\section{Limits with two half-integer radii and the proof of
Theorem~\ref{thm:main}}
\label{sec:collision}

By Theorem~\ref{thm:half-integer-limits}, every limit of an admissible
sequence of gap $g$ is a measure $\sigma(R,T)$ in which both radii are
half-integers. In this section we exclude these finitely many limits and
complete the proof of Theorem~\ref{thm:main}. The kernels of
Section~\ref{sec:kernels} cannot be used at the limiting pair $(R,T)$
itself, because their Fourier centres and their zeros collide there. The
argument has three parts.
\begin{enumerate}[(a)]
\item A quantitative form of the rigidity theorem on the common rational
 grid (Proposition~\ref{prop:grid-projection}) shows that the physical
 Fock measure $\mu_a$ agrees, on fixed compact sets and up to
 $a+b_a^{n+1}$ for every fixed $n$, times a power of $\log(1/b_a)$,
 with a combination of at most four
 circle measures reconstructed from finitely many of its own states
 (Lemma~\ref{lem:normal-form}).
\item Positivity, the vacuum term and a moment argument turn this
 combination into one comparison measure $\sigma(R,t_a)$ with an exact
 half-integer $R$ and a radius $t_a\to T$, which reproduces the states of
 every fixed cluster up to a power of $b_a$
 (Theorem~\ref{thm:comparison}); moreover $|t_a-T|$ is comparable to the
 four-scalar displacement $\delta_a$.
\item At the radius $t_a\neq T$ the kernels of
 Section~\ref{sec:kernels} are available again. Their centres and zeros
 are separated on the scale $|t_a-T|^2$
 (Lemma~\ref{lem:quadratic-collisions}), and Gaussians of this width give
 peak budgets that lose only powers of $|t_a-T|^{-1}$
 (Lemma~\ref{lem:resolved-budget}). By
 Theorem~\ref{thm:four-scalar-slow} such powers are smaller than every
 negative power of $a$, and the localization excluded by
 Theorem~\ref{thm:moving-localization} follows.
\end{enumerate}

\subsection{Conventions}\label{subsec:col-conventions}

Throughout this section $0<g\leq\frac12$ is fixed,
$R_{\max}=(2g)^{-1/2}$, and $d$, $E_d$, $\eta_d$ are the numbers of
\eqref{eq:ker-grid-denominator}. Points of $\RR^2$ are written $[u,v]$ in
light-cone coordinates and $(P,Q)$ in momentum coordinates, as in
Section~\ref{subsec:ker-conventions}. For an admissible partition
function of gap $g$ at $c=1+12a$ with $a<a_1(g)$,
\[
 \delta_a,\qquad b_a=\sqrt a+\delta_a,\qquad H_a=\log(1/b_a)
\]
are the quantities of Lemma~\ref{lem:four-scalar-cluster} and
Corollary~\ref{cor:four-scalar-localization}; thus $a\leq b_a^2$, and
$b_a\to0$ uniformly as $a\to0$.

\paragraph{The grid.}
We use the light-cone grid of denominator $d$, which is the hyperbolic
grid $d^{-1}\Lambda_1$ of Section~\ref{subsec:rig-conventions},
\begin{equation}\label{eq:col-grid}
\begin{gathered}
 \Gamma=d^{-1}\Lambda_1=\bigl\{[m/d,\,n/d]:m,n\in\ZZ\bigr\},\qquad
 \Gamma^\vee=d\Lambda_1=\bigl\{[md,\,nd]:m,n\in\ZZ\bigr\},\\
 \mathsf b_1=[d,0],\qquad\mathsf b_2=[0,d].
\end{gathered}
\end{equation}
The vectors $\mathsf b_1,\mathsf b_2$ form a basis of $\Gamma^\vee$. In
momentum coordinates $[m/d,n/d]=\bigl(\frac{m+n}{2d},\frac{m-n}{2d}\bigr)$,
so $\Gamma\subset(2d)^{-1}\ZZ^2$, and a nonzero momentum coordinate of a
point of $\Gamma$ has modulus at least $(2d)^{-1}$. The reflections
$P\mapsto-P$ and $Q\mapsto-Q$ are the maps $[u,v]\mapsto[-v,-u]$ and
$[u,v]\mapsto[v,u]$; they preserve $\Gamma$ and $\Gamma^\vee$. This is the
grid of Corollary~\ref{cor:four-scalar-localization}: the point
associated there with a state is $[m/d,n/d]\in\Gamma$. Let
\[
 \mathcal R_d=\{k/d:\ k\ \text{a positive divisor of}\ d^2\},\qquad
 \mathcal R_d^+=\{r\in\mathcal R_d:r\geq1\},\qquad
 \Sigma_d=\bigcup_{r\in\mathcal R_d}\Lambda_r\subset\Gamma
\]
be the radii and the union of circle lattices of
Lemma~\ref{lem:grid-circles} and \eqref{eq:ker-grid-circles}; we have
$\mathcal R_d\subset[1/d,d]$. Every half-integer $r$ with
$1\leq r\leq R_{\max}$ belongs to $\mathcal R_d^+$: if $r=K/2$, then
$2\leq K\leq N_g$ and $r=k/d$ with $k=KN_g!$, which divides
$d^2=4(N_g!)^2$ because $K$ divides $N_g!$.

\paragraph{States and windows.}
A \emph{state} is a pair $(\Delta,s)\in\RR\times\ZZ$ with
$\Delta\geq|s|$; its weights are $h=\frac12(\Delta+s)$ and
$\bar h=\frac12(\Delta-s)$, its \emph{state measure} is
$A_h\otimes A_{\bar h}$, and its physical points are
$(\pm\sqrt h,\pm\sqrt{\bar h})$. A \emph{family of states} is a
collection of states, with repetitions allowed, having finitely many
members below every dimension; its measure is the sum of the state
measures. Thus $\rho_a$ is the measure of the family
$(\Delta_j,s_j)_{j\in\mathcal J}$. For $(E_0,s_0)\in\RR\times\ZZ$ and
$\eta>0$ the \emph{window}
\begin{equation}\label{eq:col-window}
 \mathcal W_\eta(E_0,s_0)=\bigl\{X\in\RR^2:\ |P^2-Q^2-s_0|<\tfrac12,\ \
 \bigl||X|^2-E_0\bigr|<\eta\bigr\}
\end{equation}
is a bounded open set, invariant under the two reflections. A state is
supported in it if and only if its spin is $s_0$ and its dimension lies
in $(E_0-\eta,E_0+\eta)$. For a family of states $\mathfrak S$ we call
the window $\mathcal W_\eta(E_0,s_0)$ \emph{isolated} if no state of
$\mathfrak S$ with spin $s_0$ has dimension in
$[E_0-3\eta,E_0+3\eta]\setminus\{E_0\}$. Since a family has finitely many
states below every dimension, every pair $(E_0,s_0)$ has isolated
windows, and finitely many pairs have a common $\eta$.

\paragraph{Weight tests.}
A \emph{weight test} is a function on $\RR^2$ of the form
\begin{equation}\label{eq:col-weight-test}
 \phi(P,Q)=\Theta(P^2,Q^2),\qquad\Theta\in C_c^\infty(\RR^2).
\end{equation}
It is smooth, compactly supported and even, and
$\langle A_h\otimes A_{\bar h},\phi\rangle=\Theta(h,\bar h)$ for
$h,\bar h\geq0$. All compactly supported test functions used below are
of this form. We fix once and for all two cutoffs
\[
 \vartheta^{\rm sp}\in C_c^\infty\bigl((-\tfrac12,\tfrac12)\bigr),\quad
 \vartheta^{\rm sp}=1\ \text{on}\ [-\tfrac14,\tfrac14];\qquad
 \vartheta\in C_c^\infty\bigl((-2,2)\bigr),\quad\vartheta=1\ \text{on}\
 [-1,1].
\]
For $(E_0,s_0)\in\RR\times\ZZ$, $\eta>0$ and a smooth function
$\varphi$ on a neighbourhood of $[E_0-2\eta,E_0+2\eta]$, the function
\begin{equation}\label{eq:col-window-test}
 \phi(X)=\vartheta^{\rm sp}\bigl(P^2-Q^2-s_0\bigr)\,
 \vartheta\Bigl(\frac{|X|^2-E_0}{\eta}\Bigr)\,\varphi\bigl(|X|^2\bigr)
\end{equation}
is a weight test, with
$\Theta(h,\bar h)=\vartheta^{\rm sp}(h-\bar h-s_0)\,
\vartheta((h+\bar h-E_0)/\eta)\,\varphi(h+\bar h)$. For a state
$(\Delta,s)$,
\begin{equation}\label{eq:col-window-pairing}
 \bigl\langle A_h\otimes A_{\bar h},\phi\bigr\rangle
 =\one_{\{s=s_0\}}\,\vartheta\Bigl(\frac{\Delta-E_0}{\eta}\Bigr)\,
 \varphi(\Delta),
\end{equation}
because the spin is an integer. In particular the pairing equals
$\varphi(\Delta)$ if the state is supported in $\mathcal W_\eta(E_0,s_0)$
and vanishes if its spin differs from $s_0$ or
$|\Delta-E_0|\geq2\eta$.

\paragraph{The states of the comparison measures.}
For $t>0$ and $(m,n)\in\ZZ^2$ let
\[
 E_{m,n}(t)=\tfrac12\bigl(m^2t^2+n^2/t^2\bigr)
\]
be the squared norm of the point $[mt,n/t]\in\Lambda_t$, whose spin is
$mn$. A \emph{class} is a set $\{(m,n),(-m,-n)\}$ with $(m,n)\neq(0,0)$.

\begin{lemma}[States of $\sigma(R,t)$]\label{lem:col-states}
Let $t>0$.
\begin{enumerate}[(i)]
\item $\frac12\rhoF_t-\frac12\delta_0$ is the measure of the family that
 contains, for each class $\{\pm(m,n)\}$, one state of dimension
 $E_{m,n}(t)$ and spin $mn$.
\item Let $R\geq1$ be a half-integer. Then $\sigma(R,t)$ is the measure
 of the family $\mathfrak S(R,t)$ consisting of the following states.
 \emph{Fixed states:} one state $(E_{m,n}(R),mn)$ for each class other
 than the class of $(0,2R)$, and the two states $(1,1)$ and $(1,-1)$.
 \emph{Moving states:} one state $(E_{m,n}(t),mn)$ for each class.
 In particular $\sigma(R,t)$ is a positive measure, its states have
 integer spins, and $\sigma(R,t)(\{x\})$ is $\frac14$ times the number
 of pairs consisting of a state of $\mathfrak S(R,t)$ and one of its
 four sign patterns whose physical point is $x$.
\item Let $I\subset(0,\infty)$ be a compact interval and $E<\infty$.
 Only finitely many classes satisfy $E_{m,n}(t)\leq E$ for some
 $t\in I$, and there is $C=C(I,E)$ such that
 $|E_{m,n}(t)-E_{m,n}(t')|\leq C|t-t'|$ for these classes and all
 $t,t'\in I$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i) By \eqref{eq:circle-measures}, and since $[n/t,mt]$ runs over
$\Lambda_{1/t}$ when $(m,n)$ runs over $\ZZ^2$,
\[
 \tfrac12\rhoF_t=\tfrac14\sum_{(m,n)\in\ZZ^2}
 \bigl(\delta_{[mt,n/t]}+\delta_{[n/t,mt]}\bigr)
 =\tfrac12\delta_0+\sum_{\{\pm(m,n)\}}\tfrac14\bigl(\delta_{[mt,n/t]}
 +\delta_{-[mt,n/t]}+\delta_{[n/t,mt]}+\delta_{-[n/t,mt]}\bigr).
\]
If $[u,v]=(P,Q)$, the four points $\pm[u,v]$, $\pm[v,u]$ are the points
$(\pm P,\pm Q)$ with all combinations of signs. Hence the term of a class
is the state measure of the state with the weights of $[mt,n/t]$, that
is, with dimension $E_{m,n}(t)$ and spin $mn$.

(ii) By (i) and \eqref{eq:sigma-R-t}, $\frac12\rhoF_R+\frac12\rhoF_t$ is
$\delta_0$ plus the measure of the family of all states
$(E_{m,n}(R),mn)$ and $(E_{m,n}(t),mn)$. By
\eqref{eq:physical-measures}, $\nu_{\vac}$ is $\delta_0$ minus the state
measures of $(1,1)$ and $(1,-1)$ plus the state measure of $(2,0)$.
Since $2R$ is an integer, the class of $(0,2R)$ contributes the state
$(E_{0,2R}(R),0)=(2,0)$, which cancels the last term. The remaining
assertions are immediate.

(iii) If $E_{m,n}(t)\leq E$ with $t\in I$, then
$|m|\leq\sqrt{2E}/\min I$ and $|n|\leq\sqrt{2E}\max I$. For such classes
$\partial_tE_{m,n}(t)=m^2t-n^2/t^3$ is bounded on $I$.
\end{proof}

\begin{lemma}[Windows]\label{lem:col-windows}
Let $R,T\geq1$ be half-integers, let $(E_0,s_0)\in\RR\times\ZZ$, and let
$\mathcal W_\eta(E_0,s_0)$ be isolated for $\mathfrak S(R,T)$. Let $N_0$
be the number of states of $\mathfrak S(R,T)$ with dimension $E_0$ and
spin $s_0$.
\begin{enumerate}[(i)]
\item There is $\varepsilon_1>0$ such that for $|t-T|<\varepsilon_1$ the
 states of $\mathfrak S(R,t)$ with spin $s_0$ and dimension in
 $(E_0-2\eta,E_0+2\eta)$ are the fixed states equal to $(E_0,s_0)$ and
 the moving states of the classes with $mn=s_0$ and $E_{m,n}(T)=E_0$.
 Their number is $N_0$, and their dimensions lie in
 $(E_0-\eta,E_0+\eta)$.
\item If an admissible sequence of gap $g$ converges to $\sigma(R,T)$,
 then for all sufficiently small $a$ in the sequence exactly $N_0$
 states of $\rho_a$ have spin $s_0$ and dimension in
 $(E_0-2\eta,E_0+2\eta)$; their dimensions lie in
 $(E_0-\eta,E_0+\eta)$ and converge to $E_0$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i) By Lemma~\ref{lem:col-states}(ii) the family $\mathfrak S(R,T)$
consists of the fixed states and of the moving states at $t=T$, so $N_0$
is the number of states described in (i). The fixed states do not depend
on $t$; those with spin $s_0$ and dimension in $(E_0-2\eta,E_0+2\eta)$
are equal to $(E_0,s_0)$, by isolation. By
Lemma~\ref{lem:col-states}(iii), finitely many classes have $mn=s_0$ and
$E_{m,n}(t)\leq E_0+2\eta$ for some $t\in[\frac12T,2T]$, and for a
suitable $\varepsilon_1\leq\frac12T$ they satisfy $|E_{m,n}(t)-E_{m,n}(T)|<\eta$ when
$|t-T|<\varepsilon_1$. For such a class either $E_{m,n}(T)=E_0$, or
$|E_{m,n}(T)-E_0|>3\eta$ by isolation, and then
$|E_{m,n}(t)-E_0|>2\eta$.

(ii) The windows $\mathcal W_\eta(E_0,s_0)$ and
$\mathcal W_{2\eta}(E_0,s_0)$ are bounded, open and invariant under the
reflections. A boundary point has spin $s_0\pm\frac12$, or it has
integer spin $s_0$ and squared norm $E_0\pm\eta$, respectively
$E_0\pm2\eta$; by isolation and the integrality of the spins of
$\sigma(R,T)$, the boundaries do not meet $\supp\sigma(R,T)$. Both
windows contain the same $N_0$ states of $\sigma(R,T)$, and
Corollary~\ref{cor:cluster-stabilization} gives the claim.
\end{proof}

\begin{remark}\label{rem:col-parity}
The parity hypothesis (iv) of Definition~\ref{def:admissible} is not
used in this section. The only symmetries used are the two reflections
$P\mapsto-P$ and $Q\mapsto-Q$, under which $\rho_a$ and $\mu_a$ are
invariant by construction. In particular the spaces of
Section~\ref{subsec:col-projection} are defined by evenness alone, and
clusters of opposite spins are treated independently.
\end{remark}

\subsection{A quantitative projection onto the invariant distributions}
\label{subsec:col-projection}

Corollary~\ref{cor:rig-grid-invariants} describes the even tempered
distributions supported in $\Gamma$ that are invariant under $\mathbb S$
and $\mathcal C$. We need a stable version: a distribution on $\Gamma$
that is exactly invariant under $\mathcal C$ and almost invariant under
$\mathbb S$, as measured by finitely many tests per unit area in a
window of logarithmic size, is close to an invariant one on every fixed
compact set. In this subsection $d\geq1$ is an arbitrary integer and
$\Gamma$, $\mathsf b_1$, $\mathsf b_2$ are given by
\eqref{eq:col-grid}; $\mathsf T$ and $\mathsf U$ denote distributions.

\paragraph{Polynomial combs.}
Let $q\geq1$ be an integer. Let $\mathsf W_q$ be the space of
distributions
\begin{equation}\label{eq:col-comb}
 \mathsf U=\sum_{\mathfrak c\in\Gamma/\Gamma^\vee}\ \sum_{|\alpha|\leq q}\
 \sum_{z\in\mathfrak c}p_{\mathfrak c,\alpha}(z)\,\partial^\alpha\delta_z
\end{equation}
that are even, where each $p_{\mathfrak c,\alpha}$ is a polynomial on
$\RR^2$ of degree at most $q$ in each of the light-cone coordinates $u$
and $v$. Here $\partial^\alpha=\partial_P^{\alpha_1}\partial_Q^{\alpha_2}$
as in Section~\ref{sec:rigidity}. A polynomial is determined by its
values on a coset of $\Gamma^\vee$, so the polynomials in
\eqref{eq:col-comb} are determined by $\mathsf U$, and $\mathsf W_q$ is
a finite-dimensional space of tempered distributions of local order at
most $q$ supported in $\Gamma$. We write
$c^{\mathsf U}_\alpha(z)=p_{\mathfrak c,\alpha}(z)$, $z\in\mathfrak c$,
for the coefficients of $\mathsf U$ in \eqref{eq:rig-local-form}, fix a
norm $\|\cdot\|$ on $\mathsf W_q$, and put
\[
 \mathsf V_q=\{\mathsf U\in\mathsf W_q:\ \mathbb S\mathsf U=\mathsf U,\
 \mathcal C\mathsf U=\mathsf U\}.
\]
Neither $\mathbb S$ nor $\mathcal C$ maps $\mathsf W_q$ into itself.

\begin{lemma}[The invariant subspace]\label{lem:col-invariant-space}
The space $\mathsf V_q$ is the space of all even tempered distributions
of local order at most $q$ that are supported in $\Gamma$ and invariant
under $\mathbb S$ and $\mathcal C$. It has the basis
\eqref{eq:rig-grid-basis},
\[
 (r\partial_r)^j\rhoF_r,\qquad r\in\mathcal R_d^+,\quad0\leq j\leq q,\quad
 j\ \text{even if}\ r=1 .
\]
\end{lemma}
\begin{proof}
Every element of $\mathsf V_q$ is an even tempered distribution of local
order at most $q$, supported in $\Gamma=d^{-1}\Lambda_1$ and invariant,
so by Corollary~\ref{cor:rig-grid-invariants} it is a combination of the
displayed distributions, which are linearly independent. It remains to
show that each of them belongs to $\mathsf W_q$; it is even and
invariant by the same corollary. By \eqref{eq:rig-radius-derivative} it
suffices to show that $\mathcal K^je_r$ has the form \eqref{eq:col-comb}
for $r\in\mathcal R_d$ and $j\leq q$. The lattice $\Lambda_r$ is
contained in $\Gamma$ by Lemma~\ref{lem:grid-circles}, and it contains
$\Gamma^\vee$: for $r=k/d$ with $k\mid d^2$ we have
$[d,0]=[(d^2/k)\,r,0]$ and $[0,d]=[0,k/r]$. Hence $\Lambda_r$ is a union
of cosets of $\Gamma^\vee$. As operators on distributions,
$\mathcal K=u\partial_u-v\partial_v=\partial_u\circ u-\partial_v\circ v$,
and by induction on $j$, using
$u\circ\partial_u^i=\partial_u^i\circ u-i\,\partial_u^{i-1}$ and the
same relation for $v$, the operator $\mathcal K^j$ is a combination of
operators $\partial_u^{i}\partial_v^{l}\circ u^{i'}v^{l'}$ with
$i+l\leq j$, $i'\leq j$ and $l'\leq j$. Since $\partial_u$ and
$\partial_v$ are combinations of $\partial_P$ and $\partial_Q$,
\eqref{eq:rig-normal-action} shows that $\mathcal K^je_r$ is of the form
\eqref{eq:col-comb} with polynomials of degree at most $j\leq q$ in each
of $u$ and $v$.
\end{proof}

Thus $\mathsf V_q$ does not depend on the choice of the comb space
beyond its order: it consists of radius-derivative jets, of order at
most $q$, of the circle measures of \emph{all} radii in
$\mathcal R_d^+$, the self-dual radius included.

\begin{lemma}[A finite-dimensional projection]
\label{lem:col-finite-projection}
Let $q\geq1$. There are a finite family $\mathcal F_q$ of functions of
the form $X^\gamma e^{-2\pi|X|^2}$, a linear projection $\Pi_q$ of
$\mathsf W_q$ onto $\mathsf V_q$, and a constant $C$, all depending only
on $d$, $q$ and the norm, such that for every $\mathsf U\in\mathsf W_q$
\begin{equation}\label{eq:col-finite-projection}
 \|\mathsf U-\Pi_q\mathsf U\|\leq C\max_{f\in\mathcal F_q}
 \Bigl(\bigl|\langle(\mathbb S-I)\mathsf U,f\rangle\bigr|
 +\bigl|\langle(\mathcal C-I)\mathsf U,f\rangle\bigr|\Bigr).
\end{equation}
\end{lemma}
\begin{proof}
Let $\mathcal D$ be the linear span of the functions
$X^\gamma e^{-2\pi|X|^2}$, $\gamma\in\NN^2$. It is the span of the
Hermite functions and is dense in the Schwartz space. We claim that for
every finite-dimensional subspace $\mathsf Y$ of the space of tempered
distributions there are finitely many monomial functions
$f_1,\dots,f_k\in\mathcal D$ such that
$\mathsf U\mapsto(\langle\mathsf U,f_i\rangle)_{i\leq k}$ is injective on
$\mathsf Y$. Indeed, the image of the linear map from $\mathcal D$ to
the dual space $\mathsf Y^*$, $f\mapsto\langle\,\cdot\,,f\rangle$, is a
subspace of $\mathsf Y^*$. If it were a proper subspace, some nonzero
$\mathsf U\in\mathsf Y$ would satisfy $\langle\mathsf U,f\rangle=0$ for
all $f\in\mathcal D$, and hence, by density and continuity,
$\mathsf U=0$. So the map is onto, and finitely many of the spanning
monomial functions are mapped onto a spanning set of $\mathsf Y^*$.

