(\mathsf x_j,\mathsf y_j)\mapsto
 (\mathsf x_j,\,\mathsf y_j-\nu\mathsf x_j),\qquad j=1,2 .
\end{equation}
The boost generator \eqref{eq:boost} is
$\mathcal K=-4\pi i\,(\mathsf x_1\mathsf y_2-\mathsf x_2\mathsf y_1)$.
Since $\mathsf x_1,\mathsf y_1$ commute with $\mathsf x_2,\mathsf y_2$,
the element $\mathsf x_1\mathsf y_2-\mathsf x_2\mathsf y_1$ is fixed by
both substitutions in \eqref{eq:rig-sigma-kappa}. Therefore
\begin{equation}\label{eq:rig-K-facts}
 \mathbb S\mathcal K=\mathcal K\mathbb S,\qquad
 \mathcal C\mathcal K=\mathcal K\mathcal C,\qquad
 \mathcal K\delta_0=0 ,
\end{equation}
the last identity because $P\partial_Q\delta_0=\partial_Q(P\delta_0)=0$
and likewise for $Q\partial_P\delta_0$. Thus
$p(\mathcal K)\delta_0=p(0)\delta_0$ for every polynomial $p$. Away
from the origin the opposite holds. For $x=(P_x,Q_x)$ the identity
$f\,\partial^\alpha\delta_x=\sum_{\beta\leq\alpha}(-1)^{|\alpha-\beta|}
\binom\alpha\beta(\partial^{\alpha-\beta}f)(x)\,\partial^\beta\delta_x$
shows that $\mathcal K\partial^\alpha\delta_x
=(Q_x\partial_P+P_x\partial_Q)\partial^\alpha\delta_x$ plus terms of
order at most $|\alpha|$, and by induction
\begin{equation}\label{eq:rig-K-nonzero}
 p(\mathcal K)\delta_x=\hat p\,(Q_x\partial_P+P_x\partial_Q)^j\delta_x
 +(\text{terms of order}<j)
\end{equation}
for a polynomial $p$ of degree $j$ with leading coefficient $\hat p$.
If $x\neq0$ this distribution is nonzero and has local order exactly
$j$. In particular the powers of $\mathcal K$ are linearly independent
in $\mathcal W$.

\subsection{Polynomial comb representations}
\label{subsec:rig-combs}

\begin{lemma}[Two difference equations modulo finite support]
\label{lem:difference-newton}
Let $c:\ZZ^2\to\CC$ be an arbitrary function and $m\geq1$ an integer.
Let $\mathsf D_1,\mathsf D_2$ be the forward differences in the two
coordinate directions,
$(\mathsf D_1c)(n_1,n_2)=c(n_1+1,n_2)-c(n_1,n_2)$ and similarly for
$\mathsf D_2$. If $\mathsf D_1^mc$ and $\mathsf D_2^mc$ have finite
support, then $c=p+c_0$, where $c_0$ has finite support and $p$ is the
restriction to $\ZZ^2$ of a polynomial of degree at most $m-1$ in each
variable separately. No growth condition on $c$ is assumed.
\end{lemma}
\begin{proof}
Identify a finitely supported function $f$ on $\ZZ^2$ with the Laurent
polynomial $\sum_nf(n)z_1^{n_1}z_2^{n_2}$. Under this identification
$\mathsf D_i$ becomes multiplication by $z_i^{-1}-1$. Put
$f_i=\mathsf D_i^mc$ and $\pi_i=(z_i^{-1}-1)^m$. Since $\mathsf D_1$
and $\mathsf D_2$ commute,
$\mathsf D_2^mf_1=\mathsf D_1^mf_2$, that is, $\pi_2f_1=\pi_1f_2$ in
the ring $\CC[z_1^{\pm1},z_2^{\pm1}]$. This ring is a unique
factorization domain in which $1-z_1$ and $1-z_2$ are non-associate
primes, so $\pi_1$ divides $f_1$: there is a Laurent polynomial $c_0$
with $f_1=\pi_1c_0$. Cancelling $\pi_1$ in
$\pi_2\pi_1c_0=\pi_1f_2$ gives $f_2=\pi_2c_0$. Regarding $c_0$ as a
finitely supported function, we obtain
$\mathsf D_1^m(c-c_0)=\mathsf D_2^m(c-c_0)=0$.

Put $p=c-c_0$. For fixed $n_2$, the function $n_1\mapsto p(n_1,n_2)$
on $\ZZ$ has vanishing $m$-th difference. Such a function is
determined, in both directions, by its values at $0,\dots,m-1$, and the
Newton polynomial
$\sum_{k<m}\binom{n_1}k(\mathsf D_1^kp)(0,n_2)$ has the same values
and also vanishing $m$-th difference; hence
$p(n_1,n_2)=\sum_{k<m}\binom{n_1}k(\mathsf D_1^kp)(0,n_2)$ for all
$n_1\in\ZZ$. Each coefficient $n_2\mapsto(\mathsf D_1^kp)(0,n_2)$ has
vanishing $m$-th difference in $n_2$ and is, by the same argument, a
polynomial of degree at most $m-1$ in $n_2$.
\end{proof}

The corresponding statement in one variable is false: the indicator
function of $\NN$ has a finitely supported first difference. Two
independent directions are essential.

\begin{lemma}[Differences along one class]
\label{lem:rig-transverse-differences}
Let $(\Gamma_i)_{i\in\mathsf I}$ be finitely many full-rank lattices in
$\RR^2$ with $\Gamma_i\cap\Gamma_{i'}=\{0\}$ for $i\neq i'$, let
$S=\bigcup_i\Gamma_i$, let $m\geq1$ be an integer and
$j\in\mathsf I$.
\begin{enumerate}[(i)]
\item For any subgroups $G_i\subset\Gamma_i$ of finite index,
 $i\neq j$, there are vectors $w_i\in G_i$ such that
 \begin{equation}\label{eq:rig-avoid}
 \sum_{i\neq j}k_iw_i\notin\Gamma_j\qquad
 \text{whenever }0\leq k_i\leq m\text{ and not all }k_i\text{ vanish}.
 \end{equation}
 In particular $w_i\neq0$.
\item Let $w_i\in\Gamma_i$, $i\neq j$, satisfy \eqref{eq:rig-avoid},
 let $w\in\Gamma_j$, and put
 $\mathsf D=(\tau_w-I)^m\prod_{i\neq j}(\tau_{w_i}-I)^m$. There is a
 finite set $F_0\subset\Gamma_j$ with the following property. If $T$ is
 any distribution of the form \eqref{eq:rig-local-form} on $S$, with
 arbitrary coefficients, and $\mathsf DT=0$, then for every $\alpha$
 \begin{equation}\label{eq:rig-class-difference}
 \sum_{k=0}^m(-1)^{m-k}\binom mk\,c_\alpha(y+kw)=0
 \qquad\text{for all }y\in\Gamma_j\setminus F_0 .
 \end{equation}
\end{enumerate}
If $\mathsf I=\{j\}$, then $\mathsf D=(\tau_w-I)^m$ and
$F_0=\emptyset$.
\end{lemma}
\begin{proof}
(i) We choose the vectors one at a time. Suppose that $w_i$ has been
chosen for $i$ in a subset $\mathsf I'\subset\mathsf I\setminus\{j\}$,
in such a way that \eqref{eq:rig-avoid} holds for all tuples with
$k_i=0$ for $i\notin\mathsf I'$, and let
$i_0\in\mathsf I\setminus(\mathsf I'\cup\{j\})$. The additional
conditions for $\mathsf I'\cup\{i_0\}$ are those with $k_{i_0}\geq1$
and $k_i=0$ for $i\notin\mathsf I'\cup\{i_0\}$. Each of these finitely
many conditions is violated by at most one vector of $G_{i_0}$: if $w$
and $w'$ both violate it, then
$k_{i_0}(w-w')\in\Gamma_j\cap\Gamma_{i_0}=\{0\}$. Since $G_{i_0}$ is
infinite, an admissible $w_{i_0}$ exists. The condition with $k_i=1$
and all other coefficients zero says $w_i\notin\Gamma_j$, so
$w_i\neq0$.

(ii) Extend $c_\alpha$ by zero outside $S$. Expanding the product,
$\mathsf D=\sum_{\mathbf k}\varkappa_{\mathbf k}\,
\tau_{\mathsf w_{\mathbf k}}$, where
$\mathbf k=(k_i)_{i\in\mathsf I}$ runs over
$\{0,\dots,m\}^{\mathsf I}$,
$\varkappa_{\mathbf k}=\prod_i(-1)^{m-k_i}\binom m{k_i}$ and
$\mathsf w_{\mathbf k}=k_jw+\sum_{i\neq j}k_iw_i$. Since
$\tau_{w'}\delta_x=\delta_{x-w'}$, the coefficient of
$\partial^\alpha\delta_y$ in $\mathsf DT$ is
$\sum_{\mathbf k}\varkappa_{\mathbf k}\,
c_\alpha(y+\mathsf w_{\mathbf k})$. Let $y\in\Gamma_j$. If $k_i=0$ for
all $i\neq j$, then $\mathsf w_{\mathbf k}=k_jw$, and these terms add
up to $(-1)^{m(|\mathsf I|-1)}$ times the left side of
\eqref{eq:rig-class-difference}. For every other $\mathbf k$ we have
$y+\mathsf w_{\mathbf k}\notin\Gamma_j$ by \eqref{eq:rig-avoid}, so
$c_\alpha(y+\mathsf w_{\mathbf k})$ can be nonzero only if
$y+\mathsf w_{\mathbf k}\in\Gamma_i$ for some $i\neq j$. For fixed
$\mathbf k$ and $i$ at most one $y\in\Gamma_j$ has this property,
because the difference of two such points lies in
$\Gamma_j\cap\Gamma_i$. Let $F_0$ be the finite set of all
$y\in\Gamma_j$ obtained in this way. For $y\in\Gamma_j\setminus F_0$
the vanishing of the coefficient of $\partial^\alpha\delta_y$ in
$\mathsf DT$ is \eqref{eq:rig-class-difference}.
\end{proof}

\begin{lemma}[Polynomial combs for transverse grids]
\label{lem:transverse-combs}
Let $(\Gamma_i)_{i\in\mathsf I}$ be finitely many grids in $\RR^2$ with
$\Gamma_i\cap\Gamma_{i'}=\{0\}$ for $i\neq i'$, and let
$e_{i,\gamma}$, $\gamma\in\Gamma_i/\Gamma_i^\vee$, be the coset combs
of $\Gamma_i$.
\begin{enumerate}[(i)]
\item \emph{Faithfulness.} If operators
 $\mathsf A_{i,\gamma}\in\mathcal W$ satisfy
 $\sum_{i,\gamma}\mathsf A_{i,\gamma}e_{i,\gamma}=0$, then
 $\mathsf A_{i,\gamma}=0$ for all $i$ and $\gamma$.
\item \emph{Representation.} If $T$ is a tempered distribution such
 that $T$ and $\mathbb ST$ are supported in $\bigcup_i\Gamma_i$, then
 there are operators $\mathsf A_{i,\gamma}\in\mathcal W$, unique by
 the first part, with
 \begin{equation}\label{eq:rig-comb-form}
 T=\sum_{i\in\mathsf I}\ \sum_{\gamma\in\Gamma_i/\Gamma_i^\vee}
 \mathsf A_{i,\gamma}\,e_{i,\gamma}.
 \end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
(i) Write $\mathsf A_{i,\gamma}=\sum_\alpha\partial^\alpha\circ
p_{i,\gamma,\alpha}$ in normal form. A point $\lambda\neq0$ of
$\bigcup_i\Gamma_i$ lies in exactly one $\Gamma_i$ and in exactly one
coset $\gamma+\Gamma_i^\vee$, and by \eqref{eq:rig-normal-action} the
coefficient of $\partial^\alpha\delta_\lambda$ in the sum is
$p_{i,\gamma,\alpha}(\lambda)$. Hence each $p_{i,\gamma,\alpha}$
vanishes on the coset $\gamma+\Gamma_i^\vee$ with at most the origin
removed. In lattice coordinates $(n_1,n_2)$ on this coset, the
polynomial vanishes identically on every line $n_2=\mathrm{const}$ that
does not contain the removed point, hence on infinitely many parallel
lines, hence identically.

(ii) Let $S=\bigcup_i\Gamma_i$ and let $m$ be a common bound for the
local orders of $T$ and $\mathbb ST$; write $T$ as in
\eqref{eq:rig-local-form}. Fix $j\in\mathsf I$ and a basis
$(\mathsf b_1,\mathsf b_2)$ of $\Gamma_j^\vee$. For $\iota\in\{1,2\}$
choose vectors $w_i^{(\iota)}\in\Gamma_i^\vee$, $i\neq j$, as in
Lemma~\ref{lem:rig-transverse-differences}(i) with $m+1$ in place of
$m$, and put
\[
 \mathsf D^{(\iota)}=(\tau_{\mathsf b_\iota}-I)^{m+1}\prod_{i\neq j}
 (\tau_{w_i^{(\iota)}}-I)^{m+1},\qquad
 \omega^{(\iota)}=(\chi_{\mathsf b_\iota}-1)^{m+1}\prod_{i\neq j}
 (\chi_{w_i^{(\iota)}}-1)^{m+1}.
\]
Every point of $S$ lies in some $\Gamma_i$, on which the $i$-th factor
of $\omega^{(\iota)}$ vanishes to order $m+1$, because the vector of
that factor lies in $\Gamma_i^\vee$. The distribution
$\mathbb S^{-1}T$ is supported in $S$ and has local order at most $m$,
so $\omega^{(\iota)}\mathbb S^{-1}T=0$ by
Lemma~\ref{lem:rig-local-pairing}(ii), and
$\mathsf D^{(\iota)}T=0$ by \eqref{eq:rig-intertwine}.
Lemma~\ref{lem:rig-transverse-differences}(ii) shows that, for
every $\alpha$ and every coset $\gamma+\Gamma_j^\vee$, the function
$(n_1,n_2)\mapsto c_\alpha(\gamma+n_1\mathsf b_1+n_2\mathsf b_2)$ on
$\ZZ^2$ has finitely supported differences of order $m+1$ in both
directions. By Lemma~\ref{lem:difference-newton} it differs from a
polynomial in $(n_1,n_2)$ at finitely many points only. Thus there are
polynomials $p_{j,\gamma,\alpha}$ on $\RR^2$, zero for $|\alpha|>m$,
with $c_\alpha(\lambda)=p_{j,\gamma,\alpha}(\lambda)$ for all but
finitely many $\lambda\in\gamma+\Gamma_j^\vee$.

Carry this out for every $j$, put
$\mathsf A_{j,\gamma}=\sum_\alpha\partial^\alpha\circ
p_{j,\gamma,\alpha}$ and
$F=T-\sum_{j,\gamma}\mathsf A_{j,\gamma}e_{j,\gamma}$. By
\eqref{eq:rig-normal-action} and the previous paragraph, $F$ has finite
support. Each $\mathsf A_{j,\gamma}e_{j,\gamma}$ is tempered, and
$\mathbb S(\mathsf A_{j,\gamma}e_{j,\gamma})
=\sigma(\mathsf A_{j,\gamma})\,\mathbb Se_{j,\gamma}$ is supported in
$\Gamma_j$ by \eqref{eq:rig-poisson}. Hence $\mathbb SF$ is supported
in the closed discrete set $S$. But $F$ is a finite combination of
derivatives of point masses, so $\mathbb SF$ is a real-analytic
function, a finite sum of polynomials times characters. A continuous
function supported in a discrete set vanishes; hence $\mathbb SF=0$
and $F=0$.
\end{proof}

\begin{lemma}[Operators invariant under two shears]
\label{lem:boost-invariants}
Let $\mathsf A\in\mathcal W$ and let $\nu$ be a nonzero integer. If
$\mathsf A$ is invariant under conjugation by $\mathcal C^\nu$ and by
$\mathbb S\mathcal C^\nu\mathbb S^{-1}$, that is,
\[
 \kappa_\nu(\mathsf A)=\mathsf A\qquad\text{and}\qquad
 \sigma\kappa_\nu\sigma^{-1}(\mathsf A)=\mathsf A ,
\]
then $\mathsf A=p(\mathcal K)$ for a polynomial $p$.
\end{lemma}
\begin{proof}
For a matrix
$\mathsf g=\bigl(\begin{smallmatrix}\mathsf g_{11}&\mathsf g_{12}\\
\mathsf g_{21}&\mathsf g_{22}\end{smallmatrix}\bigr)
\in\mathrm{SL}_2(\CC)$ the substitution
\[
 \Phi_{\mathsf g}:\ (\mathsf x_j,\mathsf y_j)\mapsto
 (\mathsf g_{11}\mathsf x_j+\mathsf g_{12}\mathsf y_j,\
 \mathsf g_{21}\mathsf x_j+\mathsf g_{22}\mathsf y_j),\qquad j=1,2,
\]
preserves the relations \eqref{eq:rig-commutators}, because
$[\mathsf g_{21}\mathsf x_j+\mathsf g_{22}\mathsf y_j,\,
\mathsf g_{11}\mathsf x_j+\mathsf g_{12}\mathsf y_j]
=\det\mathsf g\,[\mathsf y_j,\mathsf x_j]$ and the generators with
different indices commute. Hence $\Phi_{\mathsf g}$ is an automorphism
of $\mathcal W$, and
$\Phi_{\mathsf g}\Phi_{\mathsf g'}=\Phi_{\mathsf g'\mathsf g}$. By
\eqref{eq:rig-sigma-kappa},
\[
 \kappa_\nu=\Phi_{\mathsf g}\ \text{with}\
 \mathsf g=\begin{pmatrix}1&0\\-\nu&1\end{pmatrix},\qquad
 \sigma\kappa_\nu\sigma^{-1}=\Phi_{\mathsf g}\ \text{with}\
 \mathsf g=\begin{pmatrix}1&\nu\\0&1\end{pmatrix};
\]
for the second, $\sigma^{-1}$ maps $(\mathsf x_j,\mathsf y_j)$ to
$(-\mathsf y_j,\mathsf x_j)$, so
$\sigma\kappa_\nu\sigma^{-1}(\mathsf x_j)
=\sigma(-\mathsf y_j+\nu\mathsf x_j)=\mathsf x_j+\nu\mathsf y_j$ and
$\sigma\kappa_\nu\sigma^{-1}(\mathsf y_j)=\sigma(\mathsf x_j)
=\mathsf y_j$.