Apply the claim to the finite-dimensional spaces
$(\mathbb S-I)\mathsf W_q$ and $(\mathcal C-I)\mathsf W_q$, and let
$\mathcal F_q$ be the union of the two families obtained. The linear
map
\[
 \mathfrak D:\mathsf W_q\to\CC^{\mathcal F_q}\times\CC^{\mathcal F_q},\qquad
 \mathfrak D\mathsf U=\Bigl(\langle(\mathbb S-I)\mathsf U,f\rangle,\
 \langle(\mathcal C-I)\mathsf U,f\rangle\Bigr)_{f\in\mathcal F_q}
\]
then has kernel exactly $\mathsf V_q$. Choose a complement
$\mathsf W_q=\mathsf V_q\oplus\mathsf Y_q$ and let $\Pi_q$ be the
projection along $\mathsf Y_q$. The map $\mathfrak D$ is injective on
the finite-dimensional space $\mathsf Y_q$, so
$\|\mathsf Y\|\leq C\|\mathfrak D\mathsf Y\|_\infty$ for
$\mathsf Y\in\mathsf Y_q$. Applying this to
$\mathsf U-\Pi_q\mathsf U\in\mathsf Y_q$ and using
$\mathfrak D\Pi_q\mathsf U=0$ gives \eqref{eq:col-finite-projection}.
\end{proof}

The lemma bounds the distance to the whole invariant subspace, and not
only a linear functional that vanishes on it. The constant is not
explicit.

\paragraph{Probes.}
For $y\in\Gamma$ and $\beta\in\mathcal A_q$ let $\psi_{y,\beta}$ be the
function $\psi$ of Lemma~\ref{lem:rig-cardinal-tests} for the single
coset $\mathfrak c_1=\Gamma$, written with $x_1=0$ and the forms
$\ell_{1,1}(X)=du$, $\ell_{1,2}(X)=dv$, for the point $y$, the order $q$
and the index $\beta$. Thus $\psi_{y,\beta}=\mathsf t\,\psi_0$ with
\[
 \psi_0(X)=e^{-2\pi|X-y|^2}\Bigl[\sinc\bigl(\pi d(u-u_y)\bigr)
 \sinc\bigl(\pi d(v-v_y)\bigr)\Bigr]^{q+1}
\]
and $\mathsf t$ the Taylor polynomial of degree $q$ at $y$ of
$(-1)^{|\beta|}(X-y)^\beta/(\beta!\,\psi_0(X))$. This construction
depends on $X$ and $y$ only through $X-y$, so
\begin{equation}\label{eq:col-probe-translate}
 \psi_{y,\beta}(X)=\psi_{0,\beta}(X-y),\qquad y\in\Gamma .
\end{equation}
By Lemma~\ref{lem:rig-cardinal-tests}, $\psi_{0,\beta}$ is a finite sum
of functions $\mathfrak m[p,z]$ as in Lemma~\ref{lem:rig-mixture-family},
and for every tempered distribution $\mathsf U$ supported in $\Gamma$
and of local order at most $q$,
\begin{equation}\label{eq:col-probe-extraction}
 \langle\mathsf U,\psi_{y,\beta}\rangle
 =\text{the coefficient of }\partial^\beta\delta_y\text{ in }\mathsf U .
\end{equation}
With the characters $\chi_w(X)=e^{2\pi iB(w,X)}$ of
Section~\ref{subsec:rig-conventions} put
\begin{equation}\label{eq:col-probes}
 \begin{gathered}
 \omega_\iota=(\chi_{\mathsf b_\iota}-1)^{q+1}\quad(\iota=1,2),\\
 \mathcal P_q(\varrho)=\bigl\{\omega_\iota\,\mathbb S\psi_{y,\beta}:\
 \iota\in\{1,2\},\ \beta\in\mathcal A_q,\ y\in\Gamma,\
 |u_y|,|v_y|\leq\varrho\bigr\}.
 \end{gathered}
\end{equation}
Since $\chi_{\mathsf b_1}(X)=e^{2\pi idv}$ and
$\chi_{\mathsf b_2}(X)=e^{2\pi idu}$, the function $\omega_\iota$
vanishes to order $q+1$ at every point of $\Gamma$.

\begin{proposition}[Quantitative grid projection]
\label{prop:grid-projection}
Let $d,q\geq1$ be integers, let $A_0\geq0$ and $A_1>0$, and put
$D_0=4A_1/\pi$. Let $\mathcal F_q$ and $\Pi_q$ be as in
Lemma~\ref{lem:col-finite-projection}. For every compact set
$K\subset\RR^2$ there is a constant $C_K$, depending only on $d$, $q$,
$A_0$, $A_1$ and $K$, with the following property. Let $0<b<e^{-1}$,
$H=\log(1/b)$, $C_0<\infty$ and $\varepsilon\geq0$, and let
\[
 \mathsf T=\sum_{z\in\Gamma}\sum_{|\alpha|\leq q}c_\alpha(z)\,
 \partial^\alpha\delta_z
\]
be a distribution such that
\begin{enumerate}[(i)]
\item $|c_\alpha(z)|\leq C_0(1+|z|)^{A_0}$ for all $z\in\Gamma$ and
 $|\alpha|\leq q$;
\item $\mathsf T$ is even and $\mathcal C\mathsf T=\mathsf T$;
\item $|\langle(\mathbb S-I)\mathsf T,f\rangle|\leq\varepsilon$ for
 every $f\in\mathcal F_q$ and every
 $f\in\mathcal P_q\bigl(\sqrt{D_0H}+(q+3)d\bigr)$.
\end{enumerate}
Then there is $\mathsf V\in\mathsf V_q$ such that
\begin{equation}\label{eq:col-grid-projection}
 \max_{z\in\Gamma\cap K}\ \max_{|\alpha|\leq q}
 \bigl|c_\alpha(z)-c^{\mathsf V}_\alpha(z)\bigr|
 \leq C_K\Bigl(\varepsilon\,(1+H)^{q+1}+C_0\,b^{A_1}\Bigr).
\end{equation}
The maximum includes the coefficients of order zero.
\end{proposition}

No reference measure is subtracted from $\mathsf T$, the coefficients of
$\mathsf T$ need not be small anywhere, and the number $D_0$, which
fixes the window of the probes, depends only on $A_1$.

\begin{proof}
Constants called $C$ depend only on $d$, $q$, $A_0$ and $A_1$. By (i),
$\mathsf T$ is a tempered distribution of local order at most $q$
supported in $\Gamma$.

\emph{Step 1: differences.} Let
$\mathsf D_\iota=(\tau_{\mathsf b_\iota}-I)^{q+1}$. Since
$\tau_w\delta_x=\delta_{x-w}$, the distribution $\mathsf D_\iota\mathsf T$
is supported in $\Gamma$, has local order at most $q$, and its
coefficient at $\partial^\beta\delta_y$ is the forward difference
\begin{equation}\label{eq:col-forward-difference}
 (\Delta_\iota^{q+1}c_\beta)(y)=\sum_{k=0}^{q+1}(-1)^{q+1-k}
 \binom{q+1}k\,c_\beta(y+k\mathsf b_\iota).
\end{equation}
By \eqref{eq:rig-intertwine},
$\mathsf D_\iota=\mathbb S\,\omega_\iota\,\mathbb S^{-1}$. As
$\mathsf T$ is even, $\mathbb S^{-1}\mathsf T=\mathbb S\mathsf T$, and
$\omega_\iota\mathsf T=0$ by Lemma~\ref{lem:rig-local-pairing}(ii).
Hence, by \eqref{eq:col-probe-extraction},
\[
 (\Delta_\iota^{q+1}c_\beta)(y)
 =\langle\mathsf D_\iota\mathsf T,\psi_{y,\beta}\rangle
 =\bigl\langle\omega_\iota\,\mathbb S\mathsf T,\mathbb S\psi_{y,\beta}
 \bigr\rangle
 =\bigl\langle(\mathbb S-I)\mathsf T,\,
 \omega_\iota\,\mathbb S\psi_{y,\beta}\bigr\rangle ,
\]
and by (iii)
\begin{equation}\label{eq:col-small-differences}
 \bigl|(\Delta_\iota^{q+1}c_\beta)(y)\bigr|\leq\varepsilon\qquad
 \text{for }y\in\Gamma,\ |u_y|,|v_y|\leq\sqrt{D_0H}+(q+3)d,\
 \iota\in\{1,2\},\ |\beta|\leq q .
\end{equation}

\emph{Step 2: interpolation in one variable.} Let $N'\geq q$ be an
integer, let $c$ be a function on $\{-N',\dots,N'\}$ whose forward
differences satisfy $|\Delta^{q+1}c(i)|\leq\varepsilon$ for
$-N'\leq i\leq N'-q-1$, and let $\mathsf p$ be the polynomial of
degree at most $q$ with $\mathsf p(i)=c(i)$ for $0\leq i\leq q$. Then
\begin{equation}\label{eq:col-newton-one}
 |c(i)-\mathsf p(i)|\leq\varepsilon\,(q+|i|)^{q+1}\qquad(|i|\leq N').
\end{equation}
To see this, let $e=c-\mathsf p$, so that $e(0)=\dots=e(q)=0$ and
$\Delta^{q+1}e=\Delta^{q+1}c$. We show $|e(i)|\leq\varepsilon\binom i{q+1}$
for $0\leq i\leq N'$ by induction on $q\geq0$. For $q=0$,
$e(i)=\sum_{l<i}\Delta e(l)$ and $e(0)=0$. For $q\geq1$ the function
$\Delta e$ vanishes at $0,\dots,q-1$ and its differences of order $q$
are bounded by $\varepsilon$, so $|\Delta e(l)|\leq\varepsilon\binom lq$
by the induction hypothesis, and
$|e(i)|\leq\varepsilon\sum_{l<i}\binom lq=\varepsilon\binom i{q+1}$.
Only the differences $\Delta^{q+1}c(l)$ with $0\leq l\leq i-q-1$ are
used. For negative $i$ apply this to $\check e(i)=e(q-i)$, which
vanishes at $0,\dots,q$ and satisfies
$\Delta^{q+1}\check e(i)=(-1)^{q+1}\Delta^{q+1}e(-i-1)$; this gives
$|e(i)|\leq\varepsilon\binom{q-i}{q+1}$ for $-N'\leq i<0$ and uses the
differences at $i,\dots,-1$. Finally
$\binom{q+|i|}{q+1}\leq(q+|i|)^{q+1}$.

\emph{Step 3: interpolation on a coset.} Put
\[
 N'=\bigl\lceil\sqrt{D_0H}/d\bigr\rceil+q+1,\qquad
 \mathsf Q_H=\bigl\{[u,v]:|u|,|v|\leq\sqrt{D_0H}\bigr\}.
\]
Fix a coset $\mathfrak c$ of $\Gamma^\vee$ in $\Gamma$, let
$z_{\mathfrak c}$ be its point with $u,v\in[0,d)$, fix $\alpha$, and let
$c(n_1,n_2)=c_\alpha(z_{\mathfrak c}+n_1\mathsf b_1+n_2\mathsf b_2)$. The
point $z_{\mathfrak c}+n_1\mathsf b_1+n_2\mathsf b_2$ has light-cone
coordinates $u_{z_{\mathfrak c}}+n_1d$ and $v_{z_{\mathfrak c}}+n_2d$. If
it lies in $\mathsf Q_H$, then $|n_1|,|n_2|\leq N'$; and if
$|n_1|,|n_2|\leq N'$, its coordinates are bounded by
$d(N'+1)\leq\sqrt{D_0H}+(q+3)d$. By \eqref{eq:col-small-differences},
the differences of order $q+1$ of $c$ in each of the two variables are
therefore bounded by $\varepsilon$ whenever their stencil lies in
$\{-N',\dots,N'\}^2$. Let $p(n_1,n_2)$ be the polynomial of degree at
most $q$ in each variable with $p=c$ on $\{0,\dots,q\}^2$, and let
$\ell_0,\dots,\ell_q$ be the Lagrange polynomials of the nodes
$0,\dots,q$; they satisfy $|\ell_k(i)|\leq(q+|i|)^q$ for integers $i$,
because the denominators $\prod_{l\neq k}|k-l|$ are at least one. For
$0\leq k\leq q$ the function $p(\cdot,k)$ is the interpolating
polynomial of $c(\cdot,k)$, so \eqref{eq:col-newton-one} gives
$|c(n_1,k)-p(n_1,k)|\leq\varepsilon(q+|n_1|)^{q+1}$. For fixed $n_1$ the
interpolating polynomial of $c(n_1,\cdot)$ is
$\sum_k\ell_k(n_2)c(n_1,k)$, and $p(n_1,n_2)=\sum_k\ell_k(n_2)p(n_1,k)$.
Hence, for $|n_1|,|n_2|\leq N'$,
\begin{equation}\label{eq:col-newton-two}
\begin{aligned}
 |c(n_1,n_2)-p(n_1,n_2)|
 &\leq\varepsilon(q+|n_2|)^{q+1}
 +\sum_{k=0}^q|\ell_k(n_2)|\,\varepsilon\,(q+|n_1|)^{q+1}\\
 &\leq(q+2)\,(q+N')^{2q+1}\,\varepsilon\leq C\,(1+H)^{q+1}\varepsilon .
\end{aligned}
\end{equation}
Moreover, by (i) the $(q+1)^2$ initial values are bounded by $CC_0$, so
\begin{equation}\label{eq:col-comb-bound}
 |p(n_1,n_2)|\leq\sum_{i,k}|\ell_i(n_1)||\ell_k(n_2)||c(i,k)|
 \leq C\,C_0\,(q+|n_1|)^q(q+|n_2|)^q .
\end{equation}
Let $p_{\mathfrak c,\alpha}$ be the polynomial on $\RR^2$ with
$p_{\mathfrak c,\alpha}(z_{\mathfrak c}+n_1\mathsf b_1+n_2\mathsf b_2)
=p(n_1,n_2)$; it has degree at most $q$ in each of $u$ and $v$.

\emph{Step 4: a comb close to $\mathsf T$.} Let $\mathsf U^\circ$ be
the distribution \eqref{eq:col-comb} formed with the polynomials
$p_{\mathfrak c,\alpha}$ of Step~3, and let
$\widetilde{\mathsf U}=\frac14\sum_{\mathsf g}\mathsf g_*\mathsf U^\circ$
be its average over the group of the four maps
$(P,Q)\mapsto(\pm P,\pm Q)$. Each of these maps preserves $\Gamma$ and
$\Gamma^\vee$, maps $\partial^\alpha\delta_z$ to
$\pm\partial^\alpha\delta_{\mathsf gz}$, and permutes the light-cone
coordinates up to signs; hence $\mathsf g_*\mathsf U^\circ$ is again of
the form \eqref{eq:col-comb}, and
$\widetilde{\mathsf U}\in\mathsf W_q$. Since $\mathsf T$ is even,
$\mathsf T-\widetilde{\mathsf U}
=\frac14\sum_{\mathsf g}\mathsf g_*(\mathsf T-\mathsf U^\circ)$, and the
square $\mathsf Q_H$ is invariant under the four maps. Writing
$\tilde c_\alpha$ for the coefficients of $\widetilde{\mathsf U}$, we
obtain from \eqref{eq:col-newton-two} and \eqref{eq:col-comb-bound}
\begin{equation}\label{eq:col-comb-close}
 |c_\alpha(z)-\tilde c_\alpha(z)|\leq C(1+H)^{q+1}\varepsilon\quad
 (z\in\Gamma\cap\mathsf Q_H),\qquad
 |\tilde c_\alpha(z)|\leq C\,C_0\,(1+|z|)^{2q}\quad(z\in\Gamma).
\end{equation}

\emph{Step 5: the defects of $\widetilde{\mathsf U}$.} Let
$G$ be one of the functions $f$, $\mathbb Sf$,
$(e^{2\pi i(P^2-Q^2)}-1)f$ with $f\in\mathcal F_q$. Each of them is a
polynomial times $e^{-2\pi|X|^2}$, times a function all of whose
derivatives are polynomially bounded on $\RR^2$; for $\mathbb Sf$ this
follows from Lemma~\ref{lem:rig-mixture-family}(ii) with $z=0$. Hence
$|\partial^\alpha G(X)|\leq Ce^{-\pi|X|^2}$ for real $X$ and
$|\alpha|\leq q$. The pairing of the tempered distribution
$\mathsf T-\widetilde{\mathsf U}$ with $G$ is the absolutely
convergent sum of the terms
$(-1)^{|\alpha|}(c_\alpha-\tilde c_\alpha)(z)\,\partial^\alpha G(z)$.
A point $z\in\Gamma\setminus\mathsf Q_H$ has
$|z|^2=\frac12(u_z^2+v_z^2)>\frac12D_0H$, hence
$e^{-\pi|z|^2}\leq e^{-\pi D_0H/4}e^{-\pi|z|^2/2}=b^{A_1}e^{-\pi|z|^2/2}$.
By (i) and \eqref{eq:col-comb-close},
\begin{equation}\label{eq:col-defect-transfer}
\begin{aligned}
 \bigl|\langle\mathsf T-\widetilde{\mathsf U},G\rangle\bigr|
 &\leq C(1+H)^{q+1}\varepsilon\sum_{z\in\Gamma}e^{-\pi|z|^2}
 +C\,C_0\,b^{A_1}\sum_{z\in\Gamma}(1+|z|)^{A_0+2q}e^{-\pi|z|^2/2}\\
 &\leq C\bigl((1+H)^{q+1}\varepsilon+C_0\,b^{A_1}\bigr)=:\varepsilon' .
\end{aligned}
\end{equation}
For $f\in\mathcal F_q$ we have
$\langle(\mathbb S-I)\widetilde{\mathsf U},f\rangle
=\langle(\mathbb S-I)\mathsf T,f\rangle
-\langle\mathsf T-\widetilde{\mathsf U},\mathbb Sf-f\rangle$ and, by
(ii),
$\langle(\mathcal C-I)\widetilde{\mathsf U},f\rangle
=-\langle\mathsf T-\widetilde{\mathsf U},(e^{2\pi i(P^2-Q^2)}-1)f\rangle$.
By (iii) and \eqref{eq:col-defect-transfer} both are at most
$\varepsilon+2\varepsilon'$ in modulus.

\emph{Step 6: projection.} Put
$\mathsf V=\Pi_q\widetilde{\mathsf U}\in\mathsf V_q$. By
\eqref{eq:col-finite-projection},
$\|\widetilde{\mathsf U}-\mathsf V\|\leq C(\varepsilon+\varepsilon')$.
For fixed $z\in\Gamma$ and $\alpha$ the coefficient
$\mathsf U\mapsto c^{\mathsf U}_\alpha(z)$ is a linear functional on the
finite-dimensional space $\mathsf W_q$, so
$|\tilde c_\alpha(z)-c^{\mathsf V}_\alpha(z)|
\leq C_K'(\varepsilon+\varepsilon')$ for the finitely many
$z\in\Gamma\cap K$. If $K\subset\mathsf Q_H$, the first bound of
\eqref{eq:col-comb-close} gives \eqref{eq:col-grid-projection}. If
$K\not\subset\mathsf Q_H$, then $b$ is bounded below by a positive
number $b_K$ depending on $K$ and $D_0$, and (i) and the second bound of
\eqref{eq:col-comb-close} give
$|c_\alpha(z)-\tilde c_\alpha(z)|\leq C_K''C_0
\leq C_K''b_K^{-A_1}C_0\,b^{A_1}$ on $K$.
\end{proof}

\subsection{The reconstruction operator}
\label{subsec:col-reconstruction}

We now construct a linear operator, fixed once $g$ is given, which
recovers every element of every space $\mathsf V_q$ from finitely many
of its local coefficients, and which sends the Fock measure of a
partition function to a positive combination of at most four circle
measures. The grid is again \eqref{eq:col-grid} with the integer $d$ of
\eqref{eq:ker-grid-denominator}.

Fix a prime $\mathfrak p>d^2$ and put
\[
 \mathsf E(t)=E_{1,\mathfrak p}(t)=\tfrac12\bigl(t^2+\mathfrak p^2/t^2\bigr),
 \qquad\mathsf E(1/t)=E_{\mathfrak p,1}(t).
\]
Since $\mathsf E'(t)=t-\mathfrak p^2/t^3<0$ for $0<t<\sqrt{\mathfrak p}$,
the function $\mathsf E$ maps $(0,\sqrt{\mathfrak p})$ diffeomorphically
onto $(\mathfrak p,\infty)$; we write
$\mathsf E^{-1}:(\mathfrak p,\infty)\to(0,\sqrt{\mathfrak p})$ for the
inverse of this restriction. The classes with $mn=\mathfrak p$ are those
of $(1,\mathfrak p)$ and $(\mathfrak p,1)$. By
Lemma~\ref{lem:col-states}(i), for every $t>0$,
\begin{equation}\label{eq:col-private-states}
 \parbox{0.82\linewidth}{the family of\/ $\frac12\rhoF_t-\frac12\delta_0$
 has exactly two states of spin $\mathfrak p$; their dimensions are
 $\mathsf E(t)$ and $\mathsf E(1/t)$.}
\end{equation}
The radii of $\mathcal R_d$ lie in
$[1/d,d]\subset(\mathfrak p^{-1/2},\mathfrak p^{1/2})$, so the numbers
$\mathsf E(r)$, $r\in\mathcal R_d$, are distinct and greater than
$\mathfrak p$; let $g_0>0$ be the smallest distance between two of them.
Choose $\eta_*\in(0,1]$ such that
\begin{enumerate}[(a)]
\item $8\eta_*\leq g_0$ and $\mathsf E(d)-2\eta_*>\mathfrak p$;
\item for every $r\in\mathcal R_d^+$ with $r>1$ and every $E$ with
 $|E-\mathsf E(r)|\leq2\eta_*$ we have
 $\bigl|\mathsf E\bigl(1/\mathsf E^{-1}(E)\bigr)-\mathsf E(1/r)\bigr|
 \leq\frac12g_0$.
\end{enumerate}
Condition (b) holds for small $\eta_*$ by continuity. For
$r\in\mathcal R_d^+$ put
\[
 \vartheta_r(E)=\vartheta\Bigl(\frac{E-\mathsf E(r)}{\eta_*}\Bigr),\qquad
 \mathsf c_r=\begin{cases}\frac12,&r>1,\\[2pt]\frac14,&r=1.\end{cases}
\]
By (a) the supports of the functions $\vartheta_r$ are pairwise
disjoint and contained in $(\mathfrak p,\infty)$. For a weight test
$\phi$ let $\mathsf f_\phi(t)=\langle\rhoF_t,\phi\rangle$, $t>0$, and
define
\begin{equation}\label{eq:col-reconstruction}
 (P_*'\phi)(X)=\vartheta^{\rm sp}\bigl(P^2-Q^2-\mathfrak p\bigr)\,
 G_\phi\bigl(|X|^2\bigr),\qquad
 G_\phi(E)=\sum_{r\in\mathcal R_d^+}\mathsf c_r\,\vartheta_r(E)\,
 \mathsf f_\phi\bigl(\mathsf E^{-1}(E)\bigr),
\end{equation}
where the term with index $r$ is zero outside the support of
$\vartheta_r$. For a distribution $\mathsf U$ on $\RR^2$ we define
$P_*\mathsf U$ on weight tests by
$\langle P_*\mathsf U,\phi\rangle=\langle\mathsf U,P_*'\phi\rangle$.
The clusters of spin $\mathfrak p$ and dimension $\mathsf E(r)$ are the
\emph{private clusters}; there is one for every radius of
$\mathcal R_d^+$, whether or not it occurs in a limit.

\begin{lemma}[The reconstruction operator]
\label{lem:reconstruction-operator}
The map $P_*'$ is linear, depends only on $g$, and has the following
properties.
\begin{enumerate}[(i)]
\item $P_*'\phi$ is a weight test for every weight test $\phi$, and for
 every state $(\Delta,s)$ with weights $(h,\bar h)$
 \begin{equation}\label{eq:col-reconstruction-pairing}
 \bigl\langle A_h\otimes A_{\bar h},P_*'\phi\bigr\rangle
 =\one_{\{s=\mathfrak p\}}\,G_\phi(\Delta).
 \end{equation}
\item $P_*\nu_{\vac}=0$.
\item For every $r\in\mathcal R_d^+$ there is an open interval
 $I_r\ni r$ such that $P_*\rhoF_t=\rhoF_t$ for $t\in I_r$, that is,
 $\langle\rhoF_t,P_*'\phi\rangle=\langle\rhoF_t,\phi\rangle$ for every
 weight test $\phi$.
\item $P_*\mathsf V=\mathsf V$ for every $q\geq1$ and every
 $\mathsf V\in\mathsf V_q$.
\item Let an admissible sequence of gap $g$ converge to
 $\sigma(R_1,R_2)$ with half-integers $R_1,R_2$, and for
 $r\in\mathcal R_d^+$ put
 \[
 n_r=\#\{i\in\{1,2\}:R_i=r\}\cdot\bigl(1+\one_{\{r=1\}}\bigr).
 \]
 For all sufficiently small $a$ in the sequence and every
 $r\in\mathcal R_d^+$, exactly $n_r$ states of $\rho_a$ have spin
 $\mathfrak p$ and dimension in
 $(\mathsf E(r)-2\eta_*,\mathsf E(r)+2\eta_*)$; their dimensions
 $\Delta_{r,1}\leq\dots\leq\Delta_{r,n_r}$ lie in
 $(\mathsf E(r)-\eta_*,\mathsf E(r)+\eta_*)$, the radii
 $t_{r,i}=\mathsf E^{-1}(\Delta_{r,i})$ converge to $r$, and
 \begin{equation}\label{eq:col-reconstructed-measure}
 P_*\mu_a=\sum_{r\in\mathcal R_d^+}\mathsf c_r\sum_{i=1}^{n_r}
 \rhoF_{t_{r,i}},\qquad
 \sum_rn_r\leq4,\qquad\sum_r\mathsf c_rn_r=1 .
 \end{equation}
\end{enumerate}
\end{lemma}

In (v) only the radii $r\in\{R_1,R_2\}$ have $n_r\neq0$. The private
clusters of all other radii of $\mathcal R_d^+$ are needed for (iv), and
they are empty for small $a$.

\begin{proof}
(i) We have $\mathsf f_\phi(t)=\frac12\sum_{m,n}\bigl(\phi([mt,n/t])
+\phi([m/t,nt])\bigr)$. Since $\phi$ has compact support, for $t$ in a
compact subset of $(0,\infty)$ only finitely many terms are different
from zero, so $\mathsf f_\phi$ is smooth. Hence $G_\phi$ is smooth and
compactly supported on $\RR$, and $P_*'\phi$ is the weight test with
$\Theta(h,\bar h)=\vartheta^{\rm sp}(h-\bar h-\mathfrak p)\,
G_\phi(h+\bar h)$. Formula \eqref{eq:col-reconstruction-pairing} follows
as \eqref{eq:col-window-pairing}.

(ii) By \eqref{eq:physical-measures} the measure $\nu_{\vac}$ is a
combination of the state measures of $(0,0)$, $(1,\pm1)$ and $(2,0)$,
none of which has spin $\mathfrak p$.

(iii) By Lemma~\ref{lem:col-states}(i), \eqref{eq:col-private-states}
and \eqref{eq:col-reconstruction-pairing}, and since
$(P_*'\phi)(0)=0$,
\begin{equation}\label{eq:col-reconstruction-circle}
 \langle\rhoF_t,P_*'\phi\rangle
 =2\,G_\phi\bigl(\mathsf E(t)\bigr)+2\,G_\phi\bigl(\mathsf E(1/t)\bigr),
 \qquad t>0 .
\end{equation}
Let $r>1$ and
$I_r=\{t\in(0,\sqrt{\mathfrak p}):|\mathsf E(t)-\mathsf E(r)|<\eta_*\}$.
For $t\in I_r$ we have $\vartheta_r(\mathsf E(t))=1$,
$\vartheta_{r'}(\mathsf E(t))=0$ for $r'\neq r$, and
$\mathsf E^{-1}(\mathsf E(t))=t$, so
$G_\phi(\mathsf E(t))=\frac12\mathsf f_\phi(t)$. By (b),
$\mathsf E(1/t)$ is within $\frac12g_0$ of $\mathsf E(1/r)$. As
$1/r\in\mathcal R_d\setminus\mathcal R_d^+$, the number
$\mathsf E(1/r)$ is at distance at least $g_0$ from every
$\mathsf E(r')$, $r'\in\mathcal R_d^+$, so $\mathsf E(1/t)$ is at
distance at least $\frac12g_0\geq4\eta_*$ from them and
$G_\phi(\mathsf E(1/t))=0$. Thus
\eqref{eq:col-reconstruction-circle} equals $\mathsf f_\phi(t)$. For
$r=1$ let $I_1$ be the set of
$t\in(\mathfrak p^{-1/2},\mathfrak p^{1/2})$ with
$|\mathsf E(t)-\mathsf E(1)|<\eta_*$ and
$|\mathsf E(1/t)-\mathsf E(1)|<\eta_*$; it is an open interval
containing $1$. For $t\in I_1$ both $\mathsf E(t)$ and $\mathsf E(1/t)$
lie where $\vartheta_1=1$ and all other cutoffs vanish, and
$\mathsf E^{-1}(\mathsf E(t))=t$, $\mathsf E^{-1}(\mathsf E(1/t))=1/t$.
Hence \eqref{eq:col-reconstruction-circle} equals
$\frac12\mathsf f_\phi(t)+\frac12\mathsf f_\phi(1/t)=\mathsf f_\phi(t)$,
because $\rhoF_{1/t}=\rhoF_t$.

(iv) By Lemma~\ref{lem:col-invariant-space} it suffices to treat
$\mathsf V=(r\partial_r)^j\rhoF_r$ with $r\in\mathcal R_d^+$. By
Theorem~\ref{thm:circle-jets}(iii), $\mathsf V$ is the distribution
$f\mapsto(t\partial_t)^j\langle\rhoF_t,f\rangle|_{t=r}$ on Schwartz
functions, so we have for every weight test $\phi$
\[
 \langle\mathsf V,P_*'\phi\rangle
 =(t\partial_t)^j\langle\rhoF_t,P_*'\phi\rangle\big|_{t=r}
 =(t\partial_t)^j\langle\rhoF_t,\phi\rangle\big|_{t=r}
 =\langle\mathsf V,\phi\rangle ,
\]
since the two functions of $t$ coincide on the open interval $I_r$ by
(iii).

(v) By Lemma~\ref{lem:col-states}(ii) and
\eqref{eq:col-private-states}, the states of spin $\mathfrak p$ of
$\sigma(R_1,R_2)$ have the dimensions $\mathsf E(R_i)$ and
$\mathsf E(1/R_i)$, $i=1,2$. Since $1\leq R_i\leq R_{\max}$ by
Theorem~\ref{thm:classification}, all four radii belong to
$\mathcal R_d$.
By (a), such a state has dimension within $3\eta_*$ of $\mathsf E(r)$,
$r\in\mathcal R_d^+$, only if its radius equals $r$, in which case the
dimension is $\mathsf E(r)$; this happens for $R_i=r$, and for $1/R_i$
when $R_i=r=1$. Hence $\mathcal W_{\eta_*}(\mathsf E(r),\mathfrak p)$ is
isolated for $\mathfrak S(R_1,R_2)$ and contains $n_r$ of its states, and
Lemma~\ref{lem:col-windows}(ii) gives the statements about the states
of $\rho_a$; the convergence $t_{r,i}\to r$ follows from the continuity
of $\mathsf E^{-1}$. By (ii) and
\eqref{eq:col-reconstruction-pairing},
\[
 \langle\mu_a,P_*'\phi\rangle=\sum_{j:\,s_j=\mathfrak p}G_\phi(\Delta_j)
 =\sum_{r\in\mathcal R_d^+}\mathsf c_r\sum_{i=1}^{n_r}
 \mathsf f_\phi\bigl(\mathsf E^{-1}(\Delta_{r,i})\bigr),
\]
because a state of spin $\mathfrak p$ with
$\vartheta_r(\Delta_j)\neq0$ is one of the $n_r$ states above, for which
$\vartheta_r(\Delta_j)=1$. Finally
$\sum_rn_r=\sum_i(1+\one_{\{R_i=1\}})\leq4$ and
$\sum_r\mathsf c_rn_r=\sum_i\mathsf c_{R_i}(1+\one_{\{R_i=1\}})=1$.
\end{proof}

\subsection{The Taylor array of the spectrum and the normal form}
\label{subsec:col-normal-form}

We expand the spectrum of an admissible partition function to a finite
order about the grid points supplied by
Corollary~\ref{cor:four-scalar-localization}. The resulting distribution
satisfies the hypotheses of Proposition~\ref{prop:grid-projection}. The
estimates must be uniform for probes centred anywhere in a window of
radius of order $\sqrt{H_a}$, and for their Fourier transforms, which
oscillate with frequencies of that order. A Gaussian envelope centred
at the origin, as in Lemma~\ref{lem:weight-derivative-bound}, would lose
a factor exponential in $H_a$ for such functions. We therefore use the
following pointwise form of that lemma.

For real $x$ let $\mathsf J(x)\subset\CC$ be the set
$\{\tau\sqrt x:-1\leq\tau\leq1\}$ if $x\leq2$ and the set
$\{\pm\sqrt x\}$ if $x>2$, where $\sqrt x=i\sqrt{|x|}$ for $x<0$.

\begin{lemma}[Pointwise bounds for weight derivatives]
\label{lem:col-pointwise}
Let $F$ be an entire function on $\CC^2$ and $k\geq0$ an integer. For
all real $h,\bar h$ and all $i+l\leq k$,
\[
 \bigl|\partial_h^i\partial_{\bar h}^l\,\ev F(h,\bar h)\bigr|
 \leq C_k\,\max\bigl\{|\partial^\alpha F(P,Q)|:\ |\alpha|\leq2k,\
 P\in\mathsf J(h),\ Q\in\mathsf J(\bar h)\bigr\},
\]
where $C_k$ depends only on $k$.
\end{lemma}
\begin{proof}
First let $\phi$ be an entire function of one variable,
$\varphi(h)=\frac12(\phi(\sqrt h)+\phi(-\sqrt h))$ and $i\geq0$. We
claim that
$|\varphi^{(i)}(h)|\leq C_i\max\{|\phi^{(l)}(P)|:l\leq2i,\ P\in\mathsf J(h)\}$
for real $h$. For $i=0$ this is clear, since $\pm\sqrt h\in\mathsf J(h)$.
For $i\geq1$ and $h\leq2$ it follows from
\eqref{eq:sq-momentum-derivative}, in which the points $t\sqrt h$ lie in
$\mathsf J(h)$ and the weight $(1-t^2)^{i-1}/(4^i(i-1)!)$ has integral
at most one.
For $i\geq1$ and $h>2$ it follows from the second statement of
Lemma~\ref{lem:sq-momentum-derivative}, in which the coefficients
$h^{-(2i-l)/2}$ are at most one. Now fix a real $\bar h$ and let
$G(P)=\partial_{\bar h}^l\bigl[\frac12(F(P,\sqrt{\bar h})
+F(P,-\sqrt{\bar h}))\bigr]$, the derivative being taken in the
variable $\bar h$ of the even evaluation in $Q$. Then $G$ is entire,
$\partial_h^i\partial_{\bar h}^l\ev F(h,\bar h)$ is the $i$-th derivative
at $h$ of $\frac12(G(\sqrt h)+G(-\sqrt h))$, and $G^{(l')}(P)$ is the
same expression as $G(P)$ with $F$ replaced by $\partial_P^{l'}F$.
Applying the claim first in the variable $P$ and then, for each fixed
$P\in\mathsf J(h)$, in the variable $Q$ gives the lemma.
\end{proof}

\begin{lemma}[Jets and weight tests]\label{lem:col-jets}
Let $X=(P,Q)$ be a real point, $k\geq0$, and let
$\phi(P,Q)=\Theta(P^2,Q^2)$ be a weight test. With the distributions
$\Upsilon^{(k)}_X$ of \eqref{eq:rig-fixed-spin-jet},
\[
 \bigl\langle\Upsilon_X^{(k)},\phi\bigr\rangle
 =\frac{d^k}{d\zeta^k}\Big|_{\zeta=0}
 \Theta\bigl(P^2+\tfrac\zeta2,\,Q^2+\tfrac\zeta2\bigr).
\]
\end{lemma}
\begin{proof}
By \eqref{eq:rig-fixed-spin-jet} and the chain rule it suffices to show
that $\langle A_h^{(i)}\otimes A_{\bar h}^{(l)},\phi\rangle
=\partial_h^i\partial_{\bar h}^l\Theta(h,\bar h)$ for $h,\bar h\geq0$,
and for this it suffices to show that
$\langle A_h^{(i)},\phi_1\rangle=\theta^{(i)}(h)$ for $h\geq0$, when
$\theta$ is smooth on $\RR$ and $\phi_1(P)=\theta(P^2)$. For $h>0$ this
is the definition of $A_h^{(i)}$, because
$\langle A_h,\phi_1\rangle=\theta(h)$. For $h=0$, Taylor's formula for
$\theta$ at $0$ gives
$\phi_1(P)=\sum_{l\leq i}\theta^{(l)}(0)P^{2l}/l!+O(P^{2i+2})$, hence
$\phi_1^{(2i)}(0)=\frac{(2i)!}{i!}\theta^{(i)}(0)$, and
$A_0^{(i)}=\frac{i!}{(2i)!}\delta_0^{(2i)}$.
\end{proof}

\paragraph{The Taylor array.}
Let $Z$ be an admissible partition function of gap $g$ at $c=1+12a$,
let $n\geq1$ be an integer and $M>0$, and suppose that
$a<a_M$, the threshold of Corollary~\ref{cor:four-scalar-localization}.
For every state with $\Delta_j\leq MH_a$ let $x_j=(P_j,Q_j)\in\Gamma$ be
the grid point $[m/d,n'/d]$ which that corollary associates with the
physical point $(\sqrt{h_j},\sqrt{\bar h_j})$, and put
\begin{equation}\label{eq:col-association}
 \zeta_j=\Delta_j-|x_j|^2;\qquad\text{thus}\qquad
 P_j^2-Q_j^2=s_j,\qquad|\zeta_j|\leq C_M\,b_aH_a^{3/2}.
\end{equation}
The \emph{Taylor array of order $n$} is the distribution
\begin{equation}\label{eq:col-array}
 \mathsf T_a=\nu_{\vac}+\sum_{j:\,\Delta_j\leq MH_a}\ \sum_{l=0}^n
 \frac{\zeta_j^l}{l!}\,\Upsilon^{(l)}_{x_j}.
\end{equation}
It differs from the distribution of
Lemma~\ref{lem:rig-taylor-distribution} in that the expansion is about
the physical weights: the central shift is not included in $\zeta_j$,
and the vacuum is the physical measure $\nu_{\vac}$. The terms with
$l=0$ are the state measures of the states $(|x_j|^2,s_j)$, so the
coefficients of order zero of $\mathsf T_a$ count the states associated
with each grid point; no comparison measure is subtracted.

\begin{lemma}[The Taylor array]\label{lem:col-array}
Let $n\geq1$, $q=2n$ and $D_0>0$, and let
\begin{equation}\label{eq:col-M-condition}
 M\geq\max\Bigl\{8D_0+8(2q+4)^2d^2+1,\ \frac{8(n+3)}\pi\Bigr\}.
\end{equation}
There are $a_0>0$ and $C<\infty$, depending only on $g$, $n$, $D_0$ and
$M$, such that for every admissible partition function of gap $g$ at
$c=1+12a$ with $0<a<a_0$ the Taylor array \eqref{eq:col-array} has the
following properties, where $b=b_a$ and $H=H_a$.
\begin{enumerate}[(i)]
\item $\mathsf T_a$ is an even distribution with finite support
 contained in $\Gamma$, of local order at most $q$, and
 $\mathcal C\mathsf T_a=\mathsf T_a$.
\item Its coefficients satisfy $|c_\alpha(z)|\leq C_0(1+|z|)^2$ for
 $z\in\Gamma$ and $|\alpha|\leq q$, with a constant $C_0$ depending only
 on $g$ and $n$.
\item For every $f\in\mathcal F_q$ and every
 $f\in\mathcal P_q\bigl(\sqrt{D_0H}+(q+3)d\bigr)$,
 \begin{equation}\label{eq:col-array-defect}
 \bigl|\langle(\mathbb S-I)\mathsf T_a,f\rangle\bigr|
 \leq C\,(a+b^{n+1})\,(1+H)^{3n+4}.
 \end{equation}
\item For every weight test $\phi$ there are $C_\phi$ and $a_\phi>0$,
 depending in addition on $\phi$, with
 $|\langle\mu_a,\phi\rangle-\langle\mathsf T_a,\phi\rangle|
 \leq C_\phi\,b^{n+1}(1+H)^{3(n+1)/2}$ for $a<a_\phi$.
\end{enumerate}
\end{lemma}
\begin{proof}
Throughout, $a<a_M$ is so small that
$a\leq\min\{\frac1{16},\frac14g^2\}$, $b<e^{-1}$ (so that $H\geq1$ and
$b^2H\leq1$), $MH\geq1$, $C_MbH^{3/2}\leq1$ and $2\pi a(MH+2)\leq1$;
this is possible because
$a\leq b^2$ and $b\to0$ uniformly. By \eqref{eq:count-bound} the number
of states with $\Delta_j\leq E$ is then at most $eA_gE$ for
$1\leq E\leq MH+2$.

(i) The sum is finite. Each $\Upsilon^{(l)}_{x_j}$ with $l\leq n$ is
even, supported in the set of the reflections of $x_j$, which is
contained in $\Gamma$, and of local order at most $2n$; the measure
$\nu_{\vac}$ is even and supported in $V\subset\Lambda_1\subset\Gamma$.
For the phase, let $X=(P,Q)$ be real with $P^2-Q^2\in\ZZ$ and let $f$ be
a polynomial. The even evaluation of $e^{2\pi i(P^2-Q^2)}f$ is
$e^{2\pi i(h-\bar h)}\ev f(h,\bar h)$, and along
$(h,\bar h)=(P^2+\frac\zeta2,Q^2+\frac\zeta2)$ the first factor is
identically one; by \eqref{eq:rig-jet-derivative},
$\langle\mathcal C\Upsilon^{(l)}_X,f\rangle
=\langle\Upsilon^{(l)}_X,f\rangle$. Both sides are distributions of
finite order with the same finite support, and polynomials realize
arbitrary jets at finitely many points; hence
$\mathcal C\Upsilon^{(l)}_X=\Upsilon^{(l)}_X$. This applies to the points
$x_j$ and to the terms of $\nu_{\vac}$, all of which have integral
spin.

(ii) For $h>0$ and $k\geq1$ the distribution $A_h^{(k)}$ is a
combination, with coefficients depending only on $k$ and $l$, of
$h^{-(2k-l)/2}\bigl(\delta^{(l)}_{\sqrt h}+(-1)^l\delta^{(l)}_{-\sqrt h}
\bigr)$, $1\leq l\leq k$, up to signs, by the second statement of
Lemma~\ref{lem:sq-momentum-derivative}; and
$A_0^{(k)}=\frac{k!}{(2k)!}\delta_0^{(2k)}$. A nonzero momentum
coordinate of a point of $\Gamma$ has square at least $(2d)^{-2}$.
Hence the coefficients of $\Upsilon^{(l)}_{x_j}$, $l\leq n$, are bounded
by a constant depending only on $d$ and $n$, and so are those of the
contribution of the state $j$ to \eqref{eq:col-array}, because
$|\zeta_j|\leq1$. A state contributes at a point $z\in\Gamma$ only if
$z$ is a reflection of $x_j$; then $|z|^2\leq\Delta_j+1\leq MH+1$ and
$\Delta_j\leq|z|^2+1\leq MH+2$, and there are at most
$eA_g(|z|^2+1)$ such states. Adding the coefficients of $\nu_{\vac}$
gives (ii).