Let $\mathcal W_n\subset\mathcal W$ be the finite-dimensional span of
all products of at most $n$ generators; every $\Phi_{\mathsf g}$
preserves $\mathcal W_n$. Choose $n$ with $\mathsf A\in\mathcal W_n$.
For $z\in\CC$ let $\alpha_z=\Phi_{\mathsf g}$ with
$\mathsf g=\bigl(\begin{smallmatrix}1&0\\z&1\end{smallmatrix}\bigr)$.
Then $\alpha_z\alpha_{z'}=\alpha_{z+z'}$, and
$z\mapsto\alpha_z(\mathsf A)$ is a polynomial map from $\CC$ to
$\mathcal W_n$. By hypothesis $\alpha_{-\nu}(\mathsf A)=\mathsf A$,
hence $\alpha_{k\nu}(\mathsf A)=\mathsf A$ for all $k\in\ZZ$, and the
polynomial $\alpha_z(\mathsf A)-\mathsf A$ has infinitely many zeros.
Thus $\alpha_z(\mathsf A)=\mathsf A$ for all $z\in\CC$. Likewise, with
$\beta_z=\Phi_{\mathsf g}$ for
$\mathsf g=\bigl(\begin{smallmatrix}1&z\\0&1\end{smallmatrix}\bigr)$,
the hypothesis $\beta_\nu(\mathsf A)=\mathsf A$ and the polynomial
dependence on $z$ give $\beta_z(\mathsf A)=\mathsf A$ for all
$z\in\CC$. The set of
$\mathsf g$ with $\Phi_{\mathsf g}(\mathsf A)=\mathsf A$ is a subgroup
of $\mathrm{SL}_2(\CC)$ that contains all elementary matrices, which
generate $\mathrm{SL}_2(\CC)$. Therefore
$\Phi_{\mathsf g}(\mathsf A)=\mathsf A$ for every
$\mathsf g\in\mathrm{SL}_2(\CC)$.

We conclude by induction on $n$; the case $n=0$ is trivial since
$\mathcal W_0=\CC$. By \eqref{eq:rig-commutators} the associated graded
algebra of the filtration $(\mathcal W_n)$ is the polynomial ring
$\CC[\mathsf x_1,\mathsf y_1,\mathsf x_2,\mathsf y_2]$, and
$\Phi_{\mathsf g}$ acts on it by the same linear substitution. Let
$\mathsf a^{\rm top}$ be the class of $\mathsf A$ in
$\mathcal W_n/\mathcal W_{n-1}$, a homogeneous polynomial of degree $n$
in the two vectors $w_1=(\mathsf x_1,\mathsf y_1)$ and
$w_2=(\mathsf x_2,\mathsf y_2)$ with
$\mathsf a^{\rm top}(\mathsf gw_1,\mathsf gw_2)
=\mathsf a^{\rm top}(w_1,w_2)$ for all
$\mathsf g\in\mathrm{SL}_2(\CC)$. If the determinant
$\mathsf k=\det(w_1,w_2)=\mathsf x_1\mathsf y_2-\mathsf x_2\mathsf y_1$
is nonzero, the matrix $\mathsf g$ with columns $w_1/\mathsf k$ and
$w_2$ has determinant one and maps $(\mathsf k,0)$ to $w_1$ and
$(0,1)$ to $w_2$; hence
$\mathsf a^{\rm top}(w_1,w_2)
=\mathsf a^{\rm top}\bigl((\mathsf k,0),(0,1)\bigr)$ is a polynomial
in $\mathsf k$ on the dense set $\{\mathsf k\neq0\}$, and therefore
everywhere. By homogeneity
$\mathsf a^{\rm top}=\vartheta\,
(\mathsf x_1\mathsf y_2-\mathsf x_2\mathsf y_1)^{n/2}$ for a constant
$\vartheta$, with $\vartheta=0$ if $n$ is odd. The element
$\mathsf x_1\mathsf y_2-\mathsf x_2\mathsf y_1=-\mathcal K/(4\pi i)$
of $\mathcal W$ is fixed by every $\Phi_{\mathsf g}$, since
$\mathsf x_1,\mathsf y_1$ commute with $\mathsf x_2,\mathsf y_2$ and
$\det\mathsf g=1$. Consequently
$\mathsf A-\vartheta\bigl(-\mathcal K/(4\pi i)\bigr)^{n/2}$ lies in
$\mathcal W_{n-1}$ and is fixed by every $\Phi_{\mathsf g}$. By the
induction hypothesis it is a polynomial in $\mathcal K$, and so is
$\mathsf A$.
\end{proof}

\begin{lemma}[Combs on one grid]\label{lem:grid-combs}
Let $\Gamma$ be a grid.
\begin{enumerate}[(i)]
\item If $T$ is a tempered distribution such that $T$ and
 $\mathbb ST$ are supported in $\Gamma$, then
 $T=\sum_{\gamma\in\Gamma/\Gamma^\vee}\mathsf A_\gamma e_\gamma$ with
 uniquely determined operators $\mathsf A_\gamma\in\mathcal W$.
\item There is an integer $\nu\geq1$ such that
 $\nu s(\lambda)\in\ZZ$ for all $\lambda\in\Gamma$, where
 $s=P^2-Q^2$; one may take $\nu=4k^2$ for $\Gamma=(2k)^{-1}\ZZ^2$ and
 $\nu=d^2$ for $\Gamma=d^{-1}\Lambda_r$. For every such $\nu$, both
 $\mathcal C^\nu$ and $\mathbb S\mathcal C^\nu\mathbb S^{-1}$ fix every
 element of $\mathcal E(\Gamma)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (i) is Lemma~\ref{lem:transverse-combs} for a single grid. For
(ii), let $n_0$ be the exponent of the finite group
$\Gamma/\Gamma^\vee$. For $\lambda\in\Gamma$ we have
$n_0\lambda\in\Gamma^\vee$, so
$2n_0s(\lambda)=B(n_0\lambda,\lambda)\in\ZZ$, and $\nu=2n_0$ has the
required property; the two special values are read off from
$s=P^2-Q^2$ and $s=mn/d^2$. If $\nu s$ is integral on $\Gamma$, then
$\mathcal C^\nu$ fixes every measure carried by $\Gamma$, in particular
every element of $\mathcal E(\Gamma)$. Since $\mathbb S^{-1}$ maps
$\mathcal E(\Gamma)$ into itself by \eqref{eq:rig-poisson},
$\mathbb S\mathcal C^\nu\mathbb S^{-1}e=\mathbb S\mathbb S^{-1}e=e$
for $e\in\mathcal E(\Gamma)$.
\end{proof}

\begin{theorem}[Rigidity on a grid]\label{thm:grid-rigidity}
Let $\Gamma$ be a grid, for instance a rational momentum grid
$(2k)^{-1}\ZZ^2$ or a hyperbolic grid $d^{-1}\Lambda_r$, and let $T$ be
a tempered distribution with $\supp T\subset\Gamma$ and
\[
 \mathbb ST=T,\qquad\mathcal CT=T .
\]
Then there are unique polynomials $p_\gamma$,
$\gamma\in\Gamma/\Gamma^\vee$, such that
\begin{equation}\label{eq:rig-grid-form}
 T=\sum_{\gamma\in\Gamma/\Gamma^\vee}p_\gamma(\mathcal K)\,e_\gamma .
\end{equation}
In particular the local expansion of $T$ at the origin is
$p_0(0)\,\delta_0$: all positive-order origin coefficients of $T$
vanish.
\end{theorem}
\begin{proof}
By Lemma~\ref{lem:grid-combs}(i),
$T=\sum_\gamma\mathsf A_\gamma e_\gamma$. Let $\nu$ be as in
Lemma~\ref{lem:grid-combs}(ii). Since $\mathcal C^\nu T=T$ and
$\mathcal C^\nu e_\gamma=e_\gamma$,
\[
 \sum_\gamma\mathsf A_\gamma e_\gamma=\mathcal C^\nu T
 =\sum_\gamma\kappa_\nu(\mathsf A_\gamma)\,\mathcal C^\nu e_\gamma
 =\sum_\gamma\kappa_\nu(\mathsf A_\gamma)\,e_\gamma ,
\]
and the uniqueness in Lemma~\ref{lem:grid-combs}(i) gives
$\kappa_\nu(\mathsf A_\gamma)=\mathsf A_\gamma$. The operator
$\mathbb S\mathcal C^\nu\mathbb S^{-1}$ also fixes $T$ and every
$e_\gamma$, and the same computation gives
$\sigma\kappa_\nu\sigma^{-1}(\mathsf A_\gamma)=\mathsf A_\gamma$. By
Lemma~\ref{lem:boost-invariants},
$\mathsf A_\gamma=p_\gamma(\mathcal K)$. The polynomials are unique by
the uniqueness of the $\mathsf A_\gamma$ and the linear independence of
the powers of $\mathcal K$. Only the coset $\gamma=0$ contains the
origin, near which $e_0=\delta_0$, and
$p_0(\mathcal K)\delta_0=p_0(0)\delta_0$ by \eqref{eq:rig-K-facts}.
\end{proof}

Neither positivity nor integrality of the coefficients of $T$, nor any
bound on its local order, enters Theorem~\ref{thm:grid-rigidity}.

\subsection{Finite unions of circle lattices}
\label{subsec:rig-circles}

Two radii $r,r'>0$ are \emph{commensurable} if $r/r'\in\QQ$. If they
are not, then $\Lambda_r\cap\Lambda_{r'}=\{0\}$, and the same holds
for $d^{-1}\Lambda_r$ and $d'^{-1}\Lambda_{r'}$: at a common point the
light-cone coordinate $u$ lies in $\QQ r\cap\QQ r'=\{0\}$ and $v$ in
$\QQ r^{-1}\cap\QQ r'^{-1}=\{0\}$.

\begin{definition}\label{def:rig-transverse-family}
A \emph{transverse family of hyperbolic grids} is a finite family
$\Gamma_i=d_i^{-1}\Lambda_{r_i}$, $i\in\mathsf I$, with integers
$d_i\geq1$ and pairwise incommensurable radii $r_i>0$. Its
\emph{support} is $S=\bigcup_i\Gamma_i$, its \emph{phase exponent} is
$\nu=\operatorname{lcm}\{d_i^2:i\in\mathsf I\}$, and its combs are the
coset combs $e_{i,\gamma}$, $\gamma\in\Gamma_i/\Gamma_i^\vee$. The
family is \emph{adapted} to a finite set $\mathcal R\subset(0,\infty)$
if $\bigcup_{r\in\mathcal R}\Lambda_r\subset S$.
\end{definition}

For such a family,
\begin{equation}\label{eq:rig-classes}
 \Gamma_i^\vee=d_i\Lambda_{r_i}\subset\Gamma_i,\qquad
 \Gamma_i\cap\Gamma_{i'}=\{0\}\ \ (i\neq i'),\qquad
 \nu s(\lambda)\in\ZZ\ \ (\lambda\in S),
\end{equation}
so that Lemmas~\ref{lem:rig-transverse-differences}
and~\ref{lem:transverse-combs} apply to it, and $\mathcal C^\nu$ fixes
every comb $e_{i,\gamma}$. Every finite set $\mathcal R$ has an adapted
family: choose one representative $r_i$ in each commensurability class
of $\mathcal R$; if $r=(k/k')r_i$ with positive integers $k,k'$, then
$\Lambda_r\subset(kk')^{-1}\Lambda_{r_i}$, and $d_i$ may be taken to be
a common multiple of the products $kk'$ occurring in the class.

\begin{theorem}[No origin jets on a finite union of circle lattices]
\label{thm:circle-union}
Let $(\Gamma_i)_{i\in\mathsf I}$ be a transverse family of hyperbolic
grids with support $S$; for instance a family adapted to a finite set
of radii $\mathcal R$, in which case
$S\supset\bigcup_{r\in\mathcal R}\Lambda_r$. Let $T$ be a tempered
distribution with $\supp T\subset S$ and
\[
 \mathbb ST=T,\qquad\mathcal CT=T .
\]
Then there are unique polynomials $p_{i,\gamma}$ with
\begin{equation}\label{eq:rig-union-form}
 T=\sum_{i\in\mathsf I}\ \sum_{\gamma\in\Gamma_i/\Gamma_i^\vee}
 p_{i,\gamma}(\mathcal K)\,e_{i,\gamma}.
\end{equation}
In particular, a tempered distribution that is invariant under
$\mathbb S$ and $\mathcal C$ and supported on a finite union of circle
lattices has a local expansion at the origin equal to a multiple of
$\delta_0$: all its positive-order origin coefficients vanish.
\end{theorem}
\begin{proof}
By \eqref{eq:rig-classes} and Lemma~\ref{lem:transverse-combs}(ii),
$T=\sum_{i,\gamma}\mathsf A_{i,\gamma}e_{i,\gamma}$. Let $\nu$ be the
phase exponent. By Lemma~\ref{lem:grid-combs}(ii), applied to each
$\Gamma_i$ with this common $\nu$, the operators $\mathcal C^\nu$ and
$\mathbb S\mathcal C^\nu\mathbb S^{-1}$ fix every $e_{i,\gamma}$; they
also fix $T$. As in the proof of Theorem~\ref{thm:grid-rigidity},
\[
 \sum_{i,\gamma}\mathsf A_{i,\gamma}e_{i,\gamma}
 =\sum_{i,\gamma}\kappa_\nu(\mathsf A_{i,\gamma})\,e_{i,\gamma}
 =\sum_{i,\gamma}\sigma\kappa_\nu\sigma^{-1}(\mathsf A_{i,\gamma})\,
 e_{i,\gamma},
\]
and the faithfulness statement of
Lemma~\ref{lem:transverse-combs}(i) shows that each
$\mathsf A_{i,\gamma}$ is fixed by $\kappa_\nu$ and by
$\sigma\kappa_\nu\sigma^{-1}$. Lemma~\ref{lem:boost-invariants} gives
$\mathsf A_{i,\gamma}=p_{i,\gamma}(\mathcal K)$. The polynomials are
unique by Lemma~\ref{lem:transverse-combs}(i) and the linear
independence of the powers of $\mathcal K$. Near the origin only the
combs $e_{i,0}$ contribute, each
equal to $\delta_0$ there, and
$p_{i,0}(\mathcal K)\delta_0=p_{i,0}(0)\delta_0$ by
\eqref{eq:rig-K-facts}.
\end{proof}

The finite-dimensional problem that remains is to find the invariant
elements of $\mathcal E(\Gamma)$ for a hyperbolic grid. The next lemma
is the special case, for such a grid, of the finite Fourier analysis in
\cite[Section~3.5]{EL}; we include the proof because the coefficients
are arbitrary complex numbers here.

\begin{lemma}[Invariant comb measures on a hyperbolic grid]
\label{lem:grid-invariants}
Let $\Gamma=d^{-1}\Lambda_r$ with an integer $d\geq1$ and $r>0$. The
elements of $\mathcal E(\Gamma)$ that are invariant under both
$\mathbb S$ and $\mathcal C$ are exactly the complex linear
combinations of the circle combs
\[
 e_{rk/d},\qquad k\ \text{a positive divisor of}\ d^2 ,
\]
and these combs are linearly independent.
\end{lemma}
\begin{proof}
Index the cosets of $\Gamma^\vee$ by
$(m,n)\in(\ZZ/d^2\ZZ)^2$, the coset $e_{m,n}$ containing the point
$(u,v)=(mr/d,\,n/(dr))$. On $\Gamma$ the spin is $mn/d^2$ and the
pairing is $B=(mn'+nm')/d^2$. By \eqref{eq:rig-poisson} and
$c_\Gamma=d^{-2}$,
\[
 \mathbb Se_{m,n}=\frac1{d^2}\sum_{m',n'}
 e^{2\pi i(mn'+nm')/d^2}\,e_{m',n'},\qquad
 \mathcal Ce_{m,n}=e^{2\pi imn/d^2}\,e_{m,n}.
\]
A measure $\sum\phi(m,n)e_{m,n}$ is therefore invariant if and only if
\begin{equation}\label{eq:rig-finite-invariance}
 \phi(m,n)=\frac1{d^2}\sum_{m',n'}\phi(m',n')\,
 e^{2\pi i(m'n+n'm)/d^2},
 \qquad\bigl(e^{2\pi imn/d^2}-1\bigr)\phi(m,n)=0 .
\end{equation}
Let $\widehat\phi(m,n)=\sum_{n'}\phi(m,n')e^{2\pi inn'/d^2}$ be the
Fourier transform in the second variable, which determines $\phi$. The
second condition in \eqref{eq:rig-finite-invariance} holds if and only
if $\widehat\phi(m,n+m)=\widehat\phi(m,n)$ for all $m,n$. The right
side of the first condition equals
$d^{-2}\sum_{m'}e^{2\pi im'n/d^2}\widehat\phi(m',m)$, whose transform
in $n$ at the point $(m,n)$ is $\widehat\phi(-n,m)$; so the first
condition holds if and only if
$\widehat\phi(m,n)=\widehat\phi(-n,m)$ for all $m,n$. The maps
$(m,n)\mapsto(m,n+m)$ and $(m,n)\mapsto(-n,m)$ are right
multiplication of the row vector $(m,n)$ by
$\bigl(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\bigr)$ and
$\bigl(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\bigr)$, which
generate $\mathrm{SL}_2(\ZZ)$. Hence $\phi$ is invariant if and only if
$\widehat\phi$ is constant on the orbits of $\mathrm{SL}_2(\ZZ)$ on
$(\ZZ/d^2\ZZ)^2$.

These orbits are the level sets of $k=\gcd(m,n,d^2)$. Indeed, the
ideal of $\ZZ$ generated by $m,n,d^2$ is unchanged by the action.
Conversely, let $\gcd(m,n,d^2)=k$ and write $(m,n)=k(m_1,n_1)$ with
integers satisfying $\gcd(m_1,n_1,d^2/k)=1$. We may take
$1\leq n_1\leq d^2/k$. By the Chinese remainder theorem there is an
integer $k'$ such that $m_2=m_1+k'd^2/k$ is not divisible by any prime
that divides $n_1$ but not $d^2/k$; the primes dividing both $n_1$ and
$d^2/k$ do not divide $m_1$, hence not $m_2$. Thus
$\gcd(m_2,n_1)=1$, and there is a matrix in $\mathrm{SL}_2(\ZZ)$ with
first row $(m_2,n_1)$. It maps $(k,0)$ to
$(km_2,kn_1)\equiv(m,n)$ modulo $d^2$, so $(m,n)$ lies in the orbit of
$(k,0)$.

Consequently the invariant $\widehat\phi$ are the functions of
$\gcd(m,n,d^2)$. A basis of this space is given by the functions
$\one_{k\mid m}\one_{k\mid n}$, $k\mid d^2$, since these are obtained
from the indicators of the level sets by a triangular change of basis
with respect to divisibility. Inverting the transform,
\[
 \widehat\phi=\one_{k\mid m}\one_{k\mid n}\quad\Longleftrightarrow\quad
 \phi(m,n)=\frac1{d^2}\sum_{n':\,k\mid n'}\one_{k\mid m}\,
 e^{-2\pi inn'/d^2}=\frac1k\,\one_{k\mid m}\,\one_{(d^2/k)\mid n}.
\]
The corresponding measure is $k^{-1}$ times the comb of the lattice of
points $(u,v)=(m_1kr/d,\,n_1d/(kr))$, $m_1,n_1\in\ZZ$, which is
$\Lambda_{rk/d}$.
\end{proof}

\begin{theorem}[Classification of the invariant distributions]
\label{thm:circle-jets}
Let $\mathcal R\subset(0,\infty)$ be finite, let $F\subset\Lambda_1$ be
finite, and let $T$ be a tempered distribution with
\[
 \mathbb ST=T,\qquad\mathcal CT=T,\qquad
 \supp T\subset\bigcup_{r\in\mathcal R}\Lambda_r\cup F .
\]
\begin{enumerate}[(i)]
\item There are polynomials $p_r$, $r\in\mathcal R$, such that
 \begin{equation}\label{eq:rig-circle-jets}
 T=\sum_{r\in\mathcal R}p_r(\mathcal K)\,e_r .
 \end{equation}
\item The polynomials $p_r$ are uniquely determined by $T$, and
 $\deg p_r\leq q$ if $T$ has local order at most $q$. Conversely,
 every distribution of the form \eqref{eq:rig-circle-jets} is tempered
 and invariant under $\mathbb S$ and $\mathcal C$.
\item Suppose that $\mathcal R$ is closed under $r\mapsto1/r$. Then
 $T$ is even if and only if $p_{1/r}(z)=p_r(-z)$ for all
 $r\in\mathcal R$, and if and only if $T$ is a finite linear
 combination of the radius derivatives
 \begin{equation}\label{eq:rig-radius-derivative}
 (r\partial_r)^j\rhoF_r
 =\frac{(-1)^j}2\,\mathcal K^j\bigl(e_r+(-1)^je_{1/r}\bigr),
 \qquad r\in\mathcal R,\ r\geq1,\ j\geq0 .
 \end{equation}
 Here $(r\partial_r)^j\rhoF_r$ denotes the tempered distribution
 $f\mapsto(t\partial_t)^j\langle\rhoF_t,f\rangle\big|_{t=r}$.
 For $r=1$ the derivatives of odd order vanish. The coefficients of
 the combination are unique if only even $j$ are used at $r=1$, and
 only orders $j\leq q$ occur if $T$ has local order at most $q$.
\end{enumerate}
\end{theorem}
\begin{proof}
\emph{Private points.} We first record a fact about circle lattices.
Let $\mathcal R'\subset(0,\infty)$ be finite, $r\in\mathcal R'$, and
let $\varpi$ be a prime with $\varpi>r/r'$ for all
$r'\in\mathcal R'$. The point $x_\varpi$ with light-cone coordinates
$(u,v)=(r,\varpi/r)$ lies in $\Lambda_r$. If it lies in
$\Lambda_{r'}$, say $(r,\varpi/r)=(m'r',n'/r')$, then $m',n'$ are
positive integers with $m'n'=\varpi$, so $m'=1$ and $r'=r$, or
$m'=\varpi$ and $r'=r/\varpi$. The second case is excluded for
$r'\in\mathcal R'$ by the choice of $\varpi$. Thus $x_\varpi$ belongs
to $\Lambda_{r'}$ for no $r'\in\mathcal R'\setminus\{r\}$; we call it
a \emph{private point} of $r$ relative to $\mathcal R'$. There are
infinitely many of them, and they are different from the origin.

\emph{Uniqueness and degrees.} Suppose \eqref{eq:rig-circle-jets}
holds, and let $x$ be a private point of $r$ relative to $\mathcal R$.
Near $x$ we have $T=p_r(\mathcal K)\delta_x$. By
\eqref{eq:rig-K-nonzero} this determines $p_r$, and its local order is
$\deg p_r$. This proves the first two assertions of (ii). The converse
follows from \eqref{eq:circle-invariance} and \eqref{eq:rig-K-facts}.

\emph{Existence.} Let $(\Gamma_i)_{i\in\mathsf I}$, with
$\Gamma_i=d_i^{-1}\Lambda_{r_i}$, be a transverse family adapted to
$\mathcal R\cup\{1\}$. Its support contains $\supp T$, and
Theorem~\ref{thm:circle-union} gives
$T=\sum_{i,\gamma}p_{i,\gamma}(\mathcal K)e_{i,\gamma}$. Collecting
powers of $\mathcal K$, write
\[
 T=\sum_{j\geq0}\mathcal K^j\eta_j,\qquad
 \eta_j=\sum_i\eta_{j,i},\quad
 \eta_{j,i}\in\mathcal E(\Gamma_i),
\]
with only finitely many $\eta_j\neq0$. Each $\mathcal E(\Gamma_i)$ is
mapped into itself by $\mathbb S$, by \eqref{eq:rig-poisson}, and by
$\mathcal C$, which multiplies $e_{i,\gamma}$ by the constant
$e^{2\pi is(\gamma)}$. Since $\mathcal K$ commutes with $\mathbb S$ and
$\mathcal C$,
\[
 0=\mathbb ST-T=\sum_j\mathcal K^j(\mathbb S\eta_j-\eta_j),\qquad
 0=\mathcal CT-T=\sum_j\mathcal K^j(\mathcal C\eta_j-\eta_j).
\]
Expanding $\mathbb S\eta_{j,i}-\eta_{j,i}$ in the combs $e_{i,\gamma}$,
the first identity has the form
$\sum_{i,\gamma}\bigl(\sum_j\vartheta_{i,\gamma,j}\mathcal K^j\bigr)
e_{i,\gamma}=0$ with complex numbers $\vartheta_{i,\gamma,j}$. By
Lemma~\ref{lem:transverse-combs}(i) each operator
$\sum_j\vartheta_{i,\gamma,j}\mathcal K^j$ vanishes, and by the linear
independence of the powers of $\mathcal K$ all
$\vartheta_{i,\gamma,j}$ vanish. Hence
$\mathbb S\eta_{j,i}=\eta_{j,i}$, and in the same way
$\mathcal C\eta_{j,i}=\eta_{j,i}$. By Lemma~\ref{lem:grid-invariants},
$\eta_{j,i}$ is a combination of circle combs $e_{r'}$ with radii
$r'=r_ik/d_i$, $k\mid d_i^2$. Therefore
\[
 T=\sum_{r'\in\mathcal R'}p_{r'}(\mathcal K)\,e_{r'}
\]
for a finite set $\mathcal R'\supset\mathcal R\cup\{1\}$ and
polynomials $p_{r'}$. Let $r'\in\mathcal R'\setminus\mathcal R$ and
suppose $p_{r'}\neq0$. Choose a private point $x$ of $r'$ relative to
$\mathcal R'$ that does not lie in the finite set $F$. Near $x$ we
have $T=p_{r'}(\mathcal K)\delta_x\neq0$ by \eqref{eq:rig-K-nonzero},
so $x\in\supp T$; but $x\notin F$, and $x\notin\Lambda_r$ for
$r\in\mathcal R$ because $\mathcal R\subset\mathcal R'\setminus\{r'\}$.
This contradicts the hypothesis on the support, so $p_{r'}=0$ for
$r'\notin\mathcal R$, which proves (i).

\emph{The even sector.} Let $\mathfrak r$ be the reflection
$Q\mapsto-Q$. It maps $e_r$ to $e_{1/r}$ and conjugates
$\mathcal K=P\partial_Q+Q\partial_P$ to $-\mathcal K$, so
$\mathfrak r_*\bigl(p(\mathcal K)e_r\bigr)=p(-\mathcal K)e_{1/r}$. The
reflection $X\mapsto-X$ fixes every $e_r$ and commutes with
$\mathcal K$, hence fixes $T$. Thus $T$ is even if and only if
$\mathfrak r_*T=T$, which by the uniqueness in (ii) means
$p_{1/r}(z)=p_r(-z)$ for all $r\in\mathcal R$. In that case $p_1$ is an
even polynomial and, writing $p_r(z)=\sum_jp_{r,j}z^j$,
\[
 T=\sum_{r\in\mathcal R,\,r>1}\ \sum_jp_{r,j}\,
 \mathcal K^j\bigl(e_r+(-1)^je_{1/r}\bigr)
 +\sum_{j\ \mathrm{even}}p_{1,j}\,\mathcal K^je_1 ,
\]
the last sum being present only if $1\in\mathcal R$. It remains to
prove \eqref{eq:rig-radius-derivative}; the uniqueness and degree
statements then follow from (ii), and each radius derivative is even
because $\rhoF_r$ is. For a Schwartz function $f$,
$\langle e_r,f\rangle=\sum_{m,n}f(mr,n/r)$ in light-cone coordinates.
Since $r\partial_r\bigl[f(mr,n/r)\bigr]
=\bigl((u\partial_u-v\partial_v)f\bigr)(mr,n/r)$ and $\mathcal Kf$ is
again a Schwartz function, the series may be differentiated termwise
any number of times, locally uniformly in $r$, and
\[
 (r\partial_r)^j\langle e_r,f\rangle
 =\langle e_r,\mathcal K^jf\rangle
 =\bigl\langle(-\mathcal K)^je_r,f\bigr\rangle ,
\]
because the transpose of $\mathcal K$ is $-\mathcal K$. Replacing $r$
by $1/r$ changes the sign of $r\partial_r$, so
$(r\partial_r)^je_{1/r}=\mathcal K^je_{1/r}$. Adding the two and
dividing by two gives \eqref{eq:rig-radius-derivative}, in view of
$\rhoF_r=\frac12(e_r+e_{1/r})$.
\end{proof}

Theorem~\ref{thm:circle-jets} applies in particular to distributions
supported on a rational grid. The link is the following elementary
fact, which is also used in Section~\ref{subsec:rig-localization}.

\begin{lemma}[Integral-spin points of a rational light-cone grid]
\label{lem:grid-circles}
Let $d\geq1$ be an integer. Every point with light-cone coordinates
$(u,v)=(m/d,n/d)$, $m,n\in\ZZ$, and integral spin $uv\in\ZZ$ lies in
\[
 \bigcup_{k\mid d^2}\Lambda_{k/d},
\]
the union over the positive divisors $k$ of $d^2$. The set
$\mathcal R_d=\{k/d:k\mid d^2\}$ of these radii contains $1$ and is
closed under $r\mapsto1/r$, and
$\Lambda_r\subset d^{-1}\Lambda_1$ for $r\in\mathcal R_d$.
\end{lemma}
\begin{proof}
Let $k=d^2/\gcd(n,d^2)$, with the convention $\gcd(0,d^2)=d^2$. Then
$d^2$ divides $nk$. Moreover $d^2\mid mn$ implies that $k$ divides
$m\cdot n/\gcd(n,d^2)$, and $k$ is coprime to $n/\gcd(n,d^2)$, so
$k\mid m$. Hence
\[
 u=\frac mk\cdot\frac kd,\qquad
 v=\frac{nk}{d^2}\Bigm/\frac kd
\]
with integers $m/k$ and $nk/d^2$, that is,
$(u,v)\in\Lambda_{k/d}$. The radius $k/d$ equals $1$ for $k=d$, and
$(k/d)^{-1}=(d^2/k)/d$. Finally a point $(m_1k/d,\,n_1d/k)$ of
$\Lambda_{k/d}$ has $v=n_1(d^2/k)/d\in d^{-1}\ZZ$.
\end{proof}

\begin{corollary}[Invariant distributions on a rational grid]
\label{cor:rig-grid-invariants}
Let $d\geq1$ and $q\geq0$ be integers, and let $\mathcal R_d$ be as in
Lemma~\ref{lem:grid-circles}. Let $T$ be a tempered distribution of
local order at most $q$, supported in the light-cone grid
$d^{-1}\Lambda_1=\{(u,v)\in d^{-1}\ZZ^2\}$, with $\mathbb ST=T$ and
$\mathcal CT=T$. Then
\[
 T=\sum_{r\in\mathcal R_d}p_r(\mathcal K)\,e_r
\]
with unique polynomials $p_r$ of degree at most $q$. If in addition
$T$ is even, then $T$ is a linear combination, with unique
coefficients, of the distributions
\begin{equation}\label{eq:rig-grid-basis}
 (r\partial_r)^j\rhoF_r,\qquad r\in\mathcal R_d,\ r\geq1,\
 0\leq j\leq q,\ \ j\ \text{even if}\ r=1 .
\end{equation}
Conversely, each distribution in \eqref{eq:rig-grid-basis} is tempered,
even, invariant under $\mathbb S$ and $\mathcal C$, supported in
$d^{-1}\Lambda_1$, and of local order at most $q$. Hence the space of
even tempered distributions of local order at most $q$ that are
supported in $d^{-1}\Lambda_1$ and invariant under $\mathbb S$ and
$\mathcal C$ has the basis \eqref{eq:rig-grid-basis}. The same
description holds, with $d=2k$, for the invariant distributions
supported in a rational momentum grid $(2k)^{-1}\ZZ^2$: they are the
elements of this space whose support lies in that grid.
\end{corollary}
\begin{proof}
The smooth function $e^{2\pi is}-1$ annihilates $T$, so $T$ vanishes
near every point at which this function is nonzero; that is, the spin
is integral on $\supp T$. By Lemma~\ref{lem:grid-circles},
$\supp T\subset\bigcup_{r\in\mathcal R_d}\Lambda_r$, and
Theorem~\ref{thm:circle-jets} applies with $\mathcal R=\mathcal R_d$
and $F=\emptyset$. The converse statements follow from
\eqref{eq:rig-radius-derivative}, Theorem~\ref{thm:circle-jets}(ii)
and the inclusion $\Lambda_r\subset d^{-1}\Lambda_1$. For the last
assertion note that $(2k)^{-1}\ZZ^2$ is contained in the light-cone
grid $(2k)^{-1}\Lambda_1$, because $u=P+Q$ and $v=P-Q$.
\end{proof}

\subsection{A finite certificate for the origin coefficients}
\label{subsec:rig-certificate}

Theorem~\ref{thm:circle-union} shows that $\mathbb S$- and
$\mathcal C$-invariance forbid positive-order origin coefficients. For
distributions of bounded local order, finitely many test functions
detect this obstruction, as we show in this subsection. The first step replaces
Lemma~\ref{lem:boost-invariants} by a statement that uses the phase
operator only.

\begin{lemma}[Operators commuting with the spin]
\label{lem:spin-centralizer}
Let $\mathsf A\in\mathcal W$ commute with multiplication by the spin
function $s=P^2-Q^2=uv$. Then
\[
 \mathsf A=\sum_{j=0}^m\mathsf q_j\,\mathcal K^j
\]
with polynomials $\mathsf q_j=\mathsf q_j(P,Q)$, and consequently
$\mathsf A\delta_0=\mathsf q_0(0)\,\delta_0$. The same conclusions hold
if $\kappa_\nu(\mathsf A)=\mathsf A$ for some nonzero integer $\nu$.
\end{lemma}
\begin{proof}
We use light-cone coordinates and induct on the differential order $m$
of $\mathsf A$; operators of order zero are polynomials. Let $m\geq1$,
write
$\mathsf A=\sum_{i+k\leq m}\mathsf a_{ik}(u,v)\,
\partial_u^i\partial_v^k$,
and let
$\mathsf a^{\rm top}=\sum_{i+k=m}\mathsf a_{ik}\,\xi_u^i\xi_v^k$ be its
principal symbol. Since $[\partial_u^i\partial_v^k,uv]
=iv\,\partial_u^{i-1}\partial_v^k+ku\,\partial_u^i\partial_v^{k-1}
+ik\,\partial_u^{i-1}\partial_v^{k-1}$, the commutator
$[\mathsf A,s]$ has order at most $m-1$, and its part of order $m-1$
has symbol
$(v\partial_{\xi_u}+u\partial_{\xi_v})\mathsf a^{\rm top}$. Hence
\[
 (v\partial_{\xi_u}+u\partial_{\xi_v})\mathsf a^{\rm top}=0 .
\]
Fix $(u,v)$ with $uv\neq0$. In the linear coordinates
$\ell=u\xi_u-v\xi_v$ and $\ell'=\xi_u/v$ on the $\xi$-plane the
derivation $v\partial_{\xi_u}+u\partial_{\xi_v}$ is
$\partial_{\ell'}$. So $\mathsf a^{\rm top}$, a homogeneous polynomial
of degree $m$ in $\xi$, depends on $\ell$ only:
$\mathsf a^{\rm top}=\mathsf q\,(u\xi_u-v\xi_v)^m$ with
$\mathsf q=\mathsf q(u,v)$. Comparing the coefficients of $\xi_u^m$
and $\xi_v^m$ gives $\mathsf a_{m0}=\mathsf q\,u^m$ and
$\mathsf a_{0m}=\mathsf q\,(-v)^m$, hence the polynomial identity
$(-v)^m\mathsf a_{m0}=u^m\mathsf a_{0m}$. As $u^m$ and $v^m$ are
coprime, $u^m$ divides $\mathsf a_{m0}$, so
$\mathsf q=\mathsf a_{m0}/u^m$ is a polynomial, and
$\mathsf a^{\rm top}=\mathsf q\,(u\xi_u-v\xi_v)^m$ holds identically.