(iii) \emph{The tests.} Let $\mathsf g$ range over the finitely many
functions $\psi_{0,\beta}$, $\mathbb S\psi_{0,\beta}$
($\beta\in\mathcal A_q$) and $f$, $\mathbb Sf$ ($f\in\mathcal F_q$).
Each is a finite sum of functions $\mathfrak m[p,z]$, by
Lemmas~\ref{lem:rig-cardinal-tests}(ii)
and~\ref{lem:rig-mixture-family}(ii), and
Lemma~\ref{lem:rig-mixture-family}(iii) gives a constant $C_1$ with
\begin{equation}\label{eq:col-fixed-envelope}
 |\partial^\alpha\mathsf g(X)|\leq C_1\,e^{-\pi|\operatorname{Re}X|^2}
 \qquad\text{for }|\alpha|\leq2n+2,\
 |\operatorname{Im}P|,|\operatorname{Im}Q|\leq1 .
\end{equation}
Let $f=\omega_\iota\mathbb S\psi_{y,\beta}$ be an element of
$\mathcal P_q(\sqrt{D_0H}+(q+3)d)$. By \eqref{eq:col-probe-translate},
$\psi_{y,\beta}=\tau_{-y}\psi_{0,\beta}$, and by the rules
$\mathbb S(\tau_{-w}f)=\chi_w\mathbb Sf$,
$\mathbb S(\chi_wf)=\tau_w\mathbb Sf$ and $\mathbb S^2f(X)=f(-X)$ of
Section~\ref{subsec:rig-conventions},
\begin{align*}
 f&=\sum_{k=0}^{q+1}(-1)^{q+1-k}\binom{q+1}k\,
 \chi_{y+k\mathsf b_\iota}\,\mathbb S\psi_{0,\beta},\\
 (\mathbb Sf)(X)&=\sum_{k=0}^{q+1}(-1)^{q+1-k}\binom{q+1}k\,
 \psi_{0,\beta}\bigl(-X-y-k\mathsf b_\iota\bigr).
\end{align*}
Thus, for every $f$ in $\mathcal F_q$ or in
$\mathcal P_q(\sqrt{D_0H}+(q+3)d)$, both $f$ and $\mathbb Sf$ are
combinations, with coefficients of total modulus at most $2^{q+1}$, of
functions $F$ of the following two types, where $\mathsf g$ is as above
and $y',\xi\in\RR^2$ satisfy
\begin{equation}\label{eq:col-centre-bound}
 |y'|,\ |\xi|\leq\varrho_H:=\sqrt{D_0H}+(2q+4)d :
\end{equation}
\begin{enumerate}[(I)]
\item real translates and reflections,
 $F(X)=\mathsf g(X-y')$ or $F(X)=\mathsf g(-X-y')$;
\item modulations, $F(X)=\chi_\xi(X)\,\mathsf g(X)
 =e^{2\pi iB(X,\xi)}\mathsf g(X)$.
\end{enumerate}
Indeed the light-cone coordinates of $y+k\mathsf b_\iota$ are bounded by
$\sqrt{D_0H}+(2q+4)d$, and $|x|^2=\frac12(u_x^2+v_x^2)
\leq\max\{u_x^2,v_x^2\}$. For type (I),
\eqref{eq:col-fixed-envelope} gives
$|\partial^\alpha F(X)|\leq C_1e^{-\pi|\operatorname{Re}X\mp y'|^2}$ on
the strip $|\operatorname{Im}P|,|\operatorname{Im}Q|\leq1$, uniformly
in $y'$. For type (II), the derivatives of order $k$ of
$e^{2\pi iB(X,\xi)}$ are bounded by
$(4\pi|\xi|)^ke^{4\pi|\operatorname{Im}X||\xi|}$, since
$B(X,\xi)=2(PP_\xi-QQ_\xi)$. On the thin strip
$|\operatorname{Im}P|,|\operatorname{Im}Q|\leq\sqrt{a/2}$ we have
$|\operatorname{Im}X||\xi|\leq\sqrt a\,\varrho_H\leq C$, because
$aH\leq b^2H\leq1$. Altogether, for both types and $|\alpha|\leq2n+2$,
\begin{equation}\label{eq:col-probe-sup}
 |\partial^\alpha F(X)|\leq C_2\,(1+H)^{n+1}\times
 \begin{cases}e^{-\pi|\operatorname{Re}X\mp y'|^2}&\text{(type I)},\\
 e^{-\pi|\operatorname{Re}X|^2}&\text{(type II)},\end{cases}
 \qquad|\operatorname{Im}P|,|\operatorname{Im}Q|\leq\sqrt{a/2}.
\end{equation}
The loss is polynomial in $H$ and uniform in $y'$ and $\xi$. All these
functions belong to $\mathcal G_{2\pi}$, which is stable under real
translations, reflections and multiplication by exponentials. By
Lemma~\ref{lem:gaussian-identity},
$\mathcal L_a(f)=\mathcal L_a(\mathbb Sf)$, and therefore
\begin{equation}\label{eq:col-defect-split}
 \langle(\mathbb S-I)\mathsf T_a,f\rangle
 =\bigl[\mathcal L_a(f)-\langle\mathsf T_a,f\rangle\bigr]
 -\bigl[\mathcal L_a(\mathbb Sf)-\langle\mathsf T_a,\mathbb Sf\rangle
 \bigr].
\end{equation}
It suffices to bound $\mathcal L_a(F)-\langle\mathsf T_a,F\rangle$ for
$F$ of type (I) or (II).

\emph{The decomposition.} For a state with $\Delta_j\leq MH$ let
$\Phi_j(\zeta)=\ev F(P_j^2+\frac\zeta2,Q_j^2+\frac\zeta2)$. Since
$P_j^2=\frac12(|x_j|^2+s_j)$ and $h_j=\frac12(\Delta_j+s_j)$, and
likewise for $Q_j^2$ and $\bar h_j$, we have
$\Phi_j(\zeta_j)=\ev F(h_j,\bar h_j)$, and by
\eqref{eq:rig-jet-derivative} the pairing of the terms of the state $j$
in \eqref{eq:col-array} with $F$ is the Taylor polynomial of degree $n$
of $\Phi_j$ at $0$, evaluated at $\zeta_j$. Hence
\begin{equation}\label{eq:col-array-split}
\begin{aligned}
 \mathcal L_a(F)-\langle\mathsf T_a,F\rangle
 ={}&\sum_{k,l\in\{0,1\}}(-1)^{k+l}\Bigl[\ev F\bigl(k-\tfrac a2,
 l-\tfrac a2\bigr)-\ev F(k,l)\Bigr]\\
 &+\sum_{j:\,\Delta_j\leq MH}\Bigl[\ev F\bigl(h_j-\tfrac a2,
 \bar h_j-\tfrac a2\bigr)-\ev F(h_j,\bar h_j)\Bigr]\\
 &+\sum_{j:\,\Delta_j\leq MH}\mathrm{Rem}_n[\Phi_j](\zeta_j)
 +\sum_{j:\,\Delta_j>MH}\ev F\bigl(h_j-\tfrac a2,\bar h_j-\tfrac a2\bigr),
\end{aligned}
\end{equation}
where $\mathrm{Rem}_n$ is the Taylor remainder of order $n$ at $0$.

\emph{Taylor remainders.} For $\zeta'$ between $0$ and $\zeta_j$ the
point $(P_j^2+\frac{\zeta'}2,Q_j^2+\frac{\zeta'}2)$ lies on the segment
from $(P_j^2,Q_j^2)$ to $(h_j,\bar h_j)$, so both its coordinates are
nonnegative, and the sets $\mathsf J$ of
Lemma~\ref{lem:col-pointwise} at these coordinates are real. The
derivative $\Phi_j^{(n+1)}(\zeta')$ is an average of the derivatives
$\partial_h^i\partial_{\bar h}^{n+1-i}\ev F$ at this point. By
Lemma~\ref{lem:col-pointwise} with $k=n+1$ and
\eqref{eq:col-probe-sup}, $|\Phi_j^{(n+1)}(\zeta')|\leq C(1+H)^{n+1}$.
With \eqref{eq:col-association} and the bound $eA_gMH$ for the number
of states,
\[
 \sum_{j:\,\Delta_j\leq MH}\bigl|\mathrm{Rem}_n[\Phi_j](\zeta_j)\bigr|
 \leq C\,H\,(1+H)^{n+1}\bigl(C_MbH^{3/2}\bigr)^{n+1}
 \leq C\,b^{n+1}(1+H)^{3n+4}.
\]

\emph{Central shifts.} For real $h,\bar h\geq0$,
\[
 \ev F\bigl(h-\tfrac a2,\bar h-\tfrac a2\bigr)-\ev F(h,\bar h)
 =-\frac a2\int_0^1(\partial_h+\partial_{\bar h})\ev F
 \bigl(h-\tfrac{\tau a}2,\bar h-\tfrac{\tau a}2\bigr)\dd\tau .
\]
The arguments are real and at least $-\frac a2$, so the points of the
sets $\mathsf J$ at these arguments have imaginary parts of modulus at
most $\sqrt{a/2}$. By Lemma~\ref{lem:col-pointwise} with $k=1$ and
\eqref{eq:col-probe-sup} the difference is at most $Ca(1+H)^{n+1}$ in
modulus. Summing over the states with $\Delta_j\leq MH$ and the four
vacuum weights, the first two sums of \eqref{eq:col-array-split} are
bounded by $Ca(1+H)^{n+2}$.

\emph{Tails.} Let $\Delta_j>MH$ and let $X=X^\sigma_{j,a}$ be a shifted
point of the state. By \eqref{eq:strip},
$|\operatorname{Im}P|,|\operatorname{Im}Q|\leq\sqrt{a/2}$ and
$|\operatorname{Re}X|^2\geq\Delta_j-a\geq MH-1$. By
\eqref{eq:col-centre-bound} and \eqref{eq:col-M-condition},
$4\varrho_H^2\leq8D_0H+8(2q+4)^2d^2\leq(M-1)H\leq MH-1$, so
$|y'|\leq\frac12|\operatorname{Re}X|$. Hence
$|\operatorname{Re}X\mp y'|\geq\frac12|\operatorname{Re}X|$, and
\eqref{eq:col-probe-sup} gives, for both types,
$|F(X)|\leq C(1+H)^{n+1}e^{-\frac\pi4(\Delta_j-a)}$. By
\eqref{eq:ker-point-evaluations} the same bound holds for
$\ev F(h_j-\frac a2,\bar h_j-\frac a2)$. By \eqref{eq:thermal-tail} with
$\beta=\frac\pi4$ and $E_0=MH$, where $\beta-2\pi a\geq\frac\pi8$,
\[
 \sum_{j:\,\Delta_j>MH}e^{-\frac\pi4\Delta_j}
 \leq C\,H\,e^{-\frac\pi8MH}=C\,H\,b^{\pi M/8}\leq C\,H\,b^{n+3},
\]
by \eqref{eq:col-M-condition}. The last sum of
\eqref{eq:col-array-split} is therefore at most $Cb^{n+2}$.

Adding the three bounds and using \eqref{eq:col-defect-split} gives
\eqref{eq:col-array-defect}.

(iv) Let $\phi(P,Q)=\Theta(P^2,Q^2)$ and let $E_\phi$ be an upper bound
for $h+\bar h$ on the support of $\Theta$. For a state with
$\Delta_j\leq MH$ put
$\Phi_j(\zeta)=\Theta(P_j^2+\frac\zeta2,Q_j^2+\frac\zeta2)$. By
Lemma~\ref{lem:col-jets} and \eqref{eq:col-array},
\[
 \langle\mu_a,\phi\rangle-\langle\mathsf T_a,\phi\rangle
 =\sum_{j:\,\Delta_j\leq MH}\mathrm{Rem}_n[\Phi_j](\zeta_j)
 +\sum_{j:\,\Delta_j>MH}\Theta(h_j,\bar h_j).
\]
Let $a$ be so small that $MH>E_\phi+1$. Then the second sum vanishes.
Since $|\zeta_j|\leq1$, a term of the first sum can be different from
zero only if $\Delta_j\leq E_\phi+1$, which holds for at most
$eA_g(E_\phi+1)$ states, and each term is at most
$|\zeta_j|^{n+1}$ times a bound for the derivatives of order $n+1$ of
$\Theta$. The claim follows from \eqref{eq:col-association}.
\end{proof}

\begin{lemma}[Normal form]\label{lem:normal-form}
Let $0<g\leq\frac12$ and let $P_*$ be the operator of
Lemma~\ref{lem:reconstruction-operator}. For every integer $n\geq1$ and
every finite set $\mathcal K$ of weight tests there are constants
$C_n$, $K_n$ and $a_n>0$, depending only on $g$, $n$ and $\mathcal K$,
such that every admissible partition function of gap $g$ at $c=1+12a$
with $0<a<a_n$ satisfies
\begin{equation}\label{eq:col-normal-form}
 \bigl|\langle\mu_a-P_*\mu_a,\phi\rangle\bigr|
 \leq C_n\,\bigl(a+b_a^{n+1}\bigr)\,(1+H_a)^{K_n}\qquad
 (\phi\in\mathcal K).
\end{equation}
One may take $K_n=5n+5$. Along an admissible sequence converging to
$\sigma(R_1,R_2)$ with half-integers $R_1,R_2$, the measure $P_*\mu_a$
is, for small $a$, the positive combination
\eqref{eq:col-reconstructed-measure} of at most four circle measures
with total weight one.
\end{lemma}
\begin{proof}
The choices are made in the following order. First, $g$ determines $d$,
the grid $\Gamma$, the prime $\mathfrak p$, the cutoffs and $P_*'$.
Secondly, $n$ determines $q=2n$ and, by
Lemma~\ref{lem:col-finite-projection}, the family $\mathcal F_q$ and the
projection $\Pi_q$. Thirdly, we apply
Proposition~\ref{prop:grid-projection} with this $q$, with $A_0=2$ and
$A_1=n+2$; this fixes $D_0=4(n+2)/\pi$. Fourthly, we fix $M$ satisfying
\eqref{eq:col-M-condition} for this $D_0$. Only then do we invoke
Corollary~\ref{cor:four-scalar-localization} with this $M$, which
provides $C_M$ and $a_M$, and Lemma~\ref{lem:col-array}. No earlier
choice depends on a later one: $\mathcal F_q$, $\Pi_q$ and $D_0$ do not
depend on $M$, and by Lemma~\ref{lem:col-array}(ii) neither do the
exponent $A_0=2$ and the constant $C_0$.

Let $a$ be small and let $\mathsf T_a$ be the Taylor array
\eqref{eq:col-array} for these $n$ and $M$. By
Lemma~\ref{lem:col-array}(i)--(iii) the hypotheses of
Proposition~\ref{prop:grid-projection} hold for $\mathsf T=\mathsf T_a$
with $b=b_a$ and $\varepsilon=C(a+b_a^{n+1})(1+H_a)^{3n+4}$. Hence there
is $\mathsf V_a\in\mathsf V_q$ such that, for every compact $K$,
\begin{equation}\label{eq:col-array-close}
 \max_{z\in\Gamma\cap K}\ \max_{|\alpha|\leq q}
 \bigl|c^{\mathsf T_a}_\alpha(z)-c^{\mathsf V_a}_\alpha(z)\bigr|
 \leq C_K\Bigl((a+b_a^{n+1})(1+H_a)^{5n+5}+b_a^{n+2}\Bigr).
\end{equation}
Let $\phi\in\mathcal K$ and $\chi=\phi-P_*'\phi$, a weight test by
Lemma~\ref{lem:reconstruction-operator}(i). Then
\[
 \langle\mu_a-P_*\mu_a,\phi\rangle=\langle\mu_a,\chi\rangle
 =\bigl[\langle\mu_a,\chi\rangle-\langle\mathsf T_a,\chi\rangle\bigr]
 +\langle\mathsf T_a-\mathsf V_a,\chi\rangle
 +\langle\mathsf V_a,\chi\rangle .
\]
The last term vanishes by Lemma~\ref{lem:reconstruction-operator}(iv).
The first is at most $C_\chi b_a^{n+1}(1+H_a)^{3(n+1)/2}$ by
Lemma~\ref{lem:col-array}(iv). The second is the finite sum of the
terms $(-1)^{|\alpha|}(c^{\mathsf T_a}_\alpha-c^{\mathsf V_a}_\alpha)(z)\,
\partial^\alpha\chi(z)$ over $z\in\Gamma\cap\supp\chi$ and
$|\alpha|\leq q$, and is bounded by \eqref{eq:col-array-close} with
$K=\supp\chi$. This proves \eqref{eq:col-normal-form}. The last
assertion is Lemma~\ref{lem:reconstruction-operator}(v).
\end{proof}

The bound \eqref{eq:col-normal-form} holds for each fixed $n$ and each
fixed finite set of tests; it is not uniform in a window growing with
$H_a$. The operator $P_*$ does not depend on $n$ or on the tests.

\subsection{Pairing, pinning and the comparison measure}
\label{subsec:col-comparison}

The combination \eqref{eq:col-reconstructed-measure} is not yet of the
form $\frac12\rhoF_R+\frac12\rhoF_t$: at the self-dual radius it
consists of circles of weight $\frac14$, and none of its radii is an
exact half-integer. Both defects are removed by comparing finitely many
moments of clusters. The moments are converted into positions by the
following elementary lemma, which requires no separation between the
points.

\begin{lemma}[Moment matching]\label{lem:moment-matching}
Let $k\geq1$ be an integer.
\begin{enumerate}[(i)]
\item There is a constant $C_k$ with the following property. Let
 $y_1\leq\dots\leq y_k$ and $y_1'\leq\dots\leq y_k'$ be real numbers in
 $[-1,1]$, let $0\leq e\leq1$, and suppose that
 \[
 \Bigl|\sum_{i=1}^ky_i^l-\sum_{i=1}^k(y_i')^l\Bigr|\leq e\qquad
 \text{for }1\leq l\leq2k-1 .
 \]
 Then $|y_i-y_i'|\leq C_k\,e^{1/(2k-1)}$ for all $i$.
\item Let $y_1\leq\dots\leq y_k$ and $y_1'\leq\dots\leq y_k'$ be real,
 let $e\geq0$, and suppose that $|y_i-y'_{\pi(i)}|\leq e$ for all $i$
 and some permutation $\pi$. Then $|y_i-y_i'|\leq e$ for all $i$.
\end{enumerate}
For $k\leq4$ the exponent in \textup{(i)} is at least $\frac17$.
\end{lemma}
\begin{proof}
(i) Suppose that $\varrho=|y_i-y_i'|>0$ for some $i$; exchanging the two
families we may assume $y_i<y_i'$. For $x\in[y_i,y_i')$ at least $i$ of
the numbers $y_l$ and at most $i-1$ of the numbers $y_l'$ are at most
$x$. At most $2k-2$ of the $2k$ numbers $y_l,y_l'$ lie in the open
interval $(y_i,y_i')$, and they divide it into at most $2k-1$ intervals.
Hence there is an interval $(\alpha,\beta)\subset(y_i,y_i')$ of length
at least $\varrho/(2k-1)$ that contains none of the $2k$ numbers. Fix a
smooth function $\mathsf s_0$ on $\RR$ with $\mathsf s_0=1$ on
$(-\infty,0]$ and $\mathsf s_0=0$ on $[1,\infty)$, and let
$\mathsf s(x)=\mathsf s_0((x-\alpha)/(\beta-\alpha))$, so that
$\|\mathsf s^{(l)}\|_\infty\leq C(\beta-\alpha)^{-l}$ for
$l\leq2k-1$. Let $\xi_1<\dots<\xi_m$, $m\leq2k$, be the distinct values
among the $2k$ numbers, and let $\mathsf q$ be the polynomial of degree
at most $m-1$ that agrees with $\mathsf s$ at these points. In Newton
form
$\mathsf q(x)=\sum_{l=0}^{m-1}\mathsf s[\xi_1,\dots,\xi_{l+1}]
\prod_{l'\leq l}(x-\xi_{l'})$, where the divided difference
$\mathsf s[\xi_1,\dots,\xi_{l+1}]$ equals $\mathsf s^{(l)}/l!$ at some
point and is therefore at most $C(\beta-\alpha)^{-l}$ in modulus. All
$\xi_{l'}$ lie in $[-1,1]$ and $\beta-\alpha\leq2$, so the coefficients
of $\mathsf q(x)=\sum_{l<2k}\mathsf q_lx^l$ satisfy
$|\mathsf q_l|\leq C_k'(\beta-\alpha)^{-(2k-1)}
\leq C_k'(2k-1)^{2k-1}\varrho^{-(2k-1)}$. Now
$\sum_l\mathsf q(y_l)-\sum_l\mathsf q(y_l')$ is the number of $y_l$ that
are at most $\alpha$ minus the number of $y_l'$ that are at most
$\alpha$, which is at least one. On the other hand it equals
$\sum_{l=1}^{2k-1}\mathsf q_l\bigl(\sum_iy_i^l-\sum_i(y_i')^l\bigr)$,
the terms with $l=0$ cancelling, and is at most
$(2k-1)C_k'(2k-1)^{2k-1}\varrho^{-(2k-1)}e$. Hence
$\varrho^{2k-1}\leq C_k^{2k-1}e$.