The operator $\mathsf q\,\mathcal K^m$, with
$\mathcal K=u\partial_u-v\partial_v$, has principal symbol
$\mathsf a^{\rm top}$, and it commutes with $s$ because
$[\mathcal K,s]=\mathcal K(uv)=0$. Thus
$\mathsf A-\mathsf q\,\mathcal K^m$ commutes with $s$ and has order
less than $m$, and the induction hypothesis applies. The statement
about $\delta_0$ follows from \eqref{eq:rig-K-facts}.

Finally let $\kappa_\nu(\mathsf A)=\mathsf A$ with $\nu\neq0$. For
$z\in\CC$,
\[
 e^{2\pi izs}\,\mathsf A\,e^{-2\pi izs}
 =\sum_{k=0}^m\frac{(2\pi iz)^k}{k!}\,
 \operatorname{ad}_s^k(\mathsf A),\qquad
 \operatorname{ad}_s(\mathsf A)=[s,\mathsf A],
\]
the sum being finite because $\operatorname{ad}_s$ lowers the order.
This polynomial in $z$ takes the value $\mathsf A$ at all integer
multiples of $\nu$, hence is constant, and the coefficient of $z$
gives $[s,\mathsf A]=0$.
\end{proof}

\begin{lemma}[Formal rigidity without growth conditions]
\label{lem:formal-rigidity}
Let $(\Gamma_i)_{i\in\mathsf I}$ be a transverse family of hyperbolic
grids with support $S$ and phase exponent $\nu$, and let $q\geq1$.
There is a finite list $\mathsf D_1,\dots,\mathsf D_{n_0}$ of
translation operators
\[
 \mathsf D_l=\prod_{i\in\mathsf I}(\tau_{w_{i,l}}-I)^{m_l},\qquad
 w_{i,l}\in\Gamma_i^\vee\setminus\{0\},\qquad
 m_l\in\{q+1,\,2q+1\},
\]
with the following property. Let $(c_\alpha(x))_{x\in S,\,
\alpha\in\mathcal A_q}$ be an arbitrary array of complex numbers,
subject to no growth condition, and let
\[
 T=\sum_{x\in S}\sum_{|\alpha|\leq q}c_\alpha(x)\,
 \partial^\alpha\delta_x .
\]
If $\mathsf D_lT=0$ for all $l$ and $\mathcal C^\nu T=T$, then
$c_\alpha(0)=0$ for all $\alpha$ with $1\leq|\alpha|\leq q$.
\end{lemma}
\begin{proof}
\emph{The list.} For each $j\in\mathsf I$ fix a basis
$(\mathsf b_{j,1},\mathsf b_{j,2})$ of $\Gamma_j^\vee$. For each $j$
and $\iota\in\{1,2\}$ the list contains the operator
\[
 \mathsf D_{j,\iota}=(\tau_{\mathsf b_{j,\iota}}-I)^{q+1}
 \prod_{i\neq j}(\tau_{w_i}-I)^{q+1},
\]
where the vectors $w_i=w_i^{(j,\iota)}\in\Gamma_i^\vee$ are chosen by
Lemma~\ref{lem:rig-transverse-differences}(i) with $m=q+1$ and
$G_i=\Gamma_i^\vee$. In addition the list contains one operator
$\mathsf D_*=\prod_{i\in\mathsf I}(\tau_{w_i^*}-I)^{2q+1}$ with
arbitrary nonzero $w_i^*\in\Gamma_i^\vee$.

\emph{Polynomial structure.} Let $T$ be as in the statement. By
Lemma~\ref{lem:rig-transverse-differences}(ii), applied to
$\mathsf D_{j,1}$ and $\mathsf D_{j,2}$, for every $j$, every coset
$\gamma+\Gamma_j^\vee$ and every $\alpha$ the function
$(n_1,n_2)\mapsto
c_\alpha(\gamma+n_1\mathsf b_{j,1}+n_2\mathsf b_{j,2})$ has finitely
supported differences of order $q+1$ in both directions.
Lemma~\ref{lem:difference-newton}, which requires no growth condition,
shows that it agrees outside a finite set with a polynomial of degree
at most $q$ in each of $n_1,n_2$. As in the proof of
Lemma~\ref{lem:transverse-combs}(ii) we obtain
\[
 T=T_0+F,\qquad
 T_0=\sum_{i,\gamma}\mathsf A_{i,\gamma}e_{i,\gamma},\qquad
 \mathsf A_{i,\gamma}=\sum_{|\alpha|\leq q}\partial^\alpha\circ
 p_{i,\gamma,\alpha},
\]
where $F$ has finite support and the polynomials $p_{i,\gamma,\alpha}$
have total degree at most $2q$.

\emph{The finite part vanishes.} For $w\in\Gamma_i^\vee$ translation by
$w$ preserves $e_{i,\gamma}$, so
$(\tau_w-I)^{2q+1}\bigl(\partial^\alpha(p\,e_{i,\gamma})\bigr)
=\partial^\alpha\bigl(((\tau_w-I)^{2q+1}p)\,e_{i,\gamma}\bigr)=0$ for
every polynomial $p$ of degree at most $2q$. Since translations
commute, $\mathsf D_*T_0=0$, and therefore
$\mathsf D_*F=\mathsf D_*T-\mathsf D_*T_0=0$. The distribution $F$ is
tempered, and by \eqref{eq:rig-intertwine}
$\omega_*\,\mathbb S^{-1}F=0$ with
$\omega_*=\prod_i(\chi_{w_i^*}-1)^{2q+1}$. Since $F$ has finite
support, $\mathbb S^{-1}F$ is a real-analytic function, and $\omega_*$ vanishes only on the locally
finite family of lines $B(w_i^*,X)\in\ZZ$. Hence $\mathbb S^{-1}F$
vanishes on a dense open set, so $\mathbb S^{-1}F=0$ and $F=0$.

\emph{Conclusion.} Thus $T=T_0$ is a polynomial comb. Since
$\mathcal C^\nu$ fixes $T$ and every $e_{i,\gamma}$,
$\sum_{i,\gamma}\mathsf A_{i,\gamma}e_{i,\gamma}
=\sum_{i,\gamma}\kappa_\nu(\mathsf A_{i,\gamma})e_{i,\gamma}$, and
Lemma~\ref{lem:transverse-combs}(i) gives
$\kappa_\nu(\mathsf A_{i,\gamma})=\mathsf A_{i,\gamma}$. Near the
origin $T=\sum_i\mathsf A_{i,0}\delta_0$, and by
Lemma~\ref{lem:spin-centralizer} each term is a multiple of
$\delta_0$.
\end{proof}

\begin{lemma}[A finite consequence of finitely supported equations]
\label{lem:finite-rows}
Let $\mathcal I$ be a set. Call an element $\mathsf r$ of the space
$\CC^{(\mathcal I)}$ of finitely supported functions on $\mathcal I$ a
\emph{row}, and let it act on arbitrary arrays $c\in\CC^{\mathcal I}$
by $\mathsf r(c)=\sum_{\imath\in\mathcal I}\mathsf r_\imath c_\imath$.
Let $\mathsf R$ be a family of rows and $\lambda$ a row. If
$\lambda(c)=0$ for every $c\in\CC^{\mathcal I}$ such that
$\mathsf r(c)=0$ for all $\mathsf r\in\mathsf R$, then $\lambda$ is a
finite linear combination of elements of $\mathsf R$.
\end{lemma}
\begin{proof}
Let $U\subset\CC^{(\mathcal I)}$ be the linear span of $\mathsf R$. If
$\lambda\notin U$, there is a linear functional $\Lambda$ on
$\CC^{(\mathcal I)}$ that vanishes on $U$ and satisfies
$\Lambda(\lambda)=1$. Let $c_\imath$ be the value of $\Lambda$ on the
indicator function of $\{\imath\}$. Then
$\mathsf r(c)=\Lambda(\mathsf r)$ for every row $\mathsf r$, so $c$
satisfies all equations of $\mathsf R$ while $\lambda(c)=1$, a
contradiction. Hence $\lambda\in U$.
\end{proof}

The array $c$ produced in this proof satisfies no growth condition.
This is the reason for stating Lemma~\ref{lem:formal-rigidity} for
arbitrary arrays: a rigidity statement for tempered arrays only would
not yield a finite combination of rows.

\paragraph{Uniform families of Gaussian mixtures.}
To control the test functions, also in families, we fix the following
notation. Let $n\geq0$ be an integer; for $n=0$ the cube $[-1,1]^n$ is
a point of unit mass. For a continuous map $z:[-1,1]^n\to\CC^2$ and a
polynomial $p(X,\xi)=\sum_{|\gamma|\leq n'}p_\gamma(\xi)X^\gamma$ whose
coefficients are continuous functions of $\xi\in[-1,1]^n$, put
\begin{gather*}
 \mathfrak m[p,z](X)=\int_{[-1,1]^n}p(X,\xi)\,
 e^{-2\pi|X|^2+z(\xi)\cdot X}\dd\xi,\\
 \|p\|=\max_\gamma\sup_\xi|p_\gamma(\xi)|,\qquad
 \|z\|=\sup_\xi|z(\xi)| .
\end{gather*}

\begin{lemma}\label{lem:rig-mixture-family}
Let $n,n'\geq0$ be integers and $\mathsf z_0<\infty$, and let $p,z$ be
as above, with $p$ of degree at most $n'$ in $X$ and
$\|z\|\leq\mathsf z_0$.
\begin{enumerate}[(i)]
\item $\mathfrak m[p,z]\in\mathcal G_{2\pi}$, and
 $e^{w\cdot X}\mathfrak m[p,z]=\mathfrak m[p,z+w]$ for $w\in\CC^2$.
\item $\mathbb S\,\mathfrak m[p,z]=\mathfrak m[\tilde p,iJz]$, where
 $J=\operatorname{diag}(1,-1)$ and $\tilde p$ is a polynomial of
 degree at most $n'$ in $X$ with
 $\|\tilde p\|\leq C\|p\|$, $C=C(n',\mathsf z_0)$.
\item For every integer $k_0\geq0$, every $\varrho<\infty$ and every
 $0<b<2\pi$ there is $C=C(n,n',\mathsf z_0,k_0,\varrho,b)$ with
 \[
 |\partial^\alpha\mathfrak m[p,z](X)|\leq C\,\|p\|\,
 e^{-b|\operatorname{Re}X|^2}\qquad
 \text{for }|\alpha|\leq k_0,\ |\operatorname{Im}X|\leq\varrho .
 \]
\end{enumerate}
\end{lemma}
\begin{proof}
(i) is immediate from Definition~\ref{def:gaussian-mixture}. For (ii),
\eqref{eq:S-gaussian} with $b=2\pi$ reads
$\mathbb S\phi_{2\pi,z}=e^{z\cdot z/(8\pi)}\phi_{2\pi,iJz}$. We have
$X^\gamma\phi_{2\pi,z}=\partial_z^\gamma\phi_{2\pi,z}$, and
$\mathbb S$ commutes with differentiation in the parameter $z$, by
differentiation under the integral \eqref{eq:S-transform}. Hence
\[
 \mathbb S\bigl(X^\gamma\phi_{2\pi,z}\bigr)
 =\partial_z^\gamma\bigl[e^{z\cdot z/(8\pi)}\phi_{2\pi,iJz}\bigr]
 =\mathsf H_\gamma(X,z)\,e^{z\cdot z/(8\pi)}\,\phi_{2\pi,iJz},
\]
where $\mathsf H_\gamma$ is a polynomial in $(X,z)$ of degree at most
$|\gamma|$ with universal coefficients. Integrating over $[-1,1]^n$,
which is permitted by Fubini's theorem, gives (ii) with
$\tilde p(X,\xi)=\sum_\gamma p_\gamma(\xi)\,
\mathsf H_\gamma(X,z(\xi))\,e^{z(\xi)\cdot z(\xi)/(8\pi)}$.
For (iii), $\partial^\alpha\bigl(X^\gamma e^{-2\pi|X|^2+z\cdot X}\bigr)$
is a polynomial in $(X,z)$ of degree at most $n'+k_0$ in $X$ times
$e^{-2\pi|X|^2+z\cdot X}$, and for $|\operatorname{Im}X|\leq\varrho$
and $|z|\leq\mathsf z_0$
\[
 \bigl|e^{-2\pi|X|^2+z\cdot X}\bigr|
 \leq e^{2\pi\varrho^2+\mathsf z_0\varrho}\,
 e^{-2\pi|\operatorname{Re}X|^2+\mathsf z_0|\operatorname{Re}X|}.
\]
The factor $e^{-(2\pi-b)|\operatorname{Re}X|^2}$ absorbs the polynomial
and $e^{\mathsf z_0|\operatorname{Re}X|}$.
\end{proof}

\begin{lemma}[Cardinal test functions]\label{lem:rig-cardinal-tests}
Let $\mathfrak c_1,\dots,\mathfrak c_{n_1}$ be cosets of lattices in
$\RR^2$, given as
\[
 \mathfrak c_i=\bigl\{X\in\RR^2:\ell_{i,1}(X-x_i)\in\ZZ,\
 \ell_{i,2}(X-x_i)\in\ZZ\bigr\}
\]
with points $x_i\in\RR^2$ and linearly independent real linear forms
$\ell_{i,1},\ell_{i,2}$. Let $y\in\RR^2$, let $q\geq0$ be an integer and
$\beta\in\mathcal A_q$. For each $i$ with $y\notin\mathfrak c_i$ choose
$\ell_i^\star\in\{\ell_{i,1},\ell_{i,2}\}$ with
$\ell_i^\star(y-x_i)\notin\ZZ$. Put
\[
\begin{aligned}
 \psi_0(X)=e^{-2\pi|X-y|^2}
 &\prod_{i:\,y\in\mathfrak c_i}
 \Bigl[\sinc\bigl(\pi\ell_{i,1}(X-y)\bigr)
 \sinc\bigl(\pi\ell_{i,2}(X-y)\bigr)\Bigr]^{q+1}\\
 &\times\prod_{i:\,y\notin\mathfrak c_i}\biggl[
 \frac{e^{2\pi i\ell_i^\star(X-x_i)}-1}
 {e^{2\pi i\ell_i^\star(y-x_i)}-1}\biggr]^{q+1},
\end{aligned}
\]
and $\psi=\mathsf t\,\psi_0$, where $\mathsf t$ is the Taylor
polynomial of degree $q$ at $y$ of the function
$(-1)^{|\beta|}(X-y)^\beta/(\beta!\,\psi_0(X))$, which is analytic near
$y$ because $\psi_0(y)=1$.
\begin{enumerate}[(i)]
\item $\psi$ vanishes to order at least $q+1$ at every point of
 $\bigcup_i\mathfrak c_i$ other than $y$, and
 $\partial^\alpha\psi(y)=(-1)^{|\beta|}\one_{\alpha=\beta}$ for
 $|\alpha|\leq q$. Consequently, for every tempered distribution $U$
 supported in $\bigcup_i\mathfrak c_i$ and of local order at most $q$,
 $\langle U,\psi\rangle$ is the coefficient of
 $\partial^\beta\delta_y$ in $U$.
\item $\psi$ is a finite sum of functions $\mathfrak m[p,z]$ as in
 Lemma~\ref{lem:rig-mixture-family}; in particular
 $\psi\in\mathcal G_{2\pi}$. If $|y|$, $|x_i|$ and the norms of the
 forms $\ell_{i,1},\ell_{i,2}$ are bounded by $\mathsf z_1$, then the
 number of terms, the integers $n$, $n'$ and a bound for $\|z\|$ can
 be chosen in terms of $q$, $n_1$ and $\mathsf z_1$ only, and
 \[
 \|p\|\leq C(q,n_1,\mathsf z_1)\,\mathfrak d^{-(q+1)^2},\qquad
 \mathfrak d=\prod_{i:\,y\notin\mathfrak c_i}
 \bigl|e^{2\pi i\ell_i^\star(y-x_i)}-1\bigr| .
 \]
\end{enumerate}
\end{lemma}
\begin{proof}
(i) Let $x\in\mathfrak c_i$, $x\neq y$. If $y\in\mathfrak c_i$, then
$\ell_{i,1}(x-y)$ and $\ell_{i,2}(x-y)$ are integers, not both zero, so
one of the two $\sinc$ factors with index $i$ vanishes at $x$ and its
$(q+1)$-st power vanishes to order $q+1$. If $y\notin\mathfrak c_i$,
the numerator of the exponential factor with index $i$ vanishes on
$\mathfrak c_i$. All factors are entire, so $\psi_0$ and $\psi$ vanish
to order at least $q+1$ at $x$. At $y$ every factor equals one, and by
the choice of $\mathsf t$,
$\psi(X)=(-1)^{|\beta|}(X-y)^\beta/\beta!+O(|X-y|^{q+1})$. The last
assertion follows from Lemma~\ref{lem:rig-local-pairing}(i) with
$F_0=\{y\}$.

(ii) Write $\ell(X)=\vec\ell\cdot X$ for a linear form $\ell$, and
$\psi_0=\mathfrak d_0^{-1}\psi_0^\circ$, where
$\mathfrak d_0=\prod_{i:\,y\notin\mathfrak c_i}
(e^{2\pi i\ell_i^\star(y-x_i)}-1)^{q+1}$ and $\psi_0^\circ$ is
$\psi_0$ with the denominators removed; thus
$|\mathfrak d_0|=\mathfrak d^{q+1}$. We have
\begin{align*}
 e^{-2\pi|X-y|^2}&=e^{-2\pi y\cdot y}\,e^{-2\pi|X|^2+4\pi y\cdot X},\\
 \sinc\bigl(\pi\ell(X-y)\bigr)&=\frac12\int_{-1}^1
 e^{-i\pi\xi\ell(y)}\,e^{i\pi\xi\vec\ell\cdot X}\dd\xi,\\
 \bigl(e^{2\pi i\ell(X-x)}-1\bigr)^{q+1}&=\sum_{k=0}^{q+1}
 (-1)^{q+1-k}\binom{q+1}k\,e^{-2\pi ik\ell(x)}\,
 e^{2\pi ik\vec\ell\cdot X}.
\end{align*}
Multiplying out, $\psi_0^\circ$ is a sum of at most $(q+2)^{n_1}$
functions $\mathfrak m[p,z]$ with $n\leq2(q+1)n_1$, with polynomials
$p$ of degree zero in $X$ and $\|p\|\leq2^{(q+1)n_1}$, and with
$\|z\|$ bounded in terms of $q,n_1,\mathsf z_1$. By
Lemma~\ref{lem:rig-mixture-family}(iii) the derivatives of
$\psi_0^\circ$ of order at most $q$ at $y$ are bounded in terms of the
same quantities. Since $\psi_0^\circ(y)=\mathfrak d_0$ and
$|\mathfrak d_0|\leq2^{(q+1)n_1}$, the Taylor coefficients at $y$, up
to order $q$, of
\[
 \frac{\mathfrak d_0}{\psi_0^\circ}
 =\sum_{k\geq0}\Bigl(-\frac{\psi_0^\circ-\mathfrak d_0}
 {\mathfrak d_0}\Bigr)^k
\]
are bounded by a constant times $|\mathfrak d_0|^{-q}$; only the terms
with $k\leq q$ contribute to them, because
$\psi_0^\circ-\mathfrak d_0$ vanishes at $y$. Hence the coefficients of
$\mathsf t$, in powers of $X-y$ and therefore also in powers of $X$,
are bounded by $C|\mathfrak d_0|^{-q}$, and
$\psi=(\mathfrak d_0^{-1}\mathsf t)\,\psi_0^\circ$ has the asserted
form with $n'=q$ and
$\|p\|\leq C|\mathfrak d_0|^{-q-1}=C\mathfrak d^{-(q+1)^2}$.
\end{proof}

\paragraph{Rows.}
Fix a transverse family $(\Gamma_i)_{i\in\mathsf I}$ of hyperbolic
grids with support $S$ and phase exponent $\nu$, an integer $q\geq1$,
and the operators $\mathsf D_1,\dots,\mathsf D_{n_0}$ of
Lemma~\ref{lem:formal-rigidity}. Expanding the products,
\begin{equation}\label{eq:rig-D-expansion}
 \mathsf D_l=\sum_{\mathbf k}\varkappa_{l,\mathbf k}\,
 \tau_{\mathsf w_{l,\mathbf k}},
 \qquad
 \omega_l:=\prod_{i\in\mathsf I}(\chi_{w_{i,l}}-1)^{m_l}
 =\sum_{\mathbf k}\varkappa_{l,\mathbf k}\,
 \chi_{\mathsf w_{l,\mathbf k}},
\end{equation}
where $\mathbf k=(k_i)_{i\in\mathsf I}$ runs over
$\{0,\dots,m_l\}^{\mathsf I}$,
$\varkappa_{l,\mathbf k}=\prod_i(-1)^{m_l-k_i}\binom{m_l}{k_i}$ and
$\mathsf w_{l,\mathbf k}=\sum_ik_iw_{i,l}$. By
\eqref{eq:rig-intertwine},
$\mathsf D_l=\mathbb S\,\omega_l\,\mathbb S^{-1}$. For an array
$c\in\CC^{S\times\mathcal A_q}$ let
$T_c=\sum_{x\in S}\sum_{|\alpha|\leq q}c_\alpha(x)\partial^\alpha\delta_x$,
and put $c_\alpha(x)=0$ for $x\notin S$. The coefficient of
$\partial^\beta\delta_y$ in $\mathsf D_lT_c$ is the
\emph{translation row}
\begin{equation}\label{eq:rig-translation-row}
 \mathsf r_{l,y,\beta}(c)=\sum_{\mathbf k}\varkappa_{l,\mathbf k}\,
 c_\beta(y+\mathsf w_{l,\mathbf k}),\qquad
 1\leq l\leq n_0,\ y\in\RR^2,\ \beta\in\mathcal A_q .
\end{equation}
Write $\partial^\gamma e^{2\pi i\nu s}=\mathsf h_\gamma\,
e^{2\pi i\nu s}$ with polynomials
$\mathsf h_\gamma\in\QQ(\pi i)[P,Q]$, $\mathsf h_0=1$. Since
$e^{2\pi i\nu s}=1$ on $S$, the coefficient of
$\partial^\beta\delta_x$ in $\mathcal C^\nu T_c-T_c$ is the
\emph{phase row}
\begin{equation}\label{eq:rig-phase-row}
 \mathsf p_{x,\beta}(c)=\sum_{\substack{\alpha\in\mathcal A_q\\
 \alpha\geq\beta,\ \alpha\neq\beta}}(-1)^{|\alpha-\beta|}
 \binom\alpha\beta\,\mathsf h_{\alpha-\beta}(x)\,c_\alpha(x),\qquad
 x\in S,\ \beta\in\mathcal A_q .
\end{equation}
Both kinds of rows are finitely supported, and an array $c$ satisfies
$\mathsf D_lT_c=0$ for all $l$ and $\mathcal C^\nu T_c=T_c$ if and only
if all translation rows and all phase rows vanish on $c$.

\begin{theorem}[A finite Gaussian-mixture certificate]
\label{thm:origin-certificate}
Let $\Sigma$ be a finite union of circle lattices and let $q\geq1$ be
an integer. There are finitely many functions $f_1,\dots,f_n$ on
$\CC^2$ and a constant $C_{\Sigma,q}$ with the following properties.
\begin{enumerate}[(i)]
\item For every $l$, both $f_l$ and $\mathbb Sf_l$ belong to the class
 $\mathcal G_{2\pi}$ of Definition~\ref{def:gaussian-mixture}. In
 particular they are entire, their restrictions to $\RR^2$ are
 Schwartz functions, they satisfy the bounds
 \eqref{eq:mixture-envelope} for every $b'<2\pi$, and
 Lemma~\ref{lem:gaussian-identity} applies to each $f_l$.
\item Every even tempered distribution $T$ with
 $\supp T\subset\Sigma$, of local order at most $q$ and with
 $\mathcal CT=T$, satisfies
 \begin{equation}\label{eq:rig-certificate}
 \max_{1\leq|\alpha|\leq q}|c_\alpha(0)|\leq C_{\Sigma,q}\,
 \max_{1\leq l\leq n}\bigl|\langle(\mathbb S-I)T,f_l\rangle\bigr| ,
 \end{equation}
 where $c_\alpha(0)$ are the origin coefficients of $T$ in
 \eqref{eq:rig-local-form}.
\end{enumerate}
The functions $f_l$ and the constant depend only on $\Sigma$ and $q$.
\end{theorem}
\begin{proof}
Let $\Sigma=\bigcup_{r\in\mathcal R}\Lambda_r$, let
$(\Gamma_i)_{i\in\mathsf I}$ be a transverse family adapted to
$\mathcal R$, and use the notation of the preceding paragraph.

\emph{Row identities.} Let $\mathsf R$ be the family of all rows
\eqref{eq:rig-translation-row} and \eqref{eq:rig-phase-row} on
$\mathcal I=S\times\mathcal A_q$. By Lemma~\ref{lem:formal-rigidity},
every array on which all rows of $\mathsf R$ vanish satisfies
$c_\alpha(0)=0$ for $1\leq|\alpha|\leq q$. By
Lemma~\ref{lem:finite-rows}, for each such $\alpha$ there are finitely
many translation rows $\mathsf r_\mu=\mathsf r_{l_\mu,y_\mu,\beta_\mu}$,
$\mu\in\mathsf M_\alpha$, finitely many phase rows $\mathsf p_{\mu'}$,
$\mu'\in\mathsf M'_\alpha$, and complex numbers
$\vartheta_\mu,\vartheta'_{\mu'}$ with
\begin{equation}\label{eq:rig-row-identity}
 c_\alpha(0)=\sum_{\mu\in\mathsf M_\alpha}\vartheta_\mu\,
 \mathsf r_\mu(c)
 +\sum_{\mu'\in\mathsf M'_\alpha}\vartheta'_{\mu'}\,
 \mathsf p_{\mu'}(c)
 \qquad\text{for all }c\in\CC^{S\times\mathcal A_q}.
\end{equation}

\emph{Extraction.} Fix $\mu\in\mathsf M_\alpha$ and write
$(l,y,\beta)=(l_\mu,y_\mu,\beta_\mu)$. For every array $c$ the
distribution $\mathsf D_lT_c$ is supported in the union of the cosets
$\Gamma_i-\mathsf w_{l,\mathbf k}$, $i\in\mathsf I$,
$\mathbf k\in\{0,\dots,m_l\}^{\mathsf I}$, and has local order at most
$q$. Each of these cosets has the form required in
Lemma~\ref{lem:rig-cardinal-tests}, with the point
$-\mathsf w_{l,\mathbf k}$ and the coordinate forms $d_iu/r_i$,
$d_ir_iv$ of $\Gamma_i$. Let $\psi_\mu$ be the function $\psi$ of that
lemma for these cosets, the point $y$, the order $q$ and the index
$\beta$. If $T_c$ is tempered, then
\begin{equation}\label{eq:rig-extraction}
 \langle\mathsf D_lT_c,\psi_\mu\rangle=\mathsf r_\mu(c)
\end{equation}
by Lemma~\ref{lem:rig-cardinal-tests}(i) and
\eqref{eq:rig-translation-row}.

\emph{The tests.} Put
\[
 f_\mu=\omega_{l_\mu}\,\mathbb S\psi_\mu ,\qquad
 \mu\in\mathsf M_\alpha,\ 1\leq|\alpha|\leq q .
\]
By Lemma~\ref{lem:rig-cardinal-tests}(ii), $\psi_\mu$ is a finite sum
of functions $\mathfrak m[p,z]$. By
Lemma~\ref{lem:rig-mixture-family}(ii) so is $\mathbb S\psi_\mu$; by
\eqref{eq:rig-D-expansion} and
Lemma~\ref{lem:rig-mixture-family}(i) so is $f_\mu$, the multiplier
$\omega_l$ being a finite sum of exponentials
$\chi_{\mathsf w}(X)=e^{2\pi iB(\mathsf w,X)}$; and by
Lemma~\ref{lem:rig-mixture-family}(ii) again so is $\mathbb Sf_\mu$.
This proves (i).

\emph{The estimate.} Let $T$ be as in (ii) and let $c$ be its
coefficient array, extended by zero to $S\supset\Sigma$, so that
$T=T_c$. Since $\mathcal CT=T$ we have $\mathcal C^\nu T=T$, and all
phase rows vanish on $c$. Every point of $S$ lies in some $\Gamma_i$,
on which the factor $(\chi_{w_{i,l}}-1)^{m_l}$ of $\omega_l$ vanishes
to order $m_l\geq q+1$, because $w_{i,l}\in\Gamma_i^\vee$. Hence
$\omega_lT=0$ by Lemma~\ref{lem:rig-local-pairing}(ii). As $T$ is
even, $\mathbb S^{-1}T=\mathbb ST$, and therefore
\[
 \mathsf D_lT=\mathbb S\,\omega_l\,\mathbb S^{-1}T
 =\mathbb S\,\omega_l\,\mathbb ST
 =\mathbb S\,\omega_l\,(\mathbb S-I)T .
\]
Pairing with $\psi_\mu$ and using \eqref{eq:rig-extraction},
\[
 \mathsf r_\mu(c)=\langle\mathsf D_{l_\mu}T,\psi_\mu\rangle
 =\bigl\langle\omega_{l_\mu}(\mathbb S-I)T,\mathbb S\psi_\mu\bigr\rangle
 =\langle(\mathbb S-I)T,f_\mu\rangle .
\]
Consequently \eqref{eq:rig-row-identity} gives
$c_\alpha(0)=\sum_{\mu\in\mathsf M_\alpha}\vartheta_\mu
\langle(\mathbb S-I)T,f_\mu\rangle$, and \eqref{eq:rig-certificate}
follows with
$C_{\Sigma,q}=\max_\alpha\sum_{\mu\in\mathsf M_\alpha}|\vartheta_\mu|$.
\end{proof}

The tests $f_l$ are not even. In \eqref{eq:rig-certificate} the
distributions $T$ and $\mathbb ST$ are even, so the pairings involve
only the even parts of $f_l$ and $\mathbb Sf_l$, that is, the functions
$\ev{f_l}$ and $\ev{(\mathbb Sf_l)}$ of \eqref{eq:natural}. The
analytic evaluation $\mathcal L_a$ and
Lemma~\ref{lem:gaussian-identity} likewise involve only these even
parts. The constant $C_{\Sigma,q}$ is not explicit, since the row
identities \eqref{eq:rig-row-identity} are obtained from
Lemma~\ref{lem:finite-rows}.