(ii) Suppose that $y_i>y_i'+e$ for some $i$. For $l\geq i$ we have
$y'_{\pi(l)}\geq y_l-e\geq y_i-e>y_i'$, hence $\pi(l)>i$. Thus $\pi$
maps the $k-i+1$ indices $l\geq i$ injectively into the $k-i$ indices
greater than $i$, which is impossible. The case $y_i'>y_i+e$ is
symmetric, with $\pi^{-1}$ in place of $\pi$.
\end{proof}

In the following theorem $m_r=1+\one_{\{r\in\{1,2\}\}}$ for a
half-integer $r\geq1$, as in \eqref{eq:cls-m-r}, and the windows are
those of \eqref{eq:col-window}. Recall from
Lemma~\ref{lem:col-states} that the states of the limit
$\sigma(R_1,R_2)$ are those of the family
$\mathfrak S(R_1,R_2)$. A family of states is determined by its
measure, so $\mathfrak S(R_1,R_2)$ and $\mathfrak S(R_2,R_1)$ consist of
the same states with the same multiplicities.

\begin{theorem}[One comparison measure]\label{thm:comparison}
Let $0<g\leq\frac12$ and let an admissible sequence of gap $g$ converge
to $\sigma(R_1,R_2)$, where $R_1\leq R_2$ are half-integers. For all
sufficiently small $a$ in the sequence there are an index
$\iota_a\in\{1,2\}$ and a radius $t_a\geq1$, defined in
\eqref{eq:col-pairing} and \eqref{eq:col-pin} below from the dimensions
of the states of $\rho_a$ in the private clusters, with the following
properties. Write $R=R_{\iota_a}$ and $T=R_{3-\iota_a}$; these depend on
$a$ only through $\iota_a$.
\begin{enumerate}[(i)]
\item $t_a-T\to0$ as $a\to0$ in the sequence.
\item For every weight test $\phi$ and every integer $n\geq1$ there are
 constants $C$, $K$ such that, for all sufficiently small $a$,
 \begin{equation}\label{eq:col-comparison-tests}
 \bigl|\langle\rho_a-\sigma(R,t_a),\phi\rangle\bigr|
 \leq C\,\bigl(a+b_a^{n+1}\bigr)^{1/12}(1+H_a)^{K}.
 \end{equation}
\item Let $(E_0,s_0)\in\RR\times\ZZ$, let
 $\mathcal W_\eta(E_0,s_0)$ be isolated for $\mathfrak S(R_1,R_2)$, and
 let $N_0$ be the number of states of $\sigma(R_1,R_2)$ with dimension
 $E_0$ and spin $s_0$. For all sufficiently small $a$, each of the
 families $(\Delta_j,s_j)_j$ and $\mathfrak S(R,t_a)$ has exactly $N_0$
 states with spin $s_0$ and dimension in $(E_0-2\eta,E_0+2\eta)$. If
 $y_1\leq\dots\leq y_{N_0}$ and $y_1'\leq\dots\leq y_{N_0}'$ are their
 dimensions, then for every $n\geq1$ there are $C$, $K$ with
 \begin{equation}\label{eq:col-comparison-matching}
 \max_{1\leq i\leq N_0}|y_i-y_i'|
 \leq C\,\bigl(a+b_a^{n+1}\bigr)^{1/84}(1+H_a)^{K}.
 \end{equation}
\item Let $\tilde t_a$ be radii with $|\tilde t_a-t_a|\leq e_a$, where
 $e_a\to0$. Then \textup{(iii)} holds for
 $\mathfrak S(R,\tilde t_a)$ in place of $\mathfrak S(R,t_a)$, with
 the right side of \eqref{eq:col-comparison-matching} increased by
 $Ce_a$.
\end{enumerate}
The index $\iota_a$ and the radius $t_a$ do not depend on $n$, on
$\phi$, or on the window; only the constants do.
\end{theorem}

Thus a single measure $\sigma(R,t_a)$, with an exact half-integer $R$,
matches the states of every fixed cluster up to
$(a+b_a^{n+1})^{1/84}$ and logarithms, for every $n$; once
$a\leq\delta_a^m$ is known for every $m$
(Theorem~\ref{thm:four-scalar-slow}), this is every order in $b_a$. No
statement is made that is uniform in a window growing with $H_a$.

\begin{proof}
Fix $n\geq1$. We use Lemma~\ref{lem:normal-form} for a finite set
$\mathcal K$ of weight tests consisting of the tests $\phi^{(1)}$,
$\phi^{(3)}$ and $\phi_1,\dots,\phi_4$ introduced below, the test
$\phi$ of (ii), and the tests \eqref{eq:col-moment-tests} of the
window considered in (iii). We write
\[
 \varepsilon_a=C_n(a+b_a^{n+1})(1+H_a)^{K_n}
\]
for the right side of \eqref{eq:col-normal-form}; it tends to zero. All
statements hold for sufficiently small $a$ in the sequence. For a weight
test $\phi$ the function $\mathsf f_\phi(t)=\langle\rhoF_t,\phi\rangle$
is smooth, so
\begin{equation}\label{eq:col-lipschitz}
 |\mathsf f_\phi(t)-\mathsf f_\phi(t')|\leq C_\phi\,|\log t-\log t'|
 \qquad\bigl(t,t'\in[\tfrac1{2d},2d]\bigr).
\end{equation}

\emph{Step 1: the reconstructed radii.} By
Lemma~\ref{lem:reconstruction-operator}(v),
$P_*\mu_a=\sum_r\mathsf c_r\sum_{i\leq n_r}\rhoF_{t_{r,i}}$ with
$t_{r,i}=\mathsf E^{-1}(\Delta_{r,i})\to r$, where only
$r\in\{R_1,R_2\}$ occur. If $R_1=1$, let
$\ell_1\leq\dots\leq\ell_{n_1}$ be the numbers $\log t_{1,i}$ in
increasing order; here $n_1=2$ if $R_2>1$ and $n_1=4$ if $R_2=1$. We
define two radii $\tau_1,\tau_2$ by
\begin{equation}\label{eq:col-pairing}
 (\tau_1,\tau_2)=\begin{cases}
 (t_{R_1,1},\,t_{R_2,1}),&1<R_1<R_2,\\
 (t_{R_1,1},\,t_{R_1,2}),&1<R_1=R_2,\\
 \bigl(e^{(\ell_2-\ell_1)/2},\,t_{R_2,1}\bigr),&1=R_1<R_2,\\
 \bigl(e^{(\ell_3-\ell_2)/2},\,e^{(\ell_4-\ell_1)/2}\bigr),&1=R_1=R_2.
 \end{cases}
\end{equation}
Then $\tau_i\to R_i$ for $i=1,2$, and $\tau_i\geq1$ if $R_i=1$. Put
$\mu_a^{\rm p}=\frac12\rhoF_{\tau_1}+\frac12\rhoF_{\tau_2}$.

\emph{Step 2: pairing at the self-dual radius.} We claim that for every
weight test $\phi$
\begin{equation}\label{eq:col-paired}
 \bigl|\langle P_*\mu_a-\mu_a^{\rm p},\phi\rangle\bigr|
 \leq C_\phi\,\varepsilon_a^{1/3}.
\end{equation}
If $R_1>1$ the two measures coincide, since $\mathsf c_r=\frac12$ for
$r>1$. Let $R_1=1$. For $k\in\{1,3\}$ let $\phi^{(k)}$ be the weight
test \eqref{eq:col-window-test} for $(E_0,s_0)=(\mathsf E(1),\mathfrak p)$,
$\eta=\eta_*$ and $\varphi(E)=(\log\mathsf E^{-1}(E))^k$; in the notation
of Section~\ref{subsec:col-reconstruction},
\[
 \phi^{(k)}(X)=\vartheta^{\rm sp}\bigl(P^2-Q^2-\mathfrak p\bigr)\,
 \vartheta_1\bigl(|X|^2\bigr)\,\bigl(\log\mathsf E^{-1}(|X|^2)\bigr)^k .
\]
It is smooth and compactly supported. By
Lemma~\ref{lem:reconstruction-operator}(v) and
\eqref{eq:col-window-pairing},
$\langle\mu_a,\phi^{(k)}\rangle=\sum_{i=1}^{n_1}\ell_i^k$. By
\eqref{eq:col-private-states} and \eqref{eq:col-window-pairing},
\[
 \langle\rhoF_t,\phi^{(k)}\rangle
 =2\,\vartheta_1(\mathsf E(t))\bigl(\log\mathsf E^{-1}(\mathsf E(t))\bigr)^k
 +2\,\vartheta_1(\mathsf E(1/t))
 \bigl(\log\mathsf E^{-1}(\mathsf E(1/t))\bigr)^k .
\]
For $t=t_{1,i}$, which lies in the interval $I_1$ of
Lemma~\ref{lem:reconstruction-operator}(iii) when $a$ is small, this is
$2(\log t)^k+2(-\log t)^k=0$, because $k$ is odd. For $t=t_{R_2,i}$ with
$R_2>1$, which lies in $I_{R_2}$, both cutoffs vanish, as shown in the
proof of Lemma~\ref{lem:reconstruction-operator}(iii). Hence
$\langle P_*\mu_a,\phi^{(k)}\rangle=0$, and
\eqref{eq:col-normal-form} gives
\begin{equation}\label{eq:col-odd-moments}
 \Bigl|\sum_{i=1}^{n_1}\ell_i\Bigr|\leq\varepsilon_a,\qquad
 \Bigl|\sum_{i=1}^{n_1}\ell_i^3\Bigr|\leq\varepsilon_a .
\end{equation}
Let $n_1=2$ and $x=\frac12(\ell_2-\ell_1)\geq0$. Then
$|\ell_1+x|=|\ell_2-x|=\frac12|\ell_1+\ell_2|\leq\varepsilon_a$. Since
$\rhoF_{e^{-x}}=\rhoF_{e^x}$, the part of $P_*\mu_a$ belonging to $r=1$
is $\frac14(\rhoF_{e^{\ell_1}}+\rhoF_{e^{\ell_2}})$ and the
corresponding part of $\mu_a^{\rm p}$ is
$\frac12\rhoF_{e^x}=\frac14(\rhoF_{e^{-x}}+\rhoF_{e^x})$; the other
parts coincide, and \eqref{eq:col-lipschitz} gives
\eqref{eq:col-paired}. Let $n_1=4$ and put
\[
 \alpha=\tfrac12(\ell_1+\ell_4),\quad\lambda=\tfrac12(\ell_4-\ell_1),
 \qquad
 \alpha'=\tfrac12(\ell_2+\ell_3),\quad\lambda'=\tfrac12(\ell_3-\ell_2).
\]
Then $\lambda\geq\lambda'\geq0$, and $\ell_1\leq\ell_2$,
$\ell_3\leq\ell_4$ give $\lambda-\lambda'\geq|\alpha-\alpha'|$. Let
$e=\alpha+\alpha'$; by \eqref{eq:col-odd-moments},
$|e|=\frac12|\sum_i\ell_i|\leq\varepsilon_a$. Since
$(\alpha-\lambda)^3+(\alpha+\lambda)^3=2\alpha^3+6\alpha\lambda^2$,
\[
 \sum_i\ell_i^3=2\alpha^3+6\alpha\lambda^2+2\alpha'^3+6\alpha'\lambda'^2
 =6\alpha(\lambda^2-\lambda'^2)+O(|e|),
\]
where we inserted $\alpha'=e-\alpha$ and used that all numbers are
bounded by one. Hence $|\alpha|(\lambda^2-\lambda'^2)\leq C\varepsilon_a$.
If $|\alpha|\geq\varepsilon_a$, then
$\lambda-\lambda'\geq|2\alpha-e|\geq|\alpha|$ and
$\lambda+\lambda'\geq\lambda-\lambda'$, so
$\lambda^2-\lambda'^2\geq\alpha^2$ and $|\alpha|^3\leq C\varepsilon_a$.
In either case $|\alpha|\leq C\varepsilon_a^{1/3}$, and
$|\alpha'|\leq|\alpha|+|e|\leq C\varepsilon_a^{1/3}$. Thus the numbers
$\ell_1,\ell_4,\ell_2,\ell_3$ are within $C\varepsilon_a^{1/3}$ of
$-\lambda,\lambda,-\lambda',\lambda'$, and
\[
 \tfrac14\bigl(\rhoF_{e^{-\lambda}}+\rhoF_{e^{\lambda}}
 +\rhoF_{e^{-\lambda'}}+\rhoF_{e^{\lambda'}}\bigr)
 =\tfrac12\rhoF_{e^{\lambda'}}+\tfrac12\rhoF_{e^{\lambda}}
 =\mu_a^{\rm p}.
\]
With \eqref{eq:col-lipschitz} this gives \eqref{eq:col-paired}.

\emph{Step 3: the scalar at dimension two pins one radius.} Put
\[
 x_i=\log(\tau_i/R_i)\to0\quad(i=1,2),\qquad k=m_{R_1}+m_{R_2}\in\{2,3,4\}.
\]
Choose $\eta_{\rm s}\in(0,\frac14]$ such that
$\mathcal W_{\eta_{\rm s}}(2,0)$ is isolated for $\mathfrak S(R_1,R_2)$.
By Lemma~\ref{lem:boundary-clusters}(iv) the limit has $k-1$ states of
dimension two and spin zero, so by Lemma~\ref{lem:col-windows}(ii)
exactly $k-1$ states of $\rho_a$ have spin zero and dimension in
$(2-2\eta_{\rm s},2+2\eta_{\rm s})$; we write their dimensions as
$2+z_i$, $1\leq i\leq k-1$, with $|z_i|<\eta_{\rm s}$. For
$1\leq l\leq4$ let $\phi_l$ be the weight test
\eqref{eq:col-window-test} for $(E_0,s_0)=(2,0)$, $\eta=\eta_{\rm s}$
and $\varphi(E)=(E-2)^l$. Among the terms of $\nu_{\vac}$, the state
$(0,0)$ lies outside the support of $\phi_l$, the states $(1,\pm1)$
have nonzero spin, and the state $(2,0)$ has $\varphi(2)=0$. Hence
\begin{equation}\label{eq:col-scalar-actual}
 \langle\mu_a,\phi_l\rangle=\sum_{i=1}^{k-1}z_i^l
 =\mathsf p_l(0,z_1,\dots,z_{k-1}),
\end{equation}
where $\mathsf p_l$ denotes the power sum of degree $l$ of $k$ numbers:
the fixed vacuum state at dimension two contributes the node $0$. By
Lemma~\ref{lem:col-states}(i), $\mu_a^{\rm p}$ is $\delta_0$ plus the
measure of the family of the states $(E_{m,n}(\tau_i),mn)$, $i=1,2$. A
class with $mn=0$ is the class of $(m,0)$ or of $(0,n)$, with
$E_{m,0}(\tau)=\frac12m^2\tau^2$ and $E_{0,n}(\tau)=\frac12n^2/\tau^2$.
At $\tau=R_i$ these equal two exactly for $n=\pm2R_i$ and, if
$R_i\in\{1,2\}$, for $m=\pm2/R_i$; this gives $m_{R_i}$ classes. Their
states have the dimensions
\[
 E_{0,2R_i}(\tau_i)=2e^{-2x_i},\qquad
 E_{2/R_i,0}(\tau_i)=2e^{2x_i}\quad(\text{if }R_i\in\{1,2\}).
\]
By Lemma~\ref{lem:col-states}(iii) and isolation, all other states of
spin zero of the family have dimension outside
$(2-2\eta_{\rm s},2+2\eta_{\rm s})$ when $a$ is small. Therefore
$\langle\mu_a^{\rm p},\phi_l\rangle=\mathsf p_l(w_1,\dots,w_k)$, where
$w_1,\dots,w_k$ are the numbers $2(e^{-2x_i}-1)$, $i=1,2$, together
with $2(e^{2x_i}-1)$ for those $i$ with $R_i\in\{1,2\}$. By
\eqref{eq:col-normal-form}, \eqref{eq:col-paired} and
\eqref{eq:col-scalar-actual},
\[
 \bigl|\mathsf p_l(w_1,\dots,w_k)-\mathsf p_l(0,z_1,\dots,z_{k-1})\bigr|
 \leq C\varepsilon_a^{1/3}\qquad(1\leq l\leq k).
\]
By Newton's identities the product of $k$ numbers is a polynomial with
rational coefficients in their power sums of degrees $1,\dots,k$. All
numbers involved are bounded by one, and the product of
$0,z_1,\dots,z_{k-1}$ vanishes. Hence
$|w_1\cdots w_k|\leq C\varepsilon_a^{1/3}$. Since
$|2(e^{\pm2x}-1)|\geq|x|$ for $|x|\leq\frac12$,
\begin{equation}\label{eq:col-pin-product}
 |x_1|^{m_{R_1}}|x_2|^{m_{R_2}}\leq C\varepsilon_a^{1/3},\qquad
 \text{hence}\qquad
 \min\{|x_1|,|x_2|\}\leq\bigl(C\varepsilon_a^{1/3}\bigr)^{1/k}
 \leq C\varepsilon_a^{1/12}.
\end{equation}
We define
\begin{equation}\label{eq:col-pin}
 \iota_a=\begin{cases}1,&|x_1|\leq|x_2|,\\2,&|x_1|>|x_2|,\end{cases}
 \qquad t_a=\tau_{3-\iota_a},\qquad R=R_{\iota_a},\quad T=R_{3-\iota_a},
\end{equation}
and $\mu_a^{\rm c}=\frac12\rhoF_R+\frac12\rhoF_{t_a}$. Thus the radius
with the smaller logarithmic displacement is replaced by its exact
limit. By \eqref{eq:col-lipschitz} and \eqref{eq:col-pin-product},
$|\langle\mu_a^{\rm p}-\mu_a^{\rm c},\phi\rangle|
=\frac12|\mathsf f_\phi(\tau_{\iota_a})-\mathsf f_\phi(R)|
\leq C_\phi\varepsilon_a^{1/12}$ for every weight test $\phi$.
Assertion (i) holds because $\tau_i\to R_i$, and $t_a\geq1$ for small
$a$: if $T>1$ this follows from (i), and if $T=1$ from Step~1.

\emph{Step 4: proof of \textup{(ii)}.} By \eqref{eq:sigma-R-t},
$\rho_a-\sigma(R,t_a)=\mu_a-\mu_a^{\rm c}$. By
\eqref{eq:col-normal-form}, \eqref{eq:col-paired} and Step~3,
\[
 \bigl|\langle\mu_a-\mu_a^{\rm c},\phi\rangle\bigr|
 \leq\varepsilon_a+C_\phi\varepsilon_a^{1/3}+C_\phi\varepsilon_a^{1/12}
 \leq C\,\varepsilon_a^{1/12},
\]
which is \eqref{eq:col-comparison-tests} with $K=K_n/12$.

\emph{Step 5: proof of \textup{(iii)}.} The statement about
$(\Delta_j,s_j)_j$ is Lemma~\ref{lem:col-windows}(ii). The statement
about $\mathfrak S(R,t_a)$ follows from Lemma~\ref{lem:col-windows}(i),
applied to the pair $(R,T)$ for each of the two possible values of
$\iota_a$, and from (i); note that
$\sigma(R,T)=\sigma(R_1,R_2)$. If $N_0=0$ there is nothing more to
prove. Otherwise $N_0\leq4$ by Lemma~\ref{lem:boundary-clusters}(iii).
For $1\leq l\leq2N_0-1$ let $\phi_{l}^{\mathcal W}$ be the weight test
\begin{equation}\label{eq:col-moment-tests}
 \phi_l^{\mathcal W}(X)=\vartheta^{\rm sp}\bigl(P^2-Q^2-s_0\bigr)\,
 \vartheta\Bigl(\frac{|X|^2-E_0}{\eta}\Bigr)
 \Bigl(\frac{|X|^2-E_0}{\eta}\Bigr)^l ,
\end{equation}
which is of the form \eqref{eq:col-window-test}. By
\eqref{eq:col-window-pairing} and the first part of (iii),
$\langle\rho_a,\phi_l^{\mathcal W}\rangle=\sum_iu_i^l$ and
$\langle\sigma(R,t_a),\phi_l^{\mathcal W}\rangle=\sum_i(u_i')^l$ with
$u_i=(y_i-E_0)/\eta$ and $u_i'=(y_i'-E_0)/\eta$, which lie in $(-1,1)$.
By (ii) these power sums differ by at most $C\varepsilon_a^{1/12}$, and
Lemma~\ref{lem:moment-matching}(i) with $k=N_0\leq4$ gives
$|y_i-y_i'|\leq C(\varepsilon_a^{1/12})^{1/7}=C\varepsilon_a^{1/84}$.
This is \eqref{eq:col-comparison-matching} with $K=K_n/84$.

\emph{Step 6: proof of \textup{(iv)}.} By
Lemma~\ref{lem:col-windows}(i), for $t$ close to $T$ the states of
$\mathfrak S(R,t)$ in the window are fixed states, which do not depend
on $t$, and the moving states of finitely many classes that do not
depend on $t$. By Lemma~\ref{lem:col-states}(iii) the dimension of a
moving state changes by at most $C|\tilde t_a-t_a|\leq Ce_a$ when $t_a$
is replaced by $\tilde t_a$. Lemma~\ref{lem:moment-matching}(ii) shows
that the ordered dimensions change by at most $Ce_a$.
\end{proof}

The exponent $\frac1{84}=\frac13\cdot\frac14\cdot\frac17$ is the product
of the losses in the pairing of four self-dual nodes, in the pinning,
where at most four scalar branches meet, and in the matching of at most
four states. The bound four in the last step is
Lemma~\ref{lem:boundary-clusters}(iii); it includes the clusters at the
points of the vacuum support, where the limit has at most three long
states.

\begin{corollary}[Matching of the low states]\label{cor:col-matching}
In the situation of Theorem~\ref{thm:comparison}, let $E_*\geq2$ and
$n\geq1$. There are constants $C$, $K$ and, for all sufficiently small
$a$ in the sequence, an injective map $j\mapsto j'$ from the set of
states of $\rho_a$ with $\Delta_j\leq E_*$ into the family
$\mathfrak S(R,t_a)$, whose image contains every state of
$\mathfrak S(R,t_a)$ of dimension at most $E_*-1$, such that $j'$ has
the spin of $j$ and the dimensions satisfy
\begin{equation}\label{eq:col-matching}
 |\Delta_j-\Delta'_{j'}|\leq C\,\bigl(a+b_a^{n+1}\bigr)^{1/84}(1+H_a)^{K}.
\end{equation}
The same holds for $\mathfrak S(R,\tilde t_a)$ when
$|\tilde t_a-t_a|\leq e_a\to0$, with the right side increased by
$Ce_a$.
\end{corollary}
\begin{proof}
Let $\mathfrak C$ be the finite set of pairs $(E_0,s_0)$ that occur as
states of $\sigma(R_1,R_2)$ with $E_0\leq E_*+1$, and choose
$\eta\in(0,\frac14]$ such that all windows $\mathcal W_\eta(E_0,s_0)$,
$(E_0,s_0)\in\mathfrak C$, are isolated for $\mathfrak S(R_1,R_2)$.
These windows are pairwise disjoint. Choose $E'\in(E_*,E_*+1)$ that is
not the dimension of a state of $\sigma(R_1,R_2)$, and let $U$ be the
set of $X$ with $|X|^2<E'$ that lie in the closure of none of the
windows $\mathcal W_{\eta/2}(E_0,s_0)$, $(E_0,s_0)\in\mathfrak C$. This
is a bounded open set, invariant under the reflections, its boundary
does not meet $\supp\sigma(R_1,R_2)$, and no state of
$\sigma(R_1,R_2)$ is supported in it. By
Corollary~\ref{cor:cluster-stabilization}, for small $a$ no state of
$\rho_a$ is supported in $U$. The four physical points of a state have
the same squared norm and the same spin, so a state with
$\Delta_j<E'$ is either supported in $U$ or supported in the closure of
one of the windows $\mathcal W_{\eta/2}(E_0,s_0)$. Hence every state of
$\rho_a$ with $\Delta_j\leq E_*$ is supported in one of the windows
$\mathcal W_\eta(E_0,s_0)$, $(E_0,s_0)\in\mathfrak C$. The same holds
for every state of $\mathfrak S(R,t_a)$ of dimension at most $E_*-1$: a
fixed state is a state of $\sigma(R_1,R_2)$, and a moving state of a
class $\{\pm(m,n)\}$ satisfies $|E_{m,n}(t_a)-E_{m,n}(T)|<\eta$ for
small $a$, by Lemma~\ref{lem:col-states}(iii), while
$(E_{m,n}(T),mn)$ is a state of $\sigma(R_1,R_2)$ of dimension at most
$E_*$. In each window, Theorem~\ref{thm:comparison}(iii) provides the
same number of states of the two families and the bound
\eqref{eq:col-comparison-matching} for their ordered dimensions. We let
$j'$ be the state of $\mathfrak S(R,t_a)$ in the window of $j$ with the
same position in the order by dimension, ties being resolved by a fixed
enumeration. The last assertion follows in the same way from
Theorem~\ref{thm:comparison}(iv).
\end{proof}

\subsection{Kernel data at a pair of half-integers and their collisions}
\label{subsec:col-data}

We now fix the kernels that will be used at the moving radius. Since the
radius is not a limit of the sequence, the hypotheses
\textup{(H1)--(H4)} and \textup{(S1)--(S5)} of
Section~\ref{subsec:ker-families} are replaced by quantitative
statements on a punctured neighbourhood of $T$.