\subsection{One moving radius}
\label{subsec:rig-moving}

The constant in Theorem~\ref{thm:origin-certificate} depends on the
union of lattices. When one radius varies, it can be controlled up to
a finite power of the distance to a limiting radius.

\begin{proposition}[A certificate for one moving radius]
\label{prop:moving-certificate}
Let $\mathcal R_0\subset(0,\infty)$ be finite, let
$\Sigma_0=\bigcup_{r\in\mathcal R_0}\Lambda_r$, and for $t>0$ put
\[
 \Sigma(t)=\Sigma_0\cup\Lambda_t\cup\Lambda_{1/t}.
\]
Fix $t_0>0$ and an integer $q\geq1$. There exist
$\varepsilon_0\in(0,t_0)$, integers $n\geq1$ and $K\geq0$, and, for
every $t$ with $0<|t-t_0|<\varepsilon_0$, functions
$f_{1,t},\dots,f_{n,t}$ on $\CC^2$ such that the following hold.
\begin{enumerate}[(i)]
\item $f_{l,t}\in\mathcal G_{2\pi}$ and
 $\mathbb Sf_{l,t}\in\mathcal G_{2\pi}$ for all $l$ and $t$.
\item For every $t$ with $0<|t-t_0|<\varepsilon_0$, every even
 tempered distribution $T$ with $\supp T\subset\Sigma(t)$, of local
 order at most $q$ and with $\mathcal CT=T$, satisfies
 \begin{equation}\label{eq:rig-moving-certificate}
 \max_{1\leq|\alpha|\leq q}|c_\alpha(0)|\leq\sum_{l=1}^n
 \bigl|\langle(\mathbb S-I)T,f_{l,t}\rangle\bigr| .
 \end{equation}
\item For every integer $k_0\geq0$, every $\varrho<\infty$ and every
 $0<b<2\pi$ there is a constant $C$ such that, for all $l$, all
 $|\alpha|\leq k_0$, all $X\in\CC^2$ with
 $|\operatorname{Im}X|\leq\varrho$ and all $t$ with
 $0<|t-t_0|<\varepsilon_0$,
 \begin{equation}\label{eq:rig-moving-envelope}
 |\partial^\alpha f_{l,t}(X)|+|\partial^\alpha\mathbb Sf_{l,t}(X)|
 \leq C\,|t-t_0|^{-K}\,e^{-b|\operatorname{Re}X|^2}.
 \end{equation}
\end{enumerate}
The numbers $\varepsilon_0$, $n$, $K$ and the functions $f_{l,t}$ can
be chosen depending only on $\mathcal R_0$, $t_0$ and $q$; the constant
$C$ depends in addition on $k_0$, $\varrho$ and $b$.
\end{proposition}

The bound \eqref{eq:rig-moving-envelope} may degenerate as $t\to t_0$;
only its rate is controlled. In the applications the order of the
choices matters: the local order $q$, and with it $K$, is fixed before
any quantity that is to be balanced against the loss $|t-t_0|^{-K}$.

\begin{proof}
\emph{Step 1: the family of grids.} Let $(\Gamma_i)_{i\in\mathsf I}$,
$\Gamma_i=d_i^{-1}\Lambda_{r_i}$, be a transverse family adapted to
$\mathcal R_0$, with phase exponent $\nu$. Put
$\mathsf I^\pm=\mathsf I\sqcup\{+,-\}$, $d_\pm=1$, and for $t>0$
\[
 \Gamma_+(t)=\Lambda_t,\qquad\Gamma_-(t)=\Lambda_{1/t},\qquad
 \Gamma_i(t)=\Gamma_i\ \ (i\in\mathsf I),\qquad
 S(t)=\bigcup_{i\in\mathsf I^\pm}\Gamma_i(t)\supset\Sigma(t).
\]
For each $i\in\mathsf I^\pm$ let
$\ell^t_{i,1},\ell^t_{i,2}$ be the linear forms
\[
 \Bigl(\frac{d_iu}{r_i},\,d_ir_iv\Bigr)\ \ (i\in\mathsf I),\qquad
 \Bigl(\frac ut,\,tv\Bigr)\ \ (i=+),\qquad
 \Bigl(tu,\,\frac vt\Bigr)\ \ (i=-),
\]
so that $\Gamma_i(t)$ is the set where both forms are integral, and let
$x_{i,m,n}(t)$ be the point of $\Gamma_i(t)$ at which they take the
values $(m,n)\in\ZZ^2$. The dual lattice $\Gamma_i(t)^\vee$ consists of
the points $x_{i,m,n}(t)$ with $m,n\in d_i^2\ZZ$, and
$\nu s\in\ZZ$ on $S(t)$, for every $t>0$.

Let $\mathbb F=\QQ(\pi i,\,r_i:i\in\mathsf I)\subset\CC$, a countable
field. A \emph{Laurent vector} is a map $y:(0,\infty)\to\RR^2$ whose
coordinates are given by elements of $\mathbb F[t,t^{-1}]$ (real-valued
for $t>0$). The maps
$t\mapsto x_{i,m,n}(t)$ are Laurent vectors, Laurent vectors form a
group under addition, and $t\mapsto\ell^t_{i,\iota}(y(t))$ is given
by an element of $\mathbb F[t,t^{-1}]$ for every Laurent vector $y$
and $\iota\in\{1,2\}$. Fix a number $t_*>0$ that is transcendental
over $\mathbb F$. Evaluation at $t_*$ is an injective ring homomorphism
$\mathbb F[t,t^{-1}]\to\CC$ and extends to an embedding of the field
$\mathbb F(t)$ of rational functions. Hence:
\begin{equation}\label{eq:rig-generic}
 \parbox{0.8\linewidth}{two elements of $\mathbb F[t,t^{-1}]$, or two
 Laurent vectors, that agree at $t_*$ agree identically.}
\end{equation}
The radii $r_i$ $(i\in\mathsf I)$, $t_*$ and $1/t_*$ are pairwise
incommensurable. Thus $(\Gamma_i(t_*))_{i\in\mathsf I^\pm}$ is a
transverse family of hyperbolic grids with phase exponent $\nu$, to
which Lemma~\ref{lem:formal-rigidity} and the constructions of
Section~\ref{subsec:rig-certificate} apply. For other values of $t$ the
lattices $\Gamma_i(t)$ may be commensurable and may share nonzero
points.

\emph{Step 2: labels.} Let
$\Xi=\{0\}\sqcup\bigl(\mathsf I^\pm\times(\ZZ^2\setminus\{0\})\bigr)$
and put $x_0(t)=0$, $x_\lambda(t)=x_{i,m,n}(t)$ for
$\lambda=(i,m,n)$. The label $0$ is regarded as the label
$(i,0,0)$ of every class $i$. For each $t$ the map
$\lambda\mapsto x_\lambda(t)$ maps $\Xi$ onto $S(t)$ and is injective
on the labels of any one class; at $t=t_*$ it is a bijection. A
\emph{label array} is an element
$\hat c\in\CC^{\Xi\times\mathcal A_q}$, and we put
\[
 T^t[\hat c]=\sum_{\lambda\in\Xi}\sum_{|\alpha|\leq q}
 \hat c_\alpha(\lambda)\,\partial^\alpha\delta_{x_\lambda(t)} ,
\]
a locally finite sum. Conversely, let $T$ be a distribution supported
in $S(t)$ of local order at most $q$, with coefficients $c_\alpha(x)$.
For each $x\in S(t)$ choose one label $\lambda(x)$ with
$x_{\lambda(x)}(t)=x$, and define
\begin{equation}\label{eq:rig-label-array}
 \hat c_\alpha(\lambda(x))=c_\alpha(x)\quad(x\in S(t)),\qquad
 \hat c_\alpha(\lambda)=0\quad
 \text{if }\lambda\neq\lambda(x_\lambda(t)).
\end{equation}
Then $T^t[\hat c]=T$ and $\hat c_\alpha(0)=c_\alpha(0)$. When several
labels are mapped to the same point, the coefficients of $T$ at that
point are thus attributed to one of them and the others receive zero.

\emph{Step 3: rows depending on the parameter.} Apply
Lemma~\ref{lem:formal-rigidity} to the family
$(\Gamma_i(t_*))_{i\in\mathsf I^\pm}$ and the order $q$. Its
translation vectors are $w_{i,l}(t_*)$ for Laurent vectors
$w_{i,l}=x_{i,m_{i,l},n_{i,l}}$ with
$(m_{i,l},n_{i,l})\in d_i^2\ZZ^2\setminus\{0\}$. For every $t>0$ we
have $w_{i,l}(t)\in\Gamma_i(t)^\vee\setminus\{0\}$, and we define
$\mathsf D_l(t)$, $\omega_l(t)$, $\varkappa_{l,\mathbf k}$ and the
Laurent vectors $\mathsf w_{l,\mathbf k}$ as in
\eqref{eq:rig-D-expansion}. For a Laurent vector $y$,
$\beta\in\mathcal A_q$ and $t>0$ define the functional on label arrays
\[
 \mathsf r^t_{l,y,\beta}(\hat c)=\sum_{\mathbf k}
 \varkappa_{l,\mathbf k}
 \sum_{\substack{\lambda\in\Xi\\
 x_\lambda(t)=y(t)+\mathsf w_{l,\mathbf k}(t)}}\hat c_\beta(\lambda) .
\]
The inner sum has at most one term from each class. Since
$\tau_w\delta_x=\delta_{x-w}$,
\begin{equation}\label{eq:rig-moving-row-meaning}
 \mathsf r^t_{l,y,\beta}(\hat c)=\text{the coefficient of }
 \partial^\beta\delta_{y(t)}\text{ in }\mathsf D_l(t)\,T^t[\hat c]
\end{equation}
for every $t>0$ and every label array, whether or not labels coincide
at $t$. For $\lambda\in\Xi$ and $\beta\in\mathcal A_q$ define
\[
 \mathsf p^t_{\lambda,\beta}(\hat c)=\sum_{\substack{\alpha\in
 \mathcal A_q\\ \alpha\geq\beta,\ \alpha\neq\beta}}
 (-1)^{|\alpha-\beta|}\binom\alpha\beta\,
 \mathsf h_{\alpha-\beta}(x_\lambda(t))\,\hat c_\alpha(\lambda),
\]
with $\mathsf h_\gamma$ as in \eqref{eq:rig-phase-row}. Its
coefficients are elements of $\mathbb F[t,t^{-1}]$. If $T$ is
supported in $S(t)$, has local order at most $q$ and satisfies
$\mathcal C^\nu T=T$, and $\hat c$ is a label array of $T$ as in
\eqref{eq:rig-label-array}, then
\begin{equation}\label{eq:rig-moving-phase-rows}
 \mathsf p^t_{\lambda,\beta}(\hat c)=0\qquad
 \text{for all }\lambda\in\Xi,\ \beta\in\mathcal A_q .
\end{equation}
Indeed, for $\lambda=\lambda(x)$ the left side is the coefficient of
$\partial^\beta\delta_x$ in $\mathcal C^\nu T-T$, because
$e^{2\pi i\nu s(x)}=1$; for all other labels
$\hat c_\alpha(\lambda)=0$.

Finally let the \emph{generic row} be
\[
 \mathsf r^{\rm gen}_{l,y,\beta}(\hat c)=\sum_{\mathbf k}
 \varkappa_{l,\mathbf k}
 \sum_{\substack{\lambda\in\Xi\\
 x_\lambda\equiv y+\mathsf w_{l,\mathbf k}}}\hat c_\beta(\lambda),
\]
where $\equiv$ denotes equality of Laurent vectors; the inner sum has
at most one term. Its coefficients are integers that do not depend on
$t$, and $\mathsf r^{t_*}_{l,y,\beta}=\mathsf r^{\rm gen}_{l,y,\beta}$
by \eqref{eq:rig-generic}.

\emph{Step 4: the row identity at $t_*$.} At $t=t_*$ label arrays are
the same as arrays on $S(t_*)\times\mathcal A_q$, the functionals
$\mathsf p^{t_*}_{\lambda,\beta}$ are the phase rows
\eqref{eq:rig-phase-row}, and the translation row
\eqref{eq:rig-translation-row} at a point of $\RR^2$ is either zero or
equal to $\mathsf r^{t_*}_{l,y,\beta}$ for a Laurent vector $y$ whose
value at $t_*$ is that point: if the row is nonzero, the point is
$x_\lambda(t_*)-\mathsf w_{l,\mathbf k}(t_*)$ for some $\lambda$ and
$\mathbf k$. Exactly as for \eqref{eq:rig-row-identity},
Lemmas~\ref{lem:formal-rigidity} and~\ref{lem:finite-rows} give, for
each $\alpha$ with $1\leq|\alpha|\leq q$, finitely many triples
$\mu=(l_\mu,y_\mu,\beta_\mu)$, $\mu\in\mathsf M_\alpha$, with Laurent
vectors $y_\mu$, finitely many pairs
$\mu'=(\lambda_{\mu'},\beta_{\mu'})$, $\mu'\in\mathsf M'_\alpha$, and
complex numbers $\vartheta^*_\mu$, $\vartheta^*_{\mu'}$ such that for
all label arrays
\[
 \hat c_\alpha(0)=\sum_{\mu\in\mathsf M_\alpha}\vartheta^*_\mu\,
 \mathsf r^{\rm gen}_\mu(\hat c)
 +\sum_{\mu'\in\mathsf M'_\alpha}\vartheta^*_{\mu'}\,
 \mathsf p^{t_*}_{\mu'}(\hat c),
\]
where $\mathsf r^{\rm gen}_\mu=\mathsf r^{\rm gen}_{l_\mu,y_\mu,
\beta_\mu}$ and $\mathsf p^t_{\mu'}=\mathsf p^t_{\lambda_{\mu'},
\beta_{\mu'}}$.

\emph{Step 5: the row identity for all parameters.} All the rows in
the last identity, and the coordinate functional
$\hat c\mapsto\hat c_\alpha(0)$, are supported in one finite subset of
$\Xi\times\mathcal A_q$ that does not depend on $t$. Restricted to
these columns, the rows $\mathsf r^{\rm gen}_\mu$ and
$\mathsf p^t_{\mu'}$ form a matrix with entries in
$\mathbb F[t,t^{-1}]$. The identity says that, after evaluation at
$t_*$, a coordinate vector lies in the row space of this matrix.
Evaluation at $t_*$ is an embedding of the field $\mathbb F(t)$ into
$\CC$, and the solvability of a linear system is not affected by an
extension of the field of scalars. Hence there are rational functions
$\vartheta_\mu,\vartheta_{\mu'}\in\mathbb F(t)$ with
\begin{equation}\label{eq:rig-moving-row-identity}
 \hat c_\alpha(0)=\sum_{\mu\in\mathsf M_\alpha}\vartheta_\mu(t)\,
 \mathsf r^{\rm gen}_\mu(\hat c)
 +\sum_{\mu'\in\mathsf M'_\alpha}\vartheta_{\mu'}(t)\,
 \mathsf p^t_{\mu'}(\hat c)
\end{equation}
as an identity over $\mathbb F(t)$, and therefore for all label arrays
and all real $t>0$ outside the finite set $\mathsf E'$ of poles of the
functions $\vartheta_\mu,\vartheta_{\mu'}$, $\mu\in\mathsf M_\alpha$,
$\mu'\in\mathsf M'_\alpha$, $1\leq|\alpha|\leq q$.

\emph{Step 6: incidences and denominators.} Fix
$\mu\in\mathsf M_\alpha$ and write $(l,y,\beta)=(l_\mu,y_\mu,
\beta_\mu)$. For $i\in\mathsf I^\pm$,
$\mathbf k\in\{0,\dots,m_l\}^{\mathsf I^\pm}$ and
$\iota\in\{1,2\}$ let
\[
 \mathsf v_{i,\mathbf k,\iota}(t)=\ell^t_{i,\iota}\bigl(y(t)
 +\mathsf w_{l,\mathbf k}(t)\bigr)\in\mathbb F[t,t^{-1}],
\]
so that $y(t)$ lies in the coset
$\Gamma_i(t)-\mathsf w_{l,\mathbf k}(t)$ if and only if
$\mathsf v_{i,\mathbf k,1}(t)$ and $\mathsf v_{i,\mathbf k,2}(t)$ are
integers. Call the pair $(i,\mathbf k)$ of \emph{type one} if both
functions are constant with integer values, and of \emph{type two}
otherwise. For a pair of type two choose
$\iota=\iota(i,\mathbf k)$ such that
$\mathsf v^\star_{i,\mathbf k}:=\mathsf v_{i,\mathbf k,\iota}$ is not
an integer constant. By \eqref{eq:rig-generic},
$\mathsf v^\star_{i,\mathbf k}(t_*)\notin\ZZ$. Consequently the
function
\[
 \mathfrak d_\mu(t)=\prod_{(i,\mathbf k)\ \text{of type two}}
 \bigl(e^{2\pi i\mathsf v^\star_{i,\mathbf k}(t)}-1\bigr)
\]
is real-analytic on $(0,\infty)$ and does not vanish at $t_*$. Its
zeros, and the zeros of the finitely many functions $\mathfrak d_\mu$
together, form a closed discrete subset $\mathsf E''$ of $(0,\infty)$.
We claim that
\begin{equation}\label{eq:rig-no-extra-incidence}
 \mathsf r^t_\mu=\mathsf r^{\rm gen}_\mu\qquad
 \text{for all }t\in(0,\infty)\setminus\mathsf E'' .
\end{equation}
Every term of $\mathsf r^{\rm gen}_\mu$ occurs in $\mathsf r^t_\mu$,
and the two functionals differ only by terms $\hat c_\beta(\lambda)$
with $x_\lambda(t)=y(t)+\mathsf w_{l,\mathbf k}(t)$ for some
$\mathbf k$. Let $\lambda$ and $\mathbf k$ satisfy this equation at
some $t\notin\mathsf E''$, and let $i$ be a class of $\lambda$. Then
$y(t)$ lies in $\Gamma_i(t)-\mathsf w_{l,\mathbf k}(t)$, so both
$\mathsf v_{i,\mathbf k,\iota}(t)$ are integers. Since
$t\notin\mathsf E''$, the pair $(i,\mathbf k)$ is of type one:
$\mathsf v_{i,\mathbf k,1}\equiv m$ and
$\mathsf v_{i,\mathbf k,2}\equiv n$ with integers $m,n$. Hence
$y+\mathsf w_{l,\mathbf k}\equiv x_{i,m,n}$ as Laurent vectors, and
$x_\lambda(t)=x_{i,m,n}(t)$. As the labels of the class $i$ are mapped
injectively, $\lambda=(i,m,n)$, and the term $\hat c_\beta(\lambda)$
occurs in $\mathsf r^{\rm gen}_\mu$ as well. This proves
\eqref{eq:rig-no-extra-incidence}.