\paragraph{Collision data.}
Let $R,T$ be half-integers with $1\leq R,T\leq R_{\max}$. We put
\begin{equation}\label{eq:col-data}
 I=\bigl[\tfrac12,\,R_{\max}+1\bigr],\qquad\ell=1,\qquad
 p=\text{the smallest odd integer greater than }(R_{\max}+1)^2+2,
\end{equation}
so that \eqref{eq:ker-p-conditions} holds, $T$ is an interior point of
$I$ also when $T=1$, and $p\geq7$. As in
Section~\ref{subsec:ker-families}, $p_k=p+k$,
$Y_\nu(t)=[t,\nu/t]$, $B_k(t)=Y_{p_k}(t)$ and
$E_\nu(t)=\frac12(t^2+\nu^2/t^2)=E_{1,\nu}(t)$; the function $E_p$ is
strictly decreasing on $I$, with $E_p'(t)=t-p^2/t^3\leq-4/t$ there.
\begin{itemize}
\item If $R\notin\{1,2\}$, the \emph{product data} are
 \[
 U=4R,\qquad V_j=2(4R+3+j),\qquad A_j=[U,V_j]\quad(j=0,1,2).
 \]
 Then $A_j=[4\cdot R,\,2R(4R+3+j)/R]\in\Lambda_R$, $U\geq6$,
 $V_j\geq18$ and $V_j-U=4R+6+2j\geq3$, so
 \eqref{eq:ker-A-conditions} holds, with $\Delta V=2$. The polynomials
 are the ten functions $\Phi^*_t$, $\Phi^{j,k}_t$ of
 \eqref{eq:ker-product-family}.
\item If $R\in\{1,2\}$, the \emph{sum data} are $w=3$, so that
 $W(t)=Y_3(t)=[t,3/t]$ and $\mathcal N=\{p_0,p_1,p_2,3\}$. The
 polynomials are the four functions
 $\Phi^{(\nu)}_t=\Omega\,S_{Y_\nu(t)}S_{Y_\nu(t)^\sharp}$,
 $\nu\in\mathcal N$. Since $T^2\neq3$ and $T^2<p$, the points
 $Y_\nu(T)$, $\nu\in\mathcal N$, are off both axes.
\end{itemize}
In both cases the polynomials vanish on the set $\Sigma(t)$ of
\eqref{eq:ker-Sigma-t} for every $t>0$, as shown in
Section~\ref{subsec:ker-families}. Their centres are written as in
\eqref{eq:ker-product-centres} and Lemma~\ref{lem:kernel-fourier}(ii),
\begin{align*}
 \xi&=\eta_1Y+\eta_2Y^\sharp+\eta_3B_k(t)+\eta_4B_k(t)^\sharp
 &&(Y\in\{C,A_0,A_1,A_2\}),\\
 \xi&=\zeta+\eta_1Y_\nu(t)+\eta_2Y_\nu(t)^\sharp
 &&(\zeta\in\{0,\pm C,\pm C^\sharp\}),
\end{align*}
with signs $\eta_i\in\{-1,0,1\}$; the number $|\eta|$ of nonzero signs
is the \emph{length} of the centre. By
Lemma~\ref{lem:kernel-fourier}, the coefficient of a centre of length
$|\eta|$ is $\frac1{16}(-\frac12)^{|\eta|}$ for the product
polynomials; for the sum polynomials it is positive for
$(\zeta=0,|\eta|\in\{0,2\})$ and $(\zeta\neq0,|\eta|=1)$ and negative
for $(\zeta=0,|\eta|=1)$ and $(\zeta\neq0,|\eta|\in\{0,2\})$.

All these centres, and the row preimages $x_{l,k,j}(t)$ of
\eqref{eq:ker-row-preimages} in the case of sum data, have the form
\eqref{eq:ker-general-centre},
\begin{equation}\label{eq:col-laurent-point}
 \xi(t)=\bigl[a_1+b_1t+b_2\nu/t,\ a_2+b_2t+b_1\nu/t\bigr],\qquad
 a_1,a_2,b_1,b_2\in\ZZ,\quad\nu\geq1,
\end{equation}
with
\begin{equation}\label{eq:col-ab}
\begin{aligned}
 &[a_1,a_2]=\eta_1Y+\eta_2Y^\sharp,\quad(b_1,b_2)=(\eta_3,\eta_4),\quad
 \nu=p_k&&\text{(product centres)},\\
 &[a_1,a_2]=\zeta,\quad(b_1,b_2)=(\eta_1,\eta_2)&&\text{(sum centres)},\\
 &[a_1,a_2]=[2,\,2-2l],\quad(b_1,b_2)=(-k,-j),\quad\nu=3
 &&\text{(row preimages)}.
\end{aligned}
\end{equation}
We call these finitely many maps $t\mapsto\xi(t)$ the \emph{objects} of
the data and denote their set by $\mathfrak O$. Let $\varrho_1\geq2$ be
a bound for $|\xi(t)|$, $\xi\in\mathfrak O$, $t\in I$, and let
\begin{equation}\label{eq:col-E-star}
 E_*=(\varrho_1+3)^2 .
\end{equation}
The \emph{reference branches} are the constant maps with value in
$(V\cup\Lambda_R\cup\Lambda_{1/R})\cap\overline{D(0,\varrho_1+2)}$ and
the maps $t\mapsto[mt,n/t]$ and $t\mapsto[m/t,nt]$ with integers
$|m|,|n|\leq N_1$, where $N_1$ is so large that every point of
$\Lambda_t\cup\Lambda_{1/t}$ of norm at most $\varrho_1+2$, for some
$t\in I$, is the value at $t$ of one of them. Their set $\mathfrak B$ is
finite, and
\begin{equation}\label{eq:col-branches}
 \Sigma(t)\cap\overline{D(0,\varrho_1+2)}
 \subset\{\lambda(t):\lambda\in\mathfrak B\}\subset\Sigma(t)
 \qquad(t\in I).
\end{equation}

\begin{lemma}[Zeros of order at most two]\label{lem:col-laurent}
Let $T>0$ and let $\mathfrak F$ be a finite set of real functions on
$(0,\infty)$, each of one of the two forms
\[
 f(t)=\alpha t+\beta+\gamma/t\qquad\text{or}\qquad
 f(t)=\alpha t^2+\beta+\gamma/t^2,\qquad\alpha,\beta,\gamma\in\RR .
\]
There are $\varepsilon_0\in(0,\frac12T]$ and $c>0$ such that for every
$f\in\mathfrak F$ and every $t$ with $0<|t-T|\leq\varepsilon_0$:
\begin{enumerate}[(i)]
\item either $f$ vanishes identically, or $|f(t)|\geq c\,|t-T|^2$;
\item either $f$ is constant with an integer value, or
 $|\sin(\pi f(t))|\geq c\,|t-T|^2$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i) Let $f\not\equiv0$. In the first case $tf(t)=\alpha t^2+\beta t+\gamma$
is a nonzero polynomial of degree at most two in $t$; in the second,
$t^2f(t)=\alpha s^2+\beta s+\gamma$ is a nonzero polynomial of degree at
most two in $s=t^2$. Such a polynomial vanishes at $T$, respectively at
$s=T^2$, to an order $k\in\{0,1,2\}$, so its quotient by $(t-T)^k$,
respectively $(s-T^2)^k$, is continuous and nonzero at that point, and
bounded below in modulus on a neighbourhood. Since
$|s-T^2|=|t-T|(t+T)\geq T|t-T|$ and $|t-T|\leq1$ may be assumed, this
gives $|f(t)|\geq c|t-T|^k\geq c|t-T|^2$ near $T$.

(ii) If $f(T)\notin\ZZ$, then $\dist(f(t),\ZZ)$ is bounded below near
$T$. If $f(T)=N\in\ZZ$, then $f-N$ is of the same form and does not
vanish identically unless $f$ is the constant $N$; by (i),
$|f(t)-N|\geq c|t-T|^2$, and $|f(t)-N|=\dist(f(t),\ZZ)$ near $T$. In
both cases \eqref{eq:ker-sine-distance} gives the claim.
\end{proof}

The exponent two cannot be improved for functions of the first form:
\begin{equation}\label{eq:col-double-zero}
 2-t-\frac1t=-\frac{(t-1)^2}{t} .
\end{equation}
Differences of this kind occur for the data above when $T=1$. For
instance, the first light-cone coordinate of the centre
$C-Y_\nu(t)-Y_\nu(t)^\sharp=[2-t-\nu/t,\,-t-\nu/t]$ differs from that of
the point $[(1-\nu)/t,\,-(1+\nu)t]\in\Lambda_{1/t}$ by
\eqref{eq:col-double-zero}. We do not decide in which cases the distance
between two points is in fact of first order, and use the quadratic
bound throughout.

\begin{lemma}[Quadratic bounds for collisions]
\label{lem:quadratic-collisions}
Let collision data be given. There are $\varepsilon_0\in(0,\frac14]$,
$c_0\in(0,1]$, $\varpi_0\in(0,1]$ and $\lambda_0>0$ such that
$[T-\varepsilon_0,T+\varepsilon_0]$ is contained in the interior of $I$
and the following hold for every $t$ with
$0<\theta:=|t-T|\leq\varepsilon_0$.
\begin{enumerate}[(i)]
\item \emph{Separations.} For every object $\xi\in\mathfrak O$ and every
 $\lambda\in\mathfrak O\cup\mathfrak B$, either $\xi$ and $\lambda$
 coincide identically on $I$, or
 $|\xi(t)-\lambda(t)|\geq c_0\,\theta^2$. Consequently
 \begin{equation}\label{eq:col-separation}
 \dist\bigl(\xi(t),\,\Sigma(t)\setminus\{\xi(t)\}\bigr)
 \geq c_0\,\theta^2\qquad(\xi\in\mathfrak O),
 \end{equation}
 and $\xi(t)\in\Sigma(t)$ if and only if $\xi$ coincides identically
 with a reference branch.
\item \emph{Sines.} For each of the phases $f$ listed in
 Table~\ref{tab:col-phases},
 $|\sin(\pi f(t))|\geq c_0\,\theta^2$; that is, the factor $S_Y$ in the
 second column satisfies
 \begin{equation}\label{eq:col-sine-bound}
 S_Y(X_0)\geq c_0^2\,\theta^4
 \end{equation}
 at the target $X_0$ in the first column.
\item \emph{Anchors.} The points $(\pm1,\pm1)$, $A_j$, $B_k(t)$ and
 $Y_\nu(t)$, $\nu\in\mathcal N$, that occur in the data have momentum
 coordinates of modulus at least $\varpi_0$, and the rates
 \eqref{eq:ker-rate} of the intended zeros satisfy
 \[
 \lambda(A_j,C),\ \lambda(A_j,C^\sharp),\
 \lambda\bigl(B_k(t),B_0(t)\bigr)\ (k=1,2),\
 \lambda\bigl(Y_\nu(t),B_0(t)\bigr)\ (\nu\in\{p_1,p_2,3\})
 \ \geq\lambda_0 .
 \]
\end{enumerate}
\end{lemma}

\begin{table}[ht]
\centering
\begin{tabular}{llll}
\toprule
data & target $X_0$ & factor $S_Y$ & phase $f(t)=B(X_0,Y)$\\
\midrule
product & $A_j$\ \ $(j=0,1,2)$ & $S_{B_0(t)}$ & $pU/t+V_jt$\\
product & $A_j$\ \ $(j=0,1,2)$ & $S_{B_0(t)^\sharp}$ & $Ut+pV_j/t$\\
product & $B_k(t)$\ \ $(k=1,2)$ & $S_C$ & $2p_k/t$\\
product & $B_k(t)$\ \ $(k=1,2)$ & $S_{C^\sharp}$ & $2t$\\
product & $B_k(t)$\ \ $(k=1,2)$ & $S_{B_0(t)^\sharp}$ & $t^2+pp_k/t^2$\\
sum & $Y_\nu(t)$\ \ $(\nu\in\{p_1,p_2,3\})$ & $S_{C^\sharp}$ & $2t$\\
sum & $Y_\nu(t)$\ \ $(\nu\in\{p_1,p_2,3\})$ & $S_{B_0(t)^\sharp}$
 & $t^2+p\nu/t^2$\\
sum & $[2,2]$ & $S_{B_0(t)}$ and $S_{B_0(t)^\sharp}$ & $2(t+p/t)$\\
\bottomrule
\end{tabular}
\caption{The factors that are required to be different from zero at the
targets of the local-zero arguments, and their phases. The intended
zeros are not listed: they are $B(A_j,C)=2V_j$, $B(A_j,C^\sharp)=2U$,
$B(B_k(t),B_0(t))=p+p_k$ and $B(Y_\nu(t),B_0(t))=p+\nu$, which are
integers for all $t$.}
\label{tab:col-phases}
\end{table}

\begin{proof}
(i) The light-cone coordinates of the elements of
$\mathfrak O\cup\mathfrak B$ are functions of the first form in
Lemma~\ref{lem:col-laurent}. We apply that lemma to the finitely many
coordinate differences of pairs $(\xi,\lambda)$. If $\xi\not\equiv\lambda$,
one coordinate difference $f$ does not vanish identically, and
$|\xi(t)-\lambda(t)|\geq|f(t)|/\sqrt2\geq c\theta^2/\sqrt2$, because
$|x|^2=\frac12(u_x^2+v_x^2)$. For \eqref{eq:col-separation}, let
$y\in\Sigma(t)$, $y\neq\xi(t)$. If $|y|>\varrho_1+2$, then
$|y-\xi(t)|>2\geq c_0\theta^2$. Otherwise $y=\lambda(t)$ for some
$\lambda\in\mathfrak B$ by \eqref{eq:col-branches}, and
$\lambda\not\equiv\xi$. The last assertion follows from
\eqref{eq:col-branches} in the same way.

(ii) The phases are computed from $B(X,Y)=u_Xv_Y+v_Xu_Y$ with
$B_0(t)=[t,p/t]$, $B_0(t)^\sharp=[p/t,t]$, $C=[2,0]$,
$C^\sharp=[0,2]$. Each is of one of the two forms of
Lemma~\ref{lem:col-laurent} and is not constant, since its coefficients
of $t$, $1/t$, respectively $t^2$, $1/t^2$, are positive integers or
vanish, and not all of them vanish. Lemma~\ref{lem:col-laurent}(ii)
applies.

(iii) The points $(\pm1,\pm1)$ and $A_j$ are fixed and off both axes,
because $U\neq V_j$. For $t\in I$ the momentum coordinates of
$B_k(t)=[t,p_k/t]$ are $\frac12(p_k/t\pm t)$, of modulus at least
$1/t$, because $p_k>t^2+2$. For $\nu=3$ the coordinates of $Y_3(t)$
are $\frac12(t\pm3/t)$, which do not vanish at $t=T$ because
$T^2\neq3$, hence are bounded away from zero near $T$. The rates are
\[
 \frac{2V_j}{|U^2-V_j^2|},\qquad\frac{2U}{|U^2-V_j^2|},\qquad
 \frac{k}{p_k^2/t^2-t^2},\qquad\frac{|p-\nu|}{|t^2-\nu^2/t^2|},
\]
by \eqref{eq:ker-rate}; they are positive and continuous near $T$,
since $p\neq\nu$ for $\nu\in\{p_1,p_2,3\}$. We choose $\varepsilon_0$
so small that all the statements hold.
\end{proof}

Two of these factors are bounded below by a positive constant near a
radius that is not a half-integer, as in Section~\ref{sec:kernels}, but
vanish at $t=T$ in the present situation. First, the phase $2t$ of
$S_{C^\sharp}$ is an integer at every half-integer $T$, so
$\sin^2(2\pi t)$ vanishes at $t=T$. Secondly, if $A_j\in\Lambda_T$ or
$A_j\in\Lambda_{1/T}$, then the phase of $S_{B_0(t)}$, respectively of
$S_{B_0(t)^\sharp}$, at $A_j$ is an integer at $t=T$, because
$B_0(T)\in\Lambda_T$, $B_0(T)^\sharp\in\Lambda_{1/T}$ and these lattices
are integral for $B$. Neither factor is bounded below by a constant near
$T$; both are covered by the bound $c_0^2\theta^4$.

The next lemma determines which incidences between the centres and the
reference lattices hold at a radius close to $T$ and different from
$T$. By Lemma~\ref{lem:quadratic-collisions}(i) these are exactly the
incidences that hold identically in $t$, and those are read off from
the integers \eqref{eq:col-ab}. We write
\[
 \mu^{\rm c}_t=\tfrac12\rhoF_R+\tfrac12\rhoF_t
 =\tfrac14\bigl(e_R+e_{1/R}+e_t+e_{1/t}\bigr),\qquad
 \sigma(R,t)=\mu^{\rm c}_t-\nu_{\vac}.
\]

\begin{lemma}[Incidences at the moving radius]
\label{lem:col-incidences}
Let collision data be given and let $0<|t-T|\leq\varepsilon_0$. Let
$\xi\in\mathfrak O$ be given by \eqref{eq:col-laurent-point}.
\begin{enumerate}[(i)]
\item $\xi(t)\in\Lambda_t$ if and only if $a_1=a_2=0$ and $b_2=0$;
 $\xi(t)\in\Lambda_{1/t}$ if and only if $a_1=a_2=0$ and $b_1=0$;
 $\xi(t)\in V\cup\Lambda_R\cup\Lambda_{1/R}$ if and only if
 $b_1=b_2=0$ and $[a_1,a_2]\in V\cup\Lambda_R\cup\Lambda_{1/R}$.
 Consequently
 \begin{equation}\label{eq:col-generic-mass}
 \mu^{\rm c}_t(\{\xi(t)\})=\tfrac14\Bigl(\one_{\{a=0,\,b_2=0\}}
 +\one_{\{a=0,\,b_1=0\}}
 +\one_{\{b=0\}}\,(e_R+e_{1/R})(\{[a_1,a_2]\})\Bigr).
 \end{equation}
\item The centres of the polynomials have the positions and masses
 listed in Table~\ref{tab:col-centres}. In particular: the only centre
 at the origin is the one of length zero; every other centre with a
 positive coefficient lies outside $V$; every centre with a negative
 coefficient either lies outside $\Sigma(t)$ and has mass zero, or is
 one of the points $\pm C$, $\pm C^\sharp$, $\pm A_j$, $\pm A_j^\sharp$,
 $\pm Y_\nu(t)$, $\pm Y_\nu(t)^\sharp$, with the mass given in the
 table.
\item The row preimages $x_{l,k,j}(t)$ with $(k,j)\neq(0,0)$ lie outside
 $\Sigma(t)$. The three others are $[2,2]=(2,0)$, $[2,0]=(1,1)$ and
 $[2,-2]=(0,2)$; for $R\in\{1,2\}$ they have
 $\mu^{\rm c}_t$-mass $\frac12$.
\end{enumerate}
\end{lemma}

\begin{table}[ht]
\centering\small
\begin{tabular}{lllll}
\toprule
centres & sign & $[a_1,a_2]$, $(b_1,b_2)$ & position at $t$
 & $\mu^{\rm c}_t$-mass\\
\midrule
\multicolumn{5}{l}{\emph{Product data}, $Y\in\{C,A_0,A_1,A_2\}$,
 $B=B_k(t)$}\\
$0$ & $+$ & $a=0$, $b=0$ & origin & $1$\\
$\pm C$, $\pm C^\sharp$ & $-$ & $a\neq0$, $b=0$
 & in $V$, off both axes & $\frac14$\\
$\pm A_j$, $\pm A_j^\sharp$ & $-$ & $a\neq0$, $b=0$
 & in $\Lambda_R\cup\Lambda_{1/R}$, not in $V$ & $\frac14n_j^0$\\
$\pm B$ & $-$ & $a=0$, $b=(\pm1,0)$ & in $\Lambda_t$ only
 & $\frac14$\\
$\pm B^\sharp$ & $-$ & $a=0$, $b=(0,\pm1)$ & in $\Lambda_{1/t}$ only
 & $\frac14$\\
$\pm Y\pm Y^\sharp$ & $+$ & $a\neq0$, $b=0$ & fixed, not in $V$
 & $\frac14(e_R+e_{1/R})$\\
$\pm B\pm B^\sharp$ & $+$ & $a=0$, $b_1b_2\neq0$
 & not in $\Sigma(t)$ & $0$\\
other centres of length $2$ & $+$ & $a\neq0$, $b\neq0$
 & not in $\Sigma(t)$ & $0$\\
length $3$ & $-$ & $a\neq0$, $b\neq0$ & not in $\Sigma(t)$ & $0$\\
length $4$ & $+$ & $a\neq0$, $b\neq0$ & not in $\Sigma(t)$ & $0$\\
\midrule
\multicolumn{5}{l}{\emph{Sum data}, $\nu\in\mathcal N$}\\
$0$ & $+$ & $a=0$, $b=0$ & origin & $1$\\
$\pm Y_\nu$ & $-$ & $a=0$, $b=(\pm1,0)$ & in $\Lambda_t$ only
 & $\frac14$\\
$\pm Y_\nu^\sharp$ & $-$ & $a=0$, $b=(0,\pm1)$ & in $\Lambda_{1/t}$ only
 & $\frac14$\\
$\pm Y_\nu\pm Y_\nu^\sharp$ & $+$ & $a=0$, $b_1b_2\neq0$
 & not in $\Sigma(t)$ & $0$\\
$\pm C$, $\pm C^\sharp$ & $-$ & $a\neq0$, $b=0$
 & in $V\cap\Lambda_R\cap\Lambda_{1/R}$ & $\frac12$\\
$\zeta\neq0$, $|\eta|=1$ & $+$ & $a\neq0$, $b\neq0$
 & not in $\Sigma(t)$ & $0$\\
$\zeta\neq0$, $|\eta|=2$ & $-$ & $a\neq0$, $b\neq0$
 & not in $\Sigma(t)$ & $0$\\
\bottomrule
\end{tabular}
\caption{The centres of the collision kernels at a radius $t$ with
$0<|t-T|\leq\varepsilon_0$. Here $n_j^0=1$ if $A_j\notin\Lambda_{1/R}$
and $n_j^0=2$ if $A_j\in\Lambda_{1/R}$, and $a=[a_1,a_2]$,
$b=(b_1,b_2)$ are the integers of \eqref{eq:col-ab}.}
\label{tab:col-centres}
\end{table}

\begin{proof}
(i) Suppose that $\xi(t)=[mt,n/t]\in\Lambda_t$. Since
$|\xi(t)|\leq\varrho_1$, the map $\lambda(t')=[mt',n/t']$ is a
reference branch with $\lambda(t)=\xi(t)$, and
Lemma~\ref{lem:quadratic-collisions}(i) shows that $\xi\equiv\lambda$
on $I$. By the first line of \eqref{eq:ker-identities}, whose proof
uses only that the two coordinate equations hold identically and that
$\nu\neq0$, this implies $a_1=a_2=0$ and $b_2=0$. Conversely, if
$a=0$ and $b_2=0$, then $\xi(t)=[b_1t,\,b_1\nu/t]\in\Lambda_t$. The
statement for $\Lambda_{1/t}$ follows in the same way from the second
line of \eqref{eq:ker-identities}, with
$\xi(t)=[b_2\nu/t,\,b_2t]$ in the converse, and the statement for fixed
points from the third line. Formula \eqref{eq:col-generic-mass}
follows, because each of the four combs gives mass one to each of its
points.