\emph{Step 7: the tests.} Choose $\varepsilon_0\in(0,t_0)$ so small
that the set $\{t:0<|t-t_0|\leq\varepsilon_0\}$ contains no point of
$\mathsf E'\cup\mathsf E''$; this is possible because both sets are
closed and discrete in $(0,\infty)$. For $0<|t-t_0|<\varepsilon_0$ and
$\mu\in\mathsf M_\alpha$, $1\leq|\alpha|\leq q$, let $\psi_{\mu,t}$ be
the function $\psi$ of Lemma~\ref{lem:rig-cardinal-tests} for the
cosets $\Gamma_i(t)-\mathsf w_{l_\mu,\mathbf k}(t)$, written with the
forms $\ell^t_{i,1},\ell^t_{i,2}$ and the points
$-\mathsf w_{l_\mu,\mathbf k}(t)$, for the point $y_\mu(t)$, the order
$q$, the index $\beta_\mu$, and the choices
$\ell^\star=\ell^t_{i,\iota(i,\mathbf k)}$ of Step~6. These choices are
legitimate: for a pair of type one, $y_\mu(t)$ lies in the coset for
every $t$; for a pair of type two and $t\notin\mathsf E''$, the chosen
form is not integral at $y_\mu(t)+\mathsf w_{l_\mu,\mathbf k}(t)$, so
$y_\mu(t)$ does not lie in the coset. The quantity $\mathfrak d$ of
Lemma~\ref{lem:rig-cardinal-tests}(ii) is $|\mathfrak d_\mu(t)|$.
Define
\[
 f_{\mu,t}=\vartheta_\mu(t)\,\omega_{l_\mu}(t)\,\mathbb S\psi_{\mu,t} .
\]
The functions $f_{l,t}$ of the proposition are the $f_{\mu,t}$,
$\mu\in\mathsf M_\alpha$, $1\leq|\alpha|\leq q$, in some enumeration.
Assertion (i) follows as in the proof of
Theorem~\ref{thm:origin-certificate}, from
Lemmas~\ref{lem:rig-cardinal-tests}(ii)
and~\ref{lem:rig-mixture-family}.

\emph{Step 8: proof of (ii).} Let $0<|t-t_0|<\varepsilon_0$ and let
$T$ be as in (ii). Then $\supp T\subset\Sigma(t)\subset S(t)$. Every
point of $S(t)$ lies in some $\Gamma_i(t)$, $i\in\mathsf I^\pm$, and
the factor $(\chi_{w_{i,l}(t)}-1)^{m_l}$ of $\omega_l(t)$ vanishes on
$\Gamma_i(t)$ to order $m_l\geq q+1$, because
$w_{i,l}(t)\in\Gamma_i(t)^\vee$. This holds for every $t$, also when
the lattices $\Gamma_i(t)$ are not transverse, since it uses only a
class to which the point belongs and the factor of that class. Hence
$\omega_l(t)T=0$ by Lemma~\ref{lem:rig-local-pairing}(ii), and as in
the proof of Theorem~\ref{thm:origin-certificate}, using that $T$ is
even,
\[
 \mathsf D_l(t)T=\mathbb S\,\omega_l(t)\,(\mathbb S-I)T .
\]
Let $\hat c$ be a label array of $T$ as in
\eqref{eq:rig-label-array}. By \eqref{eq:rig-no-extra-incidence},
\eqref{eq:rig-moving-row-meaning} and
Lemma~\ref{lem:rig-cardinal-tests}(i),
\[
 \mathsf r^{\rm gen}_\mu(\hat c)=\mathsf r^t_\mu(\hat c)
 =\langle\mathsf D_{l_\mu}(t)T,\psi_{\mu,t}\rangle
 =\bigl\langle(\mathbb S-I)T,\,
 \omega_{l_\mu}(t)\,\mathbb S\psi_{\mu,t}\bigr\rangle .
\]
Since $\mathcal C^\nu T=T$, the phase rows vanish on $\hat c$ by
\eqref{eq:rig-moving-phase-rows}. Inserting this into
\eqref{eq:rig-moving-row-identity}, which is valid because
$t\notin\mathsf E'$, and using $\hat c_\alpha(0)=c_\alpha(0)$,
\[
 c_\alpha(0)=\sum_{\mu\in\mathsf M_\alpha}
 \langle(\mathbb S-I)T,f_{\mu,t}\rangle ,
\]
which implies \eqref{eq:rig-moving-certificate}.

\emph{Step 9: proof of (iii).} For $|t-t_0|\leq\varepsilon_0$ the
points $y_\mu(t)$, the vectors $\mathsf w_{l,\mathbf k}(t)$ and the
forms $\ell^t_{i,\iota}$ are bounded, since $t$ stays in a compact
subset of $(0,\infty)$. By Lemma~\ref{lem:rig-cardinal-tests}(ii),
each $\psi_{\mu,t}$ is a sum of a bounded number of functions
$\mathfrak m[p,z]$ with fixed $n$ and $n'$, with $\|z\|$ bounded
uniformly in $t$, and with
$\|p\|\leq C|\mathfrak d_\mu(t)|^{-(q+1)^2}$. By
Lemma~\ref{lem:rig-mixture-family}(i) and (ii) the same holds for
$\mathbb S\psi_{\mu,t}$, for
$f_{\mu,t}=\vartheta_\mu(t)\omega_{l_\mu}(t)\mathbb S\psi_{\mu,t}$ and
for $\mathbb Sf_{\mu,t}$, with
\[
 \|p\|\leq C\,|\vartheta_\mu(t)|\,|\mathfrak d_\mu(t)|^{-(q+1)^2};
\]
here we use that $\omega_l(t)$ is a sum of exponentials
$e^{2\pi iB(\mathsf w_{l,\mathbf k}(t),X)}$ with fixed integer
coefficients and bounded $\mathsf w_{l,\mathbf k}(t)$. The function
$\mathfrak d_\mu$ is real-analytic near $t_0$ and not identically
zero, so it vanishes at $t_0$ to some finite order $k_\mu\geq0$. Since
it has no other zero in $[t_0-\varepsilon_0,t_0+\varepsilon_0]$, the
quotient $\mathfrak d_\mu(t)/(t-t_0)^{k_\mu}$ is continuous and
nonvanishing on this interval, and
$|\mathfrak d_\mu(t)|\geq C^{-1}|t-t_0|^{k_\mu}$ there. Similarly the
rational function $\vartheta_\mu$ has at most a pole of finite order
$k'_\mu$ at $t_0$ and no other pole in the interval, so
$|\vartheta_\mu(t)|\leq C|t-t_0|^{-k'_\mu}$. With
$K=\max_\mu\bigl((q+1)^2k_\mu+k'_\mu\bigr)$ we obtain
$\|p\|\leq C|t-t_0|^{-K}$, and
Lemma~\ref{lem:rig-mixture-family}(iii) gives
\eqref{eq:rig-moving-envelope}. The exponent $K$ is determined by the
chosen row identities and the denominators, which were fixed in terms of
$\mathcal R_0$, $t_0$ and $q$ alone; the numbers $k_0$, $\varrho$ and $b$ affect only $C$.
\end{proof}

\subsection{The localization obstruction}
\label{subsec:rig-localization}

In this subsection the certificates are applied to the spectrum of an
admissible partition function. The distribution to which they are applied is a
finite Taylor expansion of the analytic evaluation
\eqref{eq:analytic-evaluation} about real lattice points.

\paragraph{Weight derivatives of the state measures.}
For $h>0$, an integer $k\geq0$ and a smooth function $\phi$ on $\RR$,
the function $h\mapsto\langle A_h,\phi\rangle
=\frac12(\phi(\sqrt h)+\phi(-\sqrt h))$ is smooth on $(0,\infty)$, and
by the chain rule its $k$-th derivative is a linear combination of the
numbers $\phi^{(l)}(\pm\sqrt h)$, $l\leq k$. This defines a
distribution $A_h^{(k)}$ on $\RR$ which is even, supported in
$\{\pm\sqrt h\}$ and of order at most $k$. For $h=0$ we put
\[
 A_0^{(k)}=\frac{k!}{(2k)!}\,\delta_0^{(2k)} .
\]
If $\phi$ is entire, then
$\frac12(\phi(\sqrt h)+\phi(-\sqrt h))
=\sum_k\phi^{(2k)}(0)h^k/(2k)!$ is an entire function of $h$, and
$\langle A_h^{(k)},\phi\rangle$ is its $k$-th derivative for every
$h\geq0$. Applying this in each variable, for every entire function
$f$ on $\CC^2$ and all $h,\bar h\geq0$,
\begin{equation}\label{eq:rig-jet-pairing}
 \bigl\langle A_h^{(i)}\otimes A_{\bar h}^{(l)},f\bigr\rangle
 =\partial_h^i\partial_{\bar h}^l\,\ev f(h,\bar h).
\end{equation}
For a real point $X=(P,Q)$ and $k\geq0$ let
\begin{equation}\label{eq:rig-fixed-spin-jet}
 \Upsilon_X^{(k)}=2^{-k}\sum_{i+l=k}\binom ki\,
 A_{P^2}^{(i)}\otimes A_{Q^2}^{(l)} .
\end{equation}
By \eqref{eq:rig-jet-pairing}, for entire $f$,
\begin{equation}\label{eq:rig-jet-derivative}
 \bigl\langle\Upsilon_X^{(k)},f\bigr\rangle
 =\frac{d^k}{d\zeta^k}\Big|_{\zeta=0}
 \ev f\bigl(P^2+\tfrac\zeta2,\,Q^2+\tfrac\zeta2\bigr).
\end{equation}
Thus $\Upsilon_X^{(k)}$ is the $k$-th derivative of the state measure
$A_h\otimes A_{\bar h}$ along the curve of constant spin through
$(h,\bar h)=(P^2,Q^2)$, parametrized by the dimension; the variable
$\zeta$ is the displacement called $d$ in the remark following
Lemma~\ref{lem:weight-derivative-bound}. It is an even distribution
supported in $\{(\pm P,\pm Q)\}$ and of local order at most $2k$.

\begin{lemma}[The Taylor distribution]\label{lem:rig-taylor-distribution}
Let $Z$ be an admissible partition function of gap $g$ at $c=1+12a$
with $0<a\leq1$. Let $N\geq1$ be an integer, $E_*>0$ and
$0\leq\varepsilon<g$. Let $\Sigma\subset\RR^2$ be invariant under the
reflections $P\mapsto-P$ and $Q\mapsto-Q$ and contain the nine points
with $P,Q\in\{0,\pm1\}$. Suppose that to every state $j$ with
$\Delta_j\leq E_*$ there is assigned a real point
$X_j=(P_j,Q_j)\in\Sigma$ with
\[
 P_j^2-Q_j^2=s_j,\qquad\bigl|\Delta_j-|X_j|^2\bigr|\leq\varepsilon .
\]
Put $\zeta_j=\Delta_j-|X_j|^2-a$ and define
\begin{equation}\label{eq:rig-taylor-distribution}
 T_a=\sum_{k,l\in\{0,1\}}(-1)^{k+l}\sum_{n=0}^N\frac{(-a)^n}{n!}\,
 \Upsilon^{(n)}_{(\sqrt k,\sqrt l)}
 +\sum_{j:\,\Delta_j\leq E_*}\ \sum_{n=0}^N\frac{\zeta_j^n}{n!}\,
 \Upsilon^{(n)}_{X_j}.
\end{equation}
\begin{enumerate}[(i)]
\item $T_a$ is an even distribution with finite support contained in
 $\Sigma$, of local order at most $2N$, and $\mathcal CT_a=T_a$.
\item The local expansion of $T_a$ at the origin is
 $\delta_0-\frac a4(\partial_P^2+\partial_Q^2)\delta_0$ plus terms of
 order at least four. In particular the coefficient of
 $\partial_P^2\delta_0$ in $T_a$ is $-a/4$, and no state contributes
 at the origin.
\item Let $f\in\mathcal G_{b_0}$ for some $b_0>0$, let $b>0$ and
 $M<\infty$, and suppose that
 \begin{equation}\label{eq:rig-test-envelope}
 |\partial^\alpha F(X)|\leq M\,e^{-b|\operatorname{Re}X|^2}\qquad
 \text{for }F\in\{f,\mathbb Sf\},\ |\alpha|\leq2N+2,\
 |\operatorname{Im}P|,|\operatorname{Im}Q|\leq1 .
 \end{equation}
 Then, with a constant $C_N$ depending only on $N$,
 \begin{equation}\label{eq:rig-defect-bound}
 \begin{split}
 \bigl|\langle(\mathbb S-I)T_a,f\rangle\bigr|\leq C_N\,e^{5b}M
 \Bigl[&\#\{j:\Delta_j\leq E_*\}\,(\varepsilon+a)^{N+1}+a^{N+1}\\
 &+\sum_{j:\,\Delta_j>E_*}e^{-b\Delta_j}\Bigr].
 \end{split}
 \end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
(i) Finitely many states have $\Delta_j\leq E_*$, so the sum is
finite. Each $\Upsilon^{(n)}_X$ with $n\leq N$ is even, of local order
at most $2N$, and supported in $\{(\pm P,\pm Q)\}$, which is contained
in $\Sigma$ for $X=X_j$ and for the vacuum points. For the phase, let
$X=(P,Q)$ be real with $P^2-Q^2\in\ZZ$ and let $f$ be entire. The
function $e^{2\pi i(P^2-Q^2)}f$ has the even evaluation
$e^{2\pi i(h-\bar h)}\ev f(h,\bar h)$, and along the curve
$(h,\bar h)=(P^2+\frac\zeta2,Q^2+\frac\zeta2)$ the factor
$e^{2\pi i(h-\bar h)}$ is identically one. By
\eqref{eq:rig-jet-derivative},
$\langle\mathcal C\Upsilon^{(n)}_X,f\rangle
=\langle\Upsilon^{(n)}_X,f\rangle$ for all entire $f$. Both sides are
distributions of finite order with support in the same finite set, and
polynomials realize arbitrary jets at finitely many points; hence
$\mathcal C\Upsilon^{(n)}_X=\Upsilon^{(n)}_X$. This applies to $X_j$,
because $P_j^2-Q_j^2=s_j\in\ZZ$, and to the vacuum points.

(ii) For a state, $|X_j|^2\geq\Delta_j-\varepsilon\geq g-\varepsilon>0$,
so none of the points $(\pm P_j,\pm Q_j)$ is the origin. The vacuum
terms with $(k,l)\neq(0,0)$ are supported away from the origin. For
$(k,l)=(0,0)$, by the definition of $A_0^{(i)}$,
\[
 \Upsilon^{(n)}_{(0,0)}=2^{-n}\sum_{i+l=n}\binom ni
 \frac{i!\,l!}{(2i)!\,(2l)!}\,
 \partial_P^{2i}\partial_Q^{2l}\delta_0 ,
\]
which is $\delta_0$ for $n=0$, equals
$\frac14(\partial_P^2+\partial_Q^2)\delta_0$ for $n=1$, and has order
$2n\geq4$ for $n\geq2$. The term $n=1$ carries the factor $-a$.

(iii) Let $F\in\{f,\mathbb Sf\}$; recall that
$\mathbb Sf\in\mathcal G_{4\pi^2/b_0}$. For a state with
$\Delta_j\leq E_*$ put
$\Phi_j(\zeta)=\ev F(P_j^2+\frac\zeta2,Q_j^2+\frac\zeta2)$, and for
$k,l\in\{0,1\}$ put
$\Psi_{k,l}(\zeta)=\ev F(k+\frac\zeta2,l+\frac\zeta2)$. These are
entire functions of $\zeta$. Since
$P_j^2=\frac12(|X_j|^2+s_j)$ and $h_j=\frac12(\Delta_j+s_j)$, and
similarly for $Q_j^2$ and $\bar h_j$,
\[
 \Phi_j(\zeta_j)=\ev F\bigl(h_j-\tfrac a2,\bar h_j-\tfrac a2\bigr),
 \qquad
 \Psi_{k,l}(-a)=\ev F\bigl(k-\tfrac a2,l-\tfrac a2\bigr),
\]
which are the shifted evaluations in \eqref{eq:analytic-evaluation}.
By \eqref{eq:rig-jet-derivative} and
\eqref{eq:rig-taylor-distribution}, $\langle T_a,F\rangle$ is the sum
of the Taylor polynomials of degree $N$ at $\zeta=0$ of these
functions, evaluated at $\zeta_j$ and at $-a$ respectively. Therefore
\begin{equation}\label{eq:rig-taylor-split}
 \begin{split}
 \mathcal L_a(F)-\langle T_a,F\rangle
 ={}&\sum_{k,l\in\{0,1\}}(-1)^{k+l}\,\mathrm{Rem}_N[\Psi_{k,l}](-a)
 +\sum_{j:\,\Delta_j\leq E_*}\mathrm{Rem}_N[\Phi_j](\zeta_j)\\
 &+\sum_{j:\,\Delta_j>E_*}
 \ev F\bigl(h_j-\tfrac a2,\bar h_j-\tfrac a2\bigr),
 \end{split}
\end{equation}
where $\mathrm{Rem}_N[\Phi](\zeta)$ denotes the Taylor remainder of
order $N$, and where the sums defining $\mathcal L_a(F)$ converge
absolutely by Lemma~\ref{lem:gaussian-identity}. For real $\zeta$,
\[
 \bigl|\mathrm{Rem}_N[\Phi](\zeta)\bigr|\leq
 \frac{|\zeta|^{N+1}}{(N+1)!}\,
 \max_{\zeta'\ \text{between}\ 0\ \text{and}\ \zeta}
 |\Phi^{(N+1)}(\zeta')| ,
\]
and $\Phi_j^{(N+1)}(\zeta')$ is the average, with weights
$2^{-N-1}\binom{N+1}i$, of the derivatives
$\partial_h^i\partial_{\bar h}^{N+1-i}\ev F$ at the point
$(P_j^2+\frac{\zeta'}2,Q_j^2+\frac{\zeta'}2)$. For $\zeta'$ between
$0$ and $\zeta_j$, the first coordinate of this point lies between
$P_j^2\geq0$ and $h_j-\frac a2\geq-\frac a2$, the second between
$Q_j^2\geq0$ and $\bar h_j-\frac a2\geq-\frac a2$, and their sum
$|X_j|^2+\zeta'$ is at least $-a$. Thus both arguments are real and at
least $-1$, whatever their size, and
Lemma~\ref{lem:weight-derivative-bound} applies with $k=N+1$ and
$\beta=b$, by \eqref{eq:rig-test-envelope}. That lemma is uniform over
all real arguments $h,\bar h\geq-1$. Hence no separate treatment is
needed for paths along which a squared momentum is small, vanishes,
becomes slightly negative, or passes from the range in which the proof
of that lemma uses the integral formula
\eqref{eq:sq-momentum-derivative} to the range in which it uses the
endpoint formula. We obtain
$|\Phi_j^{(N+1)}(\zeta')|\leq C_{N+1}Me^{4b}e^{b}$ and, since
$|\zeta_j|\leq\varepsilon+a$,
\[
 \bigl|\mathrm{Rem}_N[\Phi_j](\zeta_j)\bigr|
 \leq C_{N+1}Me^{5b}(\varepsilon+a)^{N+1}.
\]
The same argument for $\Psi_{k,l}$ on $[-a,0]$, where the arguments are
at least $-\frac a2$ and their sum is at least $-a$, gives
$|\mathrm{Rem}_N[\Psi_{k,l}](-a)|\leq C_{N+1}Me^{5b}a^{N+1}$. For the
states with $\Delta_j>E_*$,
Lemma~\ref{lem:weight-derivative-bound} without derivatives gives
\[
 \bigl|\ev F\bigl(h_j-\tfrac a2,\bar h_j-\tfrac a2\bigr)\bigr|
 \leq C_{N+1}Me^{4b}e^{-b(\Delta_j-a)}\leq C_{N+1}Me^{5b}e^{-b\Delta_j}.
\]
Hence $|\mathcal L_a(F)-\langle T_a,F\rangle|$ is bounded by
$4C_{N+1}e^{5b}M$ times the bracket in \eqref{eq:rig-defect-bound}.
Finally, by Lemma~\ref{lem:gaussian-identity},
$\mathcal L_a(\mathbb Sf)=\mathcal L_a(f)$, so
\[
 \langle(\mathbb S-I)T_a,f\rangle
 =\langle T_a,\mathbb Sf\rangle-\langle T_a,f\rangle
 =\bigl[\mathcal L_a(f)-\langle T_a,f\rangle\bigr]
 -\bigl[\mathcal L_a(\mathbb Sf)-\langle T_a,\mathbb Sf\rangle\bigr],
\]
and \eqref{eq:rig-defect-bound} follows with $C_N=8C_{N+1}$.
\end{proof}

The individual coefficients of $T_a$ in \eqref{eq:rig-local-form} need
not be bounded: they can be large when a nonzero squared momentum
$P_j^2$ or $Q_j^2$ is small. The proof does not use them;
Lemma~\ref{lem:weight-derivative-bound} controls their combined action
on entire test functions.

\begin{theorem}[No localization on a fixed union of circle lattices]
\label{thm:fixed-localization}
Let $0<g\leq\frac12$. Let $\mathcal R_0\subset(0,\infty)$ be a finite
set of radii with $1\in\mathcal R_0$ and $1/r\in\mathcal R_0$ for all
$r\in\mathcal R_0$, so that
$\Sigma_0=\bigcup_{r\in\mathcal R_0}\Lambda_r$ contains $\Lambda_1$ and
is invariant under the reflection exchanging $u$ and $v$. Let $p>0$,
$C<\infty$, $A<\infty$ and $u_*>1/(2\pi)$. There is no admissible
sequence of gap $g$, with parameters $a\to0$ and $L=\log(1/a)$, such
that for every $a$ in the sequence every state $j$ with
$\Delta_j\leq u_*L$ has a real point $X_j=(P_j,Q_j)\in\Sigma_0$ with
\begin{equation}\label{eq:rig-localization-hypothesis}
 P_j^2-Q_j^2=s_j\qquad\text{and}\qquad
 \bigl|\Delta_j-|X_j|^2\bigr|\leq C\,a^pL^A .
\end{equation}
The assignment $j\mapsto X_j$ is not required to be injective, and no
relation between the number of states assigned to a point and any
multiplicity is assumed.
\end{theorem}
\begin{proof}
Suppose that such a sequence exists. Fix $b$ with $1/u_*<b<2\pi$ and
an integer $N\geq1$ with $(N+1)\min(p,1)>1$, and put $q=2N$. Let
$f_1,\dots,f_n$ and $C_{\Sigma_0,q}$ be given by
Theorem~\ref{thm:origin-certificate} for $\Sigma=\Sigma_0$. Since
$f_l$ and $\mathbb Sf_l$ lie in $\mathcal G_{2\pi}$, the bound
\eqref{eq:mixture-envelope} with $b'=b$ provides a constant $M$ such
that \eqref{eq:rig-test-envelope} holds for every $f=f_l$.

Let $a$ be an element of the sequence, so small that $a\leq g^2/4$,
$2\pi a<b$, $u_*L\geq1$ and $\varepsilon_a:=Ca^pL^A<g$. The set
$\Sigma_0$ is invariant under $Q\mapsto-Q$, which exchanges $u$ and
$v$, and under $X\mapsto-X$, hence under both reflections, and it
contains the points with $P,Q\in\{0,\pm1\}$, which lie in
$\Lambda_1$. Let $T_a$ be the distribution
\eqref{eq:rig-taylor-distribution} for $\Sigma=\Sigma_0$,
$E_*=u_*L$, $\varepsilon=\varepsilon_a$ and the points $X_j$ of the
hypothesis. By Lemma~\ref{lem:rig-taylor-distribution}(i), $T_a$ is an
even tempered distribution supported in $\Sigma_0$, of local order at
most $q$, with $\mathcal CT_a=T_a$. By
Lemma~\ref{lem:rig-taylor-distribution}(ii) and
\eqref{eq:rig-certificate},
\[
 \frac a4\leq C_{\Sigma_0,q}\max_l
 \bigl|\langle(\mathbb S-I)T_a,f_l\rangle\bigr| .
\]
We bound the right side by \eqref{eq:rig-defect-bound}. By
\eqref{eq:count-bound}, the number of states with
$\Delta_j\leq u_*L$ is at most
$A_gu_*L\,e^{2\pi au_*L}\leq2A_gu_*L$ for small $a$. With
$A_+=\max(A,0)$ we have
$\varepsilon_a+a\leq(C+1)a^{\min(p,1)}L^{A_+}$. By
\eqref{eq:thermal-tail} with $\beta=b$ and $E_0=u_*L$,
\[
 \sum_{\Delta_j>u_*L}e^{-b\Delta_j}\leq A_gb\Bigl(
 \frac{u_*L}{b-2\pi a}+\frac1{(b-2\pi a)^2}\Bigr)
 e^{-(b-2\pi a)u_*L}=O\bigl(L\,a^{bu_*}\bigr),
\]
because $e^{2\pi au_*L}\to1$. Altogether
\begin{equation}\label{eq:rig-fixed-defect}
 \max_l\bigl|\langle(\mathbb S-I)T_a,f_l\rangle\bigr|
 \leq C'\Bigl[L^{1+(N+1)A_+}\,a^{(N+1)\min(p,1)}+a^{N+1}
 +L\,a^{bu_*}\Bigr]
\end{equation}
with $C'$ independent of $a$. Since $(N+1)\min(p,1)>1$, $N+1>1$ and
$bu_*>1$, the right side is $o(a)$. This contradicts the lower bound
$a/4$ for small $a$.
\end{proof}

\begin{theorem}[No localization with one moving radius]
\label{thm:moving-localization}
Let $0<g\leq\frac12$. Let $\mathcal R_0\subset(0,\infty)$ be a finite
set of radii with $1\in\mathcal R_0$ and $1/r\in\mathcal R_0$ for all
$r\in\mathcal R_0$, and let
$\Sigma_0=\bigcup_{r\in\mathcal R_0}\Lambda_r$. Let
$I\subset(0,\infty)$ be a compact interval, and let $p>0$,
$C<\infty$, $A<\infty$ and $u_*>1/(2\pi)$. There is no admissible
sequence of gap $g$, with parameters $a\to0$ and $L=\log(1/a)$,
together with radii $t_a\in I$, such that for every $a$ in the
sequence every state $j$ with $\Delta_j\leq u_*L$ has a real point
\[
 X_j=(P_j,Q_j)\in\Sigma_0\cup\Lambda_{t_a}\cup\Lambda_{1/t_a}
\]
with
\[
 P_j^2-Q_j^2=s_j\qquad\text{and}\qquad
 \bigl|\Delta_j-|X_j|^2\bigr|\leq C\,a^pL^A .
\]
The assignment $j\mapsto X_j$ is not required to be injective, no
relation between the number of states assigned to a point and any
multiplicity is assumed, and
$\Sigma_0\cup\Lambda_{t_a}\cup\Lambda_{1/t_a}$ is not required to be
the support of an element of $\mathcal B_g$.
\end{theorem}
\begin{proof}
Suppose that such a sequence exists. Passing to a subsequence we may
assume $t_a\to t_0\in I$. We use the notation $\Sigma(t)$ of
Proposition~\ref{prop:moving-certificate}.

\emph{Choice of parameters.} Fix $b$ with $1/u_*<b<2\pi$ and an
integer $N\geq1$ with $(N+1)\min(p,1)>1$, and put $q=2N$. Apply
Proposition~\ref{prop:moving-certificate} to $\mathcal R_0$, $t_0$ and
$q$; this provides $\varepsilon_0$, $n$, the exponent $K$ and the
tests $f_{l,t}$. Then apply part (iii) of the proposition with
$k_0=2N+2$, the strip $|\operatorname{Im}X|\leq\sqrt2$ and the chosen
$b$; this provides the constant $C_1$ in
\eqref{eq:rig-moving-envelope}. Finally fix $\eta>0$ so small that
\begin{equation}\label{eq:rig-eta}
 K\eta<\tfrac12\min\bigl\{(N+1)\min(p,1)-1,\ bu_*-1\bigr\}.
\end{equation}
Thus $N$, $q$, the number of derivatives and $K$ are fixed before
$\eta$. Either $|t_a-t_0|\geq a^\eta$ for infinitely many $a$ in the
sequence, or $|t_a-t_0|<a^\eta$ for all sufficiently small $a$.

\emph{First case: $|t_a-t_0|\geq a^\eta$ along a subsequence.}
Restrict to this subsequence and to $a$ so small that
$|t_a-t_0|<\varepsilon_0$, $a\leq g^2/4$, $2\pi a<b$, $u_*L\geq1$ and
$\varepsilon_a:=Ca^pL^A<g$. The set $\Sigma(t_a)$ is invariant under
both reflections and contains $\Lambda_1$. Let $T_a$ be the
distribution \eqref{eq:rig-taylor-distribution} for
$\Sigma=\Sigma(t_a)$, $E_*=u_*L$, $\varepsilon=\varepsilon_a$ and the
given points $X_j$. By Lemma~\ref{lem:rig-taylor-distribution}(i) and
(ii) and by \eqref{eq:rig-moving-certificate} with $t=t_a$,
\[
 \frac a4\leq\sum_{l=1}^n
 \bigl|\langle(\mathbb S-I)T_a,f_{l,t_a}\rangle\bigr| .
\]
By \eqref{eq:rig-moving-envelope}, each $f=f_{l,t_a}$ satisfies
\eqref{eq:rig-test-envelope} with
$M=C_1|t_a-t_0|^{-K}\leq C_1a^{-K\eta}$; here
$|\operatorname{Im}P|,|\operatorname{Im}Q|\leq1$ implies
$|\operatorname{Im}X|\leq\sqrt2$. The bracket in
\eqref{eq:rig-defect-bound} is estimated exactly as in the proof of
\eqref{eq:rig-fixed-defect}, which used only \eqref{eq:count-bound},
\eqref{eq:thermal-tail} and the bound on $\varepsilon_a$. Hence
\[
 \frac a4\leq C'\,a^{-K\eta}\Bigl[L^{1+(N+1)A_+}\,a^{(N+1)\min(p,1)}
 +a^{N+1}+L\,a^{bu_*}\Bigr],\qquad A_+=\max(A,0),
\]
with $C'$ independent of $a$. By \eqref{eq:rig-eta} and
$N+1\geq(N+1)\min(p,1)$, each of the exponents
$(N+1)\min(p,1)-K\eta$, $N+1-K\eta$ and $bu_*-K\eta$ is greater than
one, so the right side is $o(a)$, a contradiction.

\emph{Second case: $|t_a-t_0|<a^\eta$ for all small $a$.} We move the
points on the variable lattices to the limiting radius. Let $j$ be a
state with $\Delta_j\leq u_*L$. If $X_j\in\Sigma_0$ put $X_j'=X_j$. If
$X_j\notin\Sigma_0$ and $X_j\in\Lambda_{t_a}$, with light-cone
coordinates $(mt_a,n/t_a)$, let $X_j'\in\Lambda_{t_0}$ be the point
$(mt_0,n/t_0)$; otherwise $X_j\in\Lambda_{1/t_a}$ has coordinates
$(m/t_a,nt_a)$, and we let $X_j'\in\Lambda_{1/t_0}$ be the point
$(m/t_0,nt_0)$. In each case $X_j'$ has the same spin $mn=s_j$ as
$X_j$, and $(m,n)\neq(0,0)$ in the last two cases. The dimension of
the point with charges $(m,n)$ at radius $t$ is
$E(t)=\frac12(m^2t^2+n^2/t^2)$, and
$|\partial_tE(t)|=|m^2t-n^2/t^3|\leq(2/t)\,E(t)$. Integrating
$\partial_t\log E$ over the interval between $t_a$ and $t_0$, which is
contained in $I$, gives
\[
 \bigl||X_j'|^2-|X_j|^2\bigr|\leq|X_j|^2
 \bigl(e^{2|t_a-t_0|/\min I}-1\bigr)\leq C_2\,|X_j|^2\,a^\eta
 \leq C_2\,(u_*L+1)\,a^\eta
\]
for small $a$, where we used
$|X_j|^2\leq\Delta_j+Ca^pL^A\leq u_*L+1$; the same bound holds for the
points of $\Lambda_{1/t}$, with $m$ and $n$ exchanged. Therefore every
state with $\Delta_j\leq u_*L$ has a real point
$X_j'=(P_j',Q_j')$ with
\[
 X_j'\in\Sigma_0':=\Sigma_0\cup\Lambda_{t_0}\cup\Lambda_{1/t_0},
 \qquad P_j'^2-Q_j'^2=s_j,\qquad
 \bigl|\Delta_j-|X_j'|^2\bigr|\leq C_3\,a^{\min(p,\eta)}L^{\max(A,1)} .
\]
The set $\mathcal R_0\cup\{t_0,1/t_0\}$ is finite, contains $1$ and is
closed under inversion. The elements of the sequence with $a$ below a
suitable threshold form an admissible sequence of gap $g$, and for it
the last display contradicts Theorem~\ref{thm:fixed-localization},
applied with $\mathcal R_0\cup\{t_0,1/t_0\}$, the exponent
$\min(p,\eta)>0$, the constants $C_3$ and $\max(A,1)$, and the same
$g$ and $u_*$.
\end{proof}

By Lemma~\ref{lem:grid-circles}, the hypothesis on $\mathcal R_0$ in
Theorems~\ref{thm:fixed-localization}
and~\ref{thm:moving-localization} is satisfied by the set
$\mathcal R_d$ of radii that carries the integral-spin points of the
light-cone grid $d^{-1}\Lambda_1$. The two theorems are equivalent to
a statement about individual partition functions.

\begin{corollary}[Uniform form]\label{cor:rig-threshold}
Let $g$, $\mathcal R_0$, $\Sigma_0$, $I$, $p$, $C$, $A$ and $u_*$ be
as in Theorem~\ref{thm:moving-localization}. There is $a_*>0$ with the
following property. If $Z$ is an admissible partition function of gap
$g$ at $c=1+12a$ with $0<a<a_*$, $L=\log(1/a)$, and $t\in I$, then
there is a state $j$ with $\Delta_j\leq u_*L$ for which no real point
$X_j=(P_j,Q_j)\in\Sigma_0\cup\Lambda_t\cup\Lambda_{1/t}$ satisfies
\[
 P_j^2-Q_j^2=s_j\qquad\text{and}\qquad
 \bigl|\Delta_j-|X_j|^2\bigr|\leq C\,a^pL^A .
\]
The same holds, with a threshold depending only on $g$,
$\mathcal R_0$, $p$, $C$, $A$ and $u_*$, when the lattices $\Lambda_t$
and $\Lambda_{1/t}$ are omitted.
\end{corollary}
\begin{proof}
If no such $a_*$ existed, there would be admissible partition
functions $Z_n$ of gap $g$ at $c_n=1+12a_n$, with $a_n$ strictly
decreasing to zero, and radii $t_n\in I$, for which every state with
$\Delta_j\leq u_*\log(1/a_n)$ has a point as described. This is an
admissible sequence excluded by Theorem~\ref{thm:moving-localization}.
The statement without the variable lattices follows in the same way
from Theorem~\ref{thm:fixed-localization}. Conversely, the corollary
implies the two theorems, because an admissible sequence contains
partition functions with $a<a_*$.
\end{proof}