(ii) For a product centre, $[a_1,a_2]=\eta_1Y+\eta_2Y^\sharp
=[\eta_1u_Y+\eta_2v_Y,\ \eta_1v_Y+\eta_2u_Y]$ with $Y=[u_Y,v_Y]$ equal
to $[2,0]$ or to $[U,V_j]$. Since $u_Y^2\neq v_Y^2$, this vector
vanishes only for $\eta_1=\eta_2=0$; for $\eta_1\eta_2\neq0$ its
coordinates are $\pm(u_Y+v_Y)$ twice or $\pm(u_Y-v_Y)$ with opposite
signs. The columns of the table for $[a_1,a_2]$ and $(b_1,b_2)$ follow
from \eqref{eq:col-ab}, and the positions and masses from (i), as
follows.

\emph{Length one, fixed.} The point $C=[2,0]$ lies in $\Lambda_r$ if and
only if $2/r\in\ZZ$ and in $\Lambda_{1/r}$ if and only if $2r\in\ZZ$.
For product data $2R\in\ZZ$ and $2/R\notin\ZZ$, because
$R\notin\{1,2\}$; so $(e_R+e_{1/R})(\{C\})=1$ and the mass is
$\frac14=\nu_{\vac}(\{C\})$. For sum data both numbers are integers and
the mass is $\frac12$. The same holds at $-C$ and $\pm C^\sharp$ by the
invariance of $\mu^{\rm c}_t$ under the reflections. The points
$A_j$ belong to $\Lambda_R$ by construction, so
$(e_R+e_{1/R})(\{A_j\})=n_j^0\in\{1,2\}$, and they are not in $V$
because $U,V_j\geq3$. By (i) none of these fixed points lies in
$\Lambda_t\cup\Lambda_{1/t}$, since $a\neq0$.

\emph{Length one, moving.} For $\pm B_k(t)$ and $\pm Y_\nu(t)$ we have
$a=0$ and $b=(\pm1,0)$, and for their images under $\sharp$,
$b=(0,\pm1)$. By (i) the first lie in $\Lambda_t$ and not in
$\Lambda_{1/t}\cup V\cup\Lambda_R\cup\Lambda_{1/R}$, the second in
$\Lambda_{1/t}$ only, and the mass is $\frac14$.

\emph{Fixed centres of length two.} For $\pm Y\pm Y^\sharp$ the
coordinates are $\pm2$ for $Y=C$, so the point is $\pm(2,0)$ or
$\pm(0,2)$ in momentum coordinates, and they have modulus at least $3$
for $Y=A_j$, by \eqref{eq:ker-A-conditions}; in both cases the point is
not in $V$. Its mass may be positive.

\emph{All other centres.} If $a=0$ and $b_1b_2\neq0$, or if $a\neq0$ and
$b\neq0$, none of the three conditions of (i) holds, so
$\xi(t)\notin\Sigma(t)$ and the mass vanishes. This applies to the
centres $\pm B\pm B^\sharp$ and $\pm Y_\nu\pm Y_\nu^\sharp$, to the
product centres of length two with one fixed and one moving vector, to
those of length three and four, in which at least one of
$\eta_1,\eta_2$ and at least one of $\eta_3,\eta_4$ is different from
zero, and to the sum centres with $\zeta\neq0$ and $\eta\neq0$.

The summary in (ii) follows by inspection of the table and of the signs
of the coefficients recalled before \eqref{eq:col-laurent-point}.

(iii) For $(k,j)\neq(0,0)$ we have $a=[2,2-2l]\neq0$ and $b\neq0$, so
(i) applies. For $k=j=0$ the points are fixed, and for $R\in\{1,2\}$
each of $[2,2]$, $[2,0]$, $[2,-2]$ lies in $\Lambda_R$ and in
$\Lambda_{1/R}$, because $2R$ and $2/R$ are integers.
\end{proof}

The hypotheses on the integer data that enter this analysis are that
$R\neq2$ and $R\neq1$ for product data (so that $C\notin\Lambda_R$),
that $U\neq V_j$, and that the integers $\nu$ are different from zero;
the parity of $p$ is used only in
Lemma~\ref{lem:three-projection}. In the limit $t\to T$ further
incidences appear: $B_k(T)$ lies in $\Lambda_R$ when $R=T$, the centres
$\pm C$ lie in $\Lambda_{1/T}$, and $\pm C^\sharp$ in $\Lambda_T$, for
every half-integer $T$ (and in both lattices when $T\in\{1,2\}$), and
$A_j$ may lie in $\Lambda_T\cup\Lambda_{1/T}$. These are the collisions.
The masses in Table~\ref{tab:col-centres} are those at the moving
radius, not the larger limiting ones.

\begin{lemma}[Mass balance at an arbitrary radius]
\label{lem:col-mass-balance}
Let $r_1,\dots,r_k>0$ and let
$\Phi=\sum_{i\in\Xi}c_ie^{2\pi iB(\cdot,\xi_i)}$ be a trigonometric
polynomial as in \eqref{eq:ker-trig-polynomial} that vanishes on
$\Lambda_{r_1}\cup\dots\cup\Lambda_{r_k}$. Then
$\sum_{i\in\Xi}c_i\,e_{r_l}(\{\xi_i\})=0$ for every $l$. In particular,
for collision data, every polynomial $\Phi_t$ of the data and every
$t>0$,
\begin{equation}\label{eq:col-mass-balance}
 \sum_{i\in\Xi}c_i\,\mu^{\rm c}_t(\{\xi_i(t)\})=0 .
\end{equation}
\end{lemma}
\begin{proof}
Let $r=r_l$ and $0<\epsilon\leq1$. The function $\Phi_\epsilon$ of
\eqref{eq:ker-broad-narrow} is a Schwartz function vanishing on
$\Lambda_r$, and $\mathbb Se_r=e_r$ by \eqref{eq:circle-invariance}.
By \eqref{eq:ker-fourier},
\[
 0=\epsilon^2\langle e_r,\Phi_\epsilon\rangle
 =\epsilon^2\langle e_r,\mathbb S\Phi_\epsilon\rangle
 =\sum_{i\in\Xi}c_i\sum_{\lambda\in\Lambda_r}
 e^{-2\pi|\lambda-\xi_i|^2/\epsilon^2}.
\]
The inner sums are dominated by the convergent sums
$\sum_\lambda e^{-2\pi|\lambda-\xi_i|^2}$ and converge to
$e_r(\{\xi_i\})$ as $\epsilon\to0$. For collision data the polynomials
vanish on $\Sigma(t)\supset\Lambda_R\cup\Lambda_{1/R}\cup\Lambda_t
\cup\Lambda_{1/t}$.
\end{proof}

\subsection{The resolved peak budget}
\label{subsec:col-budget}

The kernel estimates at the moving radius use the following three
properties of a sequence. The exponents $1000$ and $60$ are convenient
fixed numbers; what is needed of them is recorded in
Remark~\ref{rem:col-exponents}.

\begin{definition}[Resolved sequences]\label{def:col-resolved}
Let collision data be given, with $\varepsilon_0$ as in
Lemma~\ref{lem:quadratic-collisions} and $E_*$ as in
\eqref{eq:col-E-star}. A \emph{resolved sequence} for these data is an
admissible sequence of gap $g$ converging to $\sigma(R,T)$, together
with radii $t_a\in I$, such that for all sufficiently small $a$ in the
sequence, with $\theta_a=|t_a-T|$:
\begin{enumerate}[(K1)]
\item $0<\theta_a\leq\varepsilon_0$, $\theta_a\to0$, and
 $a\leq\theta_a^{1000}$;
\item there is an injective map $j\mapsto j'$ from the set of states of
 $\rho_a$ with $\Delta_j\leq E_*$ into the family
 $\mathfrak S(R,t_a)$, whose image contains every state of
 $\mathfrak S(R,t_a)$ of dimension at most $E_*-1$, such that $j'$ has
 the spin of $j$ and $|\Delta_j-\Delta'_{j'}|\leq\theta_a^{60}$;
\item some state of $\rho_a$ has spin $p$ and dimension exactly
 $E_p(t_a)$.
\end{enumerate}
We call $\theta_a$ the \emph{collision scale}, and put
\begin{equation}\label{eq:col-resolving-radius}
 \varrho_a=c_\varrho\,\theta_a^2,\qquad c_\varrho=\tfrac14c_0,\qquad
 \kappa_a=c_0^2\,\theta_a^4 ,
\end{equation}
with $c_0$ as in Lemma~\ref{lem:quadratic-collisions}.
\end{definition}

In (K2), if a state $j$ has weights $(h,\bar h)$ and $j'$ has weights
$(h',\bar h')$, then $|h-h'|=|\bar h-\bar h'|
=\frac12|\Delta_j-\Delta'_{j'}|$, and $|\sqrt x-\sqrt{x'}|\leq
\sqrt{|x-x'|}$ for $x,x'\geq0$. Hence, for each of the four sign
patterns $\sigma$,
\begin{equation}\label{eq:col-momentum-matching}
 \bigl|X_j^\sigma-X_{j'}^{\prime\,\sigma}\bigr|\leq\theta_a^{30},
\end{equation}
where $X^{\prime\,\sigma}_{j'}$ denotes the physical point of $j'$ with
the sign pattern $\sigma$. This holds also when a weight of $j'$
vanishes.

\begin{lemma}[Exact counts on the resolving scale]
\label{lem:col-counts}
Let a resolved sequence be given. For all sufficiently small $a$, every
object $\xi\in\mathfrak O$, and $x=\xi(t_a)$:
\begin{enumerate}[(i)]
\item a pair $(j,\sigma)$ satisfies $X_j^\sigma\in D(x,\varrho_a)$ if
 and only if $\Delta_j\leq E_*$ and
 $X_{j'}^{\prime\,\sigma}=x$; in this case
 $|X_j^\sigma-x|\leq\theta_a^{30}$. Consequently
 \begin{equation}\label{eq:col-exact-count}
 \rho_a\bigl(D(x,\varrho_a)\bigr)=\sigma(R,t_a)(\{x\})
 =\mu^{\rm c}_{t_a}(\{x\})-\nu_{\vac}(\{x\}),
 \end{equation}
 and $D(x,2\varrho_a)$ contains no point of $V$ other than possibly
 $x$.
\item If $x$ is off both axes, then exactly
 $N_x=4\sigma(R,t_a)(\{x\})$ states of $\rho_a$ have a physical point
 in $D(x,\varrho_a)$. They have the spin $P_x^2-Q_x^2$ of $x$, and
 their dimensions satisfy $|\Delta_j-|x|^2|\leq\theta_a^{60}$. We call
 them the \emph{states at $x$}.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $a$ be so small that (K1) and (K2) hold and
$\theta_a^{30}<\varrho_a\leq\frac12$. By \eqref{eq:col-separation} and
the choice of $c_\varrho$,
\begin{equation}\label{eq:col-buffer}
 D(x,4\varrho_a)\cap\Sigma(t_a)\subset\{x\},
\end{equation}
and $V\cup\supp\sigma(R,t_a)\subset\Sigma(t_a)$ by
\eqref{eq:ker-limit-masses}.

(i) Let $X_j^\sigma\in D(x,\varrho_a)$. Then
$\Delta_j=|X_j^\sigma|^2\leq(\varrho_1+1)^2\leq E_*$, so $j'$ is
defined, and by \eqref{eq:col-momentum-matching} the point
$X_{j'}^{\prime\,\sigma}$ of $\supp\sigma(R,t_a)$ lies in
$D(x,2\varrho_a)$; by \eqref{eq:col-buffer} it equals $x$. Conversely,
let $\mathsf s$ be a state of $\mathfrak S(R,t_a)$ with a physical point
at $x$. Its dimension is $|x|^2\leq\varrho_1^2\leq E_*-1$, so
$\mathsf s=j'$ for a unique state $j$ with $\Delta_j\leq E_*$, and
$|X_j^\sigma-x|\leq\theta_a^{30}<\varrho_a$ for the corresponding sign
pattern. Thus $(j,\sigma)\mapsto(j',\sigma)$ is a bijection between the
pairs with $X_j^\sigma\in D(x,\varrho_a)$ and the pairs consisting of a
state of $\mathfrak S(R,t_a)$ and a sign pattern with physical point
$x$. Each pair carries mass $\frac14$ in $\rho_a$, respectively in
$\sigma(R,t_a)$ by Lemma~\ref{lem:col-states}(ii), which gives
\eqref{eq:col-exact-count}. The statement about $V$ is contained in
\eqref{eq:col-buffer}.

(ii) If $x$ is off both axes, the four physical points of a state with
a physical point at $x$ are distinct, so exactly one sign pattern of
such a state gives $x$, its weights are $(P_x^2,Q_x^2)$, and there are
$N_x$ such states in $\mathfrak S(R,t_a)$. By the bijection in (i), the
states of $\rho_a$ with a physical point in $D(x,\varrho_a)$ are the
preimages of these $N_x$ states under $j\mapsto j'$, and each of them
has exactly one physical point in the disc. They have the spin of $x$,
and the dimension bound is (K2).
\end{proof}

\begin{lemma}[Resolved peak budget]\label{lem:resolved-budget}
Let a resolved sequence be given, and let $\Phi_t$ be one of the
polynomials of the data, with centres $\xi_i(t)$ and coefficients
$c_i$, $i\in\Xi$. For $a$ in the sequence put
\[
 \Phi=\Phi_{t_a},\qquad\varrho=\varrho_a,\qquad
 m_i=\mu^{\rm c}_{t_a}\bigl(\{\xi_i(t_a)\}\bigr).
\]
For all sufficiently small $a$ the following hold.
\begin{enumerate}[(i)]
\item The numbers $m_i$ lie in $[0,1]$ and satisfy the balance
 \eqref{eq:ker-mass-balance}. Conditions \textup{(B0)}, \textup{(B1)}
 and \textup{(B2)} of Lemma~\ref{lem:peak-budget} hold, with equality
 $\rho_a(D(\xi_i(t_a),\varrho_a))=m_i$ in \textup{(B2)}, and condition
 \textup{(B3)(b)} holds at every index with $c_i<0$ and $m_i=0$.
\item At every index with $c_i<0$ and $m_i>0$ the point
 $x=\xi_i(t_a)$ is one of $\pm C$, $\pm C^\sharp$, $\pm A_j$,
 $\pm A_j^\sharp$, $\pm Y_\nu(t_a)$, $\pm Y_\nu(t_a)^\sharp$. The
 geometric requirements of \textup{(B3)(a)} hold at $x$, and with
 $w_i=\nu_{\vac}(\{x\})$ and $n_i=N_x$ the number of states at $x$ in
 the sense of Lemma~\ref{lem:col-counts}(ii),
 \[
 w_i+\tfrac14n_i=m_i,\qquad
 n_i=\begin{cases}
 0&\text{at }\pm C,\pm C^\sharp\text{ for product data},\\
 1&\text{at }\pm C,\pm C^\sharp\text{ for sum data},\\
 n_j^0&\text{at }\pm A_j,\pm A_j^\sharp,\\
 1&\text{at }\pm Y_\nu(t_a),\ \pm Y_\nu(t_a)^\sharp .
 \end{cases}
 \]
\item Let $\varepsilon\geq0$ be such that every state $j$ at a centre
 $x$ of \textup{(ii)} satisfies $|\Delta_j-|x|^2|\leq\varepsilon$, and
 suppose that $\varepsilon\leq\varrho_a^2$. Then for every $\epsilon$
 with $0<\epsilon\leq\frac12$ and $a\leq\frac14\epsilon^2$,
 \begin{equation}\label{eq:col-resolved-budget}
 \sum_j\ev{\Phi_{t_a,\epsilon}}(h_j,\bar h_j)
 \leq\frac{C_{\rm B}}{\theta_a^{4}}\Bigl(\bigl(a+\varepsilon^2\bigr)
 \epsilon^{-4}+e^{-\pi\varrho_a^2/(4\epsilon^2)}\Bigr),
 \end{equation}
 where $C_{\rm B}$ depends only on $g$ and the data. All terms on the
 left are nonnegative.
\end{enumerate}
\end{lemma}

The masses and their cancellation are taken at the moving radius $t_a$.
At the limiting radius $T$ the masses at some centres are larger. The
additional states of $\rho_a$ are, by \eqref{eq:col-momentum-matching}
and \eqref{eq:col-buffer}, within $\theta_a^{30}$ of points of
$\Sigma(t_a)$ at distance at least $4\varrho_a$ from the centre, and
they are counted at no centre. The atoms of
$\nu_{\vac}$ enter only at the origin and at the centres $\pm C$,
$\pm C^\sharp$, where they are positive; no centre comes within
$4\varrho_a$ of a negative atom. Lemma~\ref{lem:peak-budget} has no
separate hypothesis on the points outside the discs
$D(\xi_i,\varrho)$: they are estimated by
Lemma~\ref{lem:narrow-tail} in its proof, where the displacement of
the shifted vacuum point near $(1,1)$, which is of order $a$, is
accounted for as well.

\begin{proof}
Let $a$ be so small that Lemma~\ref{lem:col-counts} applies,
$\varrho_a\leq\min\{\frac14,\frac12\sqrt g,\frac12\varpi_0\}$ and
$a\leq\min\{\frac14g^2,\frac14\varrho_a^2\}$; the last condition holds
by (K1).

(i) Each of the four combs gives mass at most one to a point, so
$0\leq m_i\leq1$, and \eqref{eq:ker-mass-balance} is
\eqref{eq:col-mass-balance} at $t=t_a$. (B0) holds because the
polynomials vanish on $\Sigma(t)\supset V$. By
Lemma~\ref{lem:col-incidences}(ii) the only centre at the origin has
length zero, a positive coefficient and mass one, which is (B1). Let
$c_i>0$ and $x=\xi_i(t_a)\neq0$. Then $x\notin V$ by
Lemma~\ref{lem:col-incidences}(ii), hence
$\dist(x,V)\geq4\varrho_a$ by \eqref{eq:col-buffer}, and
$\rho_a(D(x,\varrho_a))=\mu^{\rm c}_{t_a}(\{x\})=m_i$ by
\eqref{eq:col-exact-count}; this is (B2). Let $c_i<0$ and $m_i=0$. By
Lemma~\ref{lem:col-incidences}(ii), $x=\xi_i(t_a)\notin\Sigma(t_a)$, so
$\dist(x,V)\geq4\varrho_a$ by \eqref{eq:col-buffer} and
$\rho_a(D(x,\varrho_a))=0$ by \eqref{eq:col-exact-count}; this is
(B3)(b).

(ii) The list of points is that of
Lemma~\ref{lem:col-incidences}(ii). By
Lemma~\ref{lem:quadratic-collisions}(iii) their momentum coordinates
have modulus at least $\varpi_0\geq2\varrho_a$. By
Lemma~\ref{lem:col-counts}(i) the disc $D(x,\varrho_a)$ contains no
point of $V$ other than possibly $x$, and $x\in V$ only for
$x\in\{\pm C,\pm C^\sharp\}=\{(\pm1,\pm1)\}$, where
$w_i=\frac14$ by \eqref{eq:ker-vacuum-masses}; otherwise $w_i=0$. By
\eqref{eq:col-exact-count},
$\frac14N_x=\sigma(R,t_a)(\{x\})=m_i-w_i$, and the values of $n_i=N_x$
follow from Table~\ref{tab:col-centres}. By
Lemma~\ref{lem:col-counts}(ii) the $n_i$ states at $x$ are distinct and
have the spin of $x$.

(iii) By (ii) and the hypothesis on $\varepsilon$, condition (B3)(a)
holds at every index with $c_i<0$ and $m_i>0$, with the states at
$\xi_i(t_a)$. Together with (i), all hypotheses of
Lemma~\ref{lem:peak-budget} hold for $\Phi=\Phi_{t_a}$, with
$\varrho=\varrho_a$, the bound $\varrho_1$ for the centres, $m_*=1$ and
the fixed coefficients. The range \eqref{eq:ker-budget-range} holds by
the choice of $a$ and the hypotheses on $\epsilon$ and $\varepsilon$.
Hence \eqref{eq:ker-peak-budget} holds with a constant independent of
$\varrho$, and $\varrho_a^{-2}=c_\varrho^{-2}\theta_a^{-4}$. The terms
are nonnegative by Lemma~\ref{lem:kernel-positivity}(ii) and (iii).
\end{proof}

The local-zero arguments below use Lemma~\ref{lem:local-zero}(iii) in
the following form.

\begin{lemma}[Anchoring at the moving radius]\label{lem:col-anchor}
Let a resolved sequence be given. Let $X_0(t)$ be one of the maps
$t\mapsto A_j$, $t\mapsto B_k(t)$ $(k=1,2)$, $t\mapsto Y_\nu(t)$
$(\nu\in\{p_1,p_2,3\})$, and let $\Psi_t$ be a function on $\RR^2$ with
\[
 \Psi_t\geq\prod_{i=1}^kS_{Y_i}\prod_{i=1}^{k'}S_{Y_i'},
\]
where $k\geq1$, the factors $S_{Y_i}$ are intended zeros at $X_0(t)$
listed in the caption of Table~\ref{tab:col-phases}, and the factors
$S_{Y_i'}$ are factors listed in that table for the target $X_0(t)$.
There is a constant $C$ such that, for all sufficiently small $a$,
every state $j$ at $X_0(t_a)$, with dimension
$|X_0(t_a)|^2+\delta$ and physical point $X$ in
$D(X_0(t_a),\varrho_a)$, satisfies
\begin{equation}\label{eq:col-anchor}
 |\delta|\leq C\,\theta_a^{-2k'/k}\,\Psi_{t_a}(X)^{1/(2k)} .
\end{equation}
\end{lemma}
\begin{proof}
We apply Lemma~\ref{lem:local-zero}(iii) with $X_0=X_0(t_a)$,
$\varrho_*=\varrho_1$, which bounds $X_0$ and the vectors $C$,
$C^\sharp$, $B_0(t_a)$, $B_0(t_a)^\sharp$ because these are centres of
the data, the constant $\varpi=\varpi_0$ and the rate
$\lambda=\lambda_0$ of Lemma~\ref{lem:quadratic-collisions}(iii), and
the lower bound $\varkappa=\kappa_a=c_0^2\theta_a^4$ of
\eqref{eq:col-sine-bound} for the factors $S_{Y_i'}$ at $X_0$. The
number $\delta_1$ of that lemma is then at least $c\,\theta_a^4$ with a
constant $c>0$. By Lemma~\ref{lem:col-counts}(ii) the state has the spin
of $X_0$ and $|\delta|\leq\theta_a^{60}\leq\delta_1$ for small $a$, and
its physical point in $D(X_0,\varrho_a)$ has the signs of $X_0$, so
that it is the point $X(\delta)$ of \eqref{eq:ker-fixed-spin-curve}.
The lemma gives
$|\delta|\leq\lambda_0^{-1}\bigl((2/\kappa_a)^{k'}\Psi_{t_a}(X)
\bigr)^{1/(2k)}$, which is \eqref{eq:col-anchor}.
\end{proof}

\begin{remark}[The exponents]\label{rem:col-exponents}
The geometric input is the exponent two of
Lemma~\ref{lem:quadratic-collisions}: discs of radius
$\varrho_a\asymp\theta_a^2$ and factors bounded below by
$\kappa_a\asymp\theta_a^4$. All further losses are derived from it:
the prefactor $\theta_a^{-4}$ in \eqref{eq:col-resolved-budget}, the
factor $\epsilon^{-4}\asymp\theta_a^{-8}H^2$ for a resolving width
$\epsilon^2\asymp\varrho_a^2/H$, and the powers of $\kappa_a^{-1}$ in
\eqref{eq:col-anchor}. In the proofs of
Propositions~\ref{prop:col-product} and~\ref{prop:col-sum} the largest
loss is $\theta_a^{-36}$. Of the exponents in
Definition~\ref{def:col-resolved}, the arguments use that the matching
accuracy $\theta_a^{60}$ is smaller than $\varrho_a^2$, than the radius
$\delta_1\asymp\theta_a^4$ of validity of the local-zero estimate, and
than $\theta_a^{52}$, which is needed in the scalar row; and that
$a\leq\theta_a^{1000}$ makes $a^{1/16}\theta_a^{-9/2}$ and
$a^{1/9}\theta_a^{-16/3}$ positive powers of $a$.
\end{remark}

\subsection{Exclusion of resolved sequences}
\label{subsec:col-exclusion}

In the two propositions of this subsection a resolved sequence is given,
$L=\log(1/a)$, constants called $C$ depend only on $g$ and the data, and
all statements hold for sufficiently small $a$ in the sequence. By
(K1),
\begin{equation}\label{eq:col-a-theta}
 \theta_a^{-1}\leq a^{-1/1000},\qquad aL\leq a^{1/2}\leq\theta_a^{500}.
\end{equation}
For a logarithmic scale $H\geq1$ we shall use resolving widths
$\epsilon$ with $\epsilon^2$ a fixed multiple of $\varrho_a^2/H$. For
such a width and a state with $\Delta_j\leq E_*$ the broad Gaussian
$e^{-2\pi\epsilon^2\Delta_j}$ is at least $\frac12$ when $a$ is small,
because $\epsilon\to0$. All kernels of the data are even, so their value
at a physical point $X_j^\sigma$ of a state does not depend on $\sigma$
and equals the even evaluation at $(h_j,\bar h_j)$; the left side of
\eqref{eq:col-resolved-budget} is written
$\sum_j\Phi_{t_a,\epsilon}(X_j^\sigma)$ below.

\begin{proposition}[Pinned radius different from one and two]
\label{prop:col-product}
Let collision data with $R\notin\{1,2\}$ be given. There is no resolved
sequence for these data.
\end{proposition}
\begin{proof}
Suppose that a resolved sequence exists. We use the width
\begin{equation}\label{eq:col-width-L}
 \epsilon^2=\frac{\pi\varrho_a^2}{4L},\qquad\text{so that}\qquad
 e^{-\pi\varrho_a^2/(4\epsilon^2)}=a,\qquad
 \epsilon^{-4}=\frac{16L^2}{\pi^2c_\varrho^4\,\theta_a^{8}} .
\end{equation}
By \eqref{eq:col-a-theta}, $\epsilon\leq\frac12$ and
$a\leq\frac14\epsilon^2$ for small $a$.

\emph{Step 1: the budget of $\Phi^*$.} By
Lemma~\ref{lem:resolved-budget}(ii) the centres of $\Phi^*_{t_a}$ with
negative coefficient and positive mass are $\pm C$, $\pm C^\sharp$, at
which there are no states, and $\pm B_0(t_a)$, $\pm B_0(t_a)^\sharp$, at
which there is exactly one state. The state of (K3) has spin $p$ and
dimension $E_p(t_a)=|B_0(t_a)|^2$, so one of its physical points is
$B_0(t_a)$; it is therefore the state at $B_0(t_a)$, and its
displacement is zero. Lemma~\ref{lem:resolved-budget}(iii) with
$\varepsilon=0$ and \eqref{eq:col-width-L} give
\[
 \sum_j\Phi^*_{t_a,\epsilon}(X_j^\sigma)
 \leq C\theta_a^{-4}\bigl(a\,\theta_a^{-8}L^2+a\bigr),
\]
and hence, all terms being nonnegative,
\begin{equation}\label{eq:col-product-initial}
 \Phi^*_{t_a}(X_j^\sigma)\leq\Gamma_0:=C\,a\,\theta_a^{-12}L^2
 \qquad(\Delta_j\leq E_*).
\end{equation}

\emph{Step 2: anchoring.} Let $j$ be one of the $n_i^0$ states at
$A_i$, $i\in\{0,1,2\}$, with dimension $|A_i|^2+\delta$. We apply
Lemma~\ref{lem:col-anchor} to $\Psi_t=\Phi^*_t
=S_CS_{C^\sharp}S_{B_0(t)}S_{B_0(t)^\sharp}$ with the two intended zeros
$S_C$, $S_{C^\sharp}$ and the two factors $S_{B_0(t)}$,
$S_{B_0(t)^\sharp}$ of Table~\ref{tab:col-phases}; thus $k=k'=2$. By
\eqref{eq:col-anchor} and \eqref{eq:col-product-initial},
\begin{equation}\label{eq:col-anchor-A}
 |\delta|\leq C\,\theta_a^{-2}\,\Gamma_0^{1/4}
 \leq C\,a^{1/4}\theta_a^{-5}L^{1/2}\qquad\text{for every state at }A_i .
\end{equation}
Let $k\in\{1,2\}$ and let $E_{p_k}(t_a)+\delta_{k,a}$ be the dimension
of the state at $B_k(t_a)$. We apply Lemma~\ref{lem:col-anchor} to the
same function with the intended zero $S_{B_0(t)}$ and the three factors
$S_C$, $S_{C^\sharp}$, $S_{B_0(t)^\sharp}$ of
Table~\ref{tab:col-phases}: one intended zero and three factors, so the
exponents in \eqref{eq:col-anchor} are $6$ and $\frac12$, and
\begin{equation}\label{eq:col-anchor-B}
 |\delta_{k,a}|\leq C\,\theta_a^{-6}\,\Gamma_0^{1/2}
 \leq C\,a^{1/2}\theta_a^{-12}L\qquad(k=1,2).
\end{equation}
Put $\varepsilon_a=C\,a^{1/4}\theta_a^{-12}L$, which bounds the right
sides of \eqref{eq:col-anchor-A} and \eqref{eq:col-anchor-B}. By
\eqref{eq:col-a-theta},
$\varepsilon_a\leq Ca^{1/4-12/1000}L\leq\varrho_a^2$ for small $a$.

\emph{Step 3: the nine budgets.} Fix $i,k\in\{0,1,2\}$. By
Lemma~\ref{lem:resolved-budget}(ii) the centres of
$\Phi^{i,k}_{t_a}$ with negative coefficient and positive mass are
$\pm A_i$, $\pm A_i^\sharp$, $\pm B_k(t_a)$,
$\pm B_k(t_a)^\sharp$, and by Steps~1 and~2 the states at these centres
have displacements at most $\varepsilon_a$.
Lemma~\ref{lem:resolved-budget}(iii) with
$\varepsilon=\varepsilon_a$ gives
\begin{equation}\label{eq:col-nine-budgets}
 \sum_j\Phi^{i,k}_{t_a,\epsilon}(X_j^\sigma)
 \leq C\theta_a^{-4}\Bigl(\bigl(a+a^{1/2}\theta_a^{-24}L^2\bigr)
 \theta_a^{-8}L^2+a\Bigr)\leq C\,a^{1/2}\theta_a^{-36}L^4 .
\end{equation}

\emph{Step 4: localization.} Let $j$ be a state with $\Delta_j\leq L$
and $X=[u,v]$ one of its physical points. By \eqref{eq:col-width-L},
$2\pi\epsilon^2\Delta_j\leq\frac12\pi^2\varrho_a^2\leq1$, so the broad
Gaussian at $X$ is at least $e^{-1}$, and by
\eqref{eq:col-nine-budgets}
\[
 S_{A_i}(X)S_{A_i^\sharp}(X)\,S_{B_k(t_a)}(X)S_{B_k(t_a)^\sharp}(X)
 \leq\Gamma_a:=C\,a^{1/2}\theta_a^{-36}L^4\qquad(0\leq i,k\leq2).
\]
By \eqref{eq:col-a-theta}, $\Gamma_a\leq Ca^{1/2-36/1000}L^4\leq1$.
Moreover $uv=s_j\in\ZZ$ and $|u|,|v|\leq\Upsilon:=\sqrt{2L}$. We apply
Lemma~\ref{lem:three-projection}(c) with $t=t_a$, $t_-=\frac12$,
$t_+=R_{\max}+1$, $\ell=1$, the integer $p$ of \eqref{eq:col-data},
$\Delta V=2$, $D=2U\Delta V=16R$, and
\[
 \zeta=\tfrac12\Gamma_a^{1/8}=C\,a^{1/16}\theta_a^{-9/2}L^{1/2},
 \qquad\zeta\,\Upsilon\leq C\,a^{1/16}\theta_a^{-9/2}L
 \leq C\,a^{29/500}L ,
\]
where the last inequality is \eqref{eq:col-a-theta}. Since
$\zeta\Upsilon\to0$, the smallness conditions of parts (a) and (b) of
that lemma hold. Hence there is a real point $X_j'$, equal either to a
point of $\Lambda_{t_a}\cup\Lambda_{1/t_a}$ or to a point $[m/D,n/D]$,
whose spin is exactly $s_j$, with
\begin{equation}\label{eq:col-product-localization}
 \bigl|\Delta_j-|X_j'|^2\bigr|\leq C\,\zeta\,\Upsilon
 \leq C\,a^{1/16}\theta_a^{-9/2}L\leq C\,a^{29/500}L .
\end{equation}
In the second case $X_j'$ lies in the set $\Sigma_D$ of
\eqref{eq:ker-grid-circles}.

\emph{Step 5: contradiction.} The set $\Sigma_D$ is the union of the
circle lattices with radii in $\{k/D:k\mid D^2\}$, which contains $1$
and is closed under inversion by Lemma~\ref{lem:grid-circles}; the radii
$t_a$ lie in the compact interval $I$; and by Step~4 every state with
$\Delta_j\leq L$ has a real point
$X_j'\in\Sigma_D\cup\Lambda_{t_a}\cup\Lambda_{1/t_a}$ of spin exactly
$s_j$ satisfying \eqref{eq:col-product-localization}, for all
sufficiently small $a$ in the sequence. This contradicts
Theorem~\ref{thm:moving-localization}, applied with $u_*=1$ to the
elements of the sequence below a suitable threshold.
\end{proof}

Theorem~\ref{thm:moving-localization} makes no assumption on the limit
of the radii $t_a$. Here $t_a\to T$ is a half-integer, so that
$\Lambda_{t_a}$ approaches a lattice contained in the rational grid,
and for $T=1$ the squared transverse momenta of points of
$\Lambda_{t_a}$ near the momentum axes tend to zero. Both situations are
covered by that theorem, whose statement and proof place no restriction
on the limit $t_0$ of the radii: if $|t_a-T|\geq a^\eta$ along a
subsequence it uses the moving certificate of
Proposition~\ref{prop:moving-certificate} at $t_0=T$, and otherwise it
transports the points to the radius $T$. For the sequences produced in
the proof of Theorem~\ref{thm:main} the first alternative occurs,
because $|t_a-T|$ is comparable to $\delta_a$.

\begin{proposition}[Pinned radius one or two]
\label{prop:col-sum}
Let collision data with $R\in\{1,2\}$ be given. There is no resolved
sequence for these data.
\end{proposition}
\begin{proof}
Suppose that a resolved sequence exists.

\emph{Step 0: the states used.} We apply
Lemmas~\ref{lem:col-incidences} and~\ref{lem:col-counts} to several
objects. At $C=(1,1)$ the measure $\mu^{\rm c}_{t_a}$ has mass
$\frac12$ and $\nu_{\vac}$ has mass $\frac14$, so there is exactly one
state at $(1,1)$; it has spin zero and dimension $2+z_a$ with
\[
 \varsigma_a:=|z_a|\leq\theta_a^{60}.
\]
We call it the \emph{long scalar}; its physical point in
$D((1,1),\varrho_a)$ is $(\sqrt h,\sqrt h)$ with $h=1+\frac12z_a$, and
this is the only physical point of a state in that disc. For
$\nu\in\mathcal N$ there is exactly one state at $Y_\nu(t_a)$; it has
spin $\nu$ and dimension $E_\nu(t_a)+\delta_{\nu,a}$, and
$\delta_{p,a}=0$ as in Step~1 of the proof of
Proposition~\ref{prop:col-product}. The row preimage
$x_{0,0,0}=[2,2]=(2,0)$ lies on the $P$-axis and outside $V$, and
$\sigma(R,t_a)(\{(2,0)\})=\frac12$ by
Lemma~\ref{lem:col-incidences}(iii). A state with a physical point at
$(2,0)$ has weights $(4,0)$ and gives mass $\frac12$ to this point, so
$\mathfrak S(R,t_a)$ contains exactly one such state, whose sign
patterns $(+,+)$ and $(+,-)$ give the point $(2,0)$. By
Lemma~\ref{lem:col-counts}(i) there is one state $j_+$ of $\rho_a$ such
that the physical points of states in $D((2,0),\varrho_a)$ are exactly
the two points $(\sqrt h,\pm\sqrt{\bar h})$ of $j_+$; it has spin $4$
and dimension $4+\delta_a^+$, where $0<\delta_a^+\leq\theta_a^{60}$,
the lower bound because $\bar h>0$. In the same way there is one state
$j_-$, of spin $-4$ and dimension $4+\delta_a^-$, such that the
physical points of states in $D((0,2),\varrho_a)$ are the two points
$(\pm\sqrt h,\sqrt{\bar h})$ of $j_-$. Finally, for $(k,j)\neq(0,0)$
the preimage $x_{l,k,j}(t_a)$ is not in $\Sigma(t_a)$, and by
Lemma~\ref{lem:col-counts}(i) the disc
$D(x_{l,k,j}(t_a),\varrho_a)$ contains no physical point of a state and
no point of $V$.

\emph{Step 1: the scale.} Put
\begin{equation}\label{eq:col-scale-H}
 b=\varsigma_a+a,\qquad H=\log(1/b)\leq L,\qquad\beta=a+\varsigma_a^2,
 \qquad\epsilon^2=\frac{\pi\varrho_a^2}{16H}.
\end{equation}
Then $b\leq2\theta_a^{60}\to0$,
\[
 e^{-\pi\varrho_a^2/(4\epsilon^2)}=b^4\leq8\beta,\qquad
 e^{-\pi\varrho_a^2/(2\epsilon^2)}=b^8,\qquad
 \epsilon^{-4}=\frac{256H^2}{\pi^2c_\varrho^4\,\theta_a^8},
\]
and, since $x\log^{5/2}(1/x)\leq C\sqrt x$ for $0<x\leq1$,
\begin{equation}\label{eq:col-flat}
 \varsigma_aH^{5/2}\leq b\log^{5/2}(1/b)\leq C\,b^{1/2}
 \leq C\,\theta_a^{30}.
\end{equation}
By \eqref{eq:col-a-theta}, $aH\leq aL\leq\theta_a^{500}$, so
$\epsilon\leq\frac12$ and $a\leq\frac18\epsilon^2$ for small $a$; thus
Lemma~\ref{lem:resolved-budget}(iii) and
Lemma~\ref{lem:row-multiplier}(iii) apply with the widths $\epsilon$
and $\epsilon/\sqrt2$.

\emph{Step 2: the budget of $\Phi^{(p)}$.} By
Lemma~\ref{lem:resolved-budget}(ii) the centres of
$\Phi^{(p)}_{t_a}$ with negative coefficient and positive mass are
$\pm C$, $\pm C^\sharp$, at which the only state is the long scalar,
with displacement $\varsigma_a\leq\varrho_a^2$, and $\pm B_0(t_a)$,
$\pm B_0(t_a)^\sharp$, at which the state has displacement zero.
Lemma~\ref{lem:resolved-budget}(iii) with
$\varepsilon=\varsigma_a$ gives
\[
 \sum_j\Phi^{(p)}_{t_a,\epsilon}(X_j^\sigma)
 \leq C\theta_a^{-4}\bigl((a+\varsigma_a^2)\,\theta_a^{-8}H^2+b^4\bigr)
 \leq C\,\beta\,\theta_a^{-12}H^2 ,
\]
and therefore
\begin{equation}\label{eq:col-sum-initial}
 \Phi^{(p)}_{t_a}(X_j^\sigma)\leq\Gamma_0:=C\,\beta\,\theta_a^{-12}H^2
 \qquad(\Delta_j\leq E_*).
\end{equation}

\emph{Step 3: the state at $W(t_a)$.} Since $\Omega\geq S_{C^\sharp}$,
we have $\Phi^{(p)}_t\geq S_{C^\sharp}S_{B_0(t)}S_{B_0(t)^\sharp}$. We
apply Lemma~\ref{lem:col-anchor} at $X_0(t)=Y_3(t)=W(t)$ with the
intended zero $S_{B_0(t)}$ and the two factors $S_{C^\sharp}$,
$S_{B_0(t)^\sharp}$ of Table~\ref{tab:col-phases}; thus $k=1$, $k'=2$,
and by \eqref{eq:col-sum-initial}
\begin{equation}\label{eq:col-anchor-W}
 |\delta_{3,a}|\leq C\,\theta_a^{-4}\,\Gamma_0^{1/2}
 \leq C\,\beta^{1/2}\theta_a^{-10}H .
\end{equation}

\emph{Step 4: the state near $(2,0)$.} Let $X=[u,v]$ be the physical
point $(\sqrt h,\sqrt{\bar h})$ of $j_+$, so that $u,v>0$, $uv=4$ and
$|X-(2,0)|\leq\theta_a^{30}$ by Lemma~\ref{lem:col-counts}(i). Then, as
$|u-2|,|v-2|\leq\frac14$,
\[
 \delta_a^+=\tfrac12(u^2+v^2)-uv=\tfrac12(u-v)^2\leq(u-2)^2+(v-2)^2
 \leq\tfrac1{16}\,\Omega(X),
\]
by \eqref{eq:ker-sine-distance}. By \eqref{eq:col-sine-bound} for the
target $[2,2]$, both $S_{B_0(t_a)}$ and $S_{B_0(t_a)^\sharp}$ are at
least $\kappa_a$ at $(2,0)$. For real $Y$ the function $S_Y$ has
gradient bounded by $2\pi|Y|$, since $|B(Z,Y)|\leq2|Z||Y|$; hence these
two factors are at least
$\kappa_a-2\pi\varrho_1\theta_a^{30}\geq\frac12\kappa_a$ at $X$. With
\eqref{eq:col-sum-initial},
\begin{equation}\label{eq:col-axis-bound}
 0<\delta_a^+\leq\frac1{16}\,\frac{4\,\Gamma_0}{\kappa_a^2}
 \leq C\,\beta\,\theta_a^{-20}H^2 .
\end{equation}

\emph{Step 5: the budget of $\Phi^{(3)}$ at the width
$\epsilon/\sqrt2$.} By Lemma~\ref{lem:resolved-budget}(ii) the centres
of $\Phi^{(3)}_{t_a}=\Omega\,S_{W(t_a)}S_{W(t_a)^\sharp}$ with negative
coefficient and positive mass are $\pm C$, $\pm C^\sharp$ and
$\pm W(t_a)$, $\pm W(t_a)^\sharp$. Put
$\varepsilon=C\beta^{1/2}\theta_a^{-10}H$ with the constant of
\eqref{eq:col-anchor-W}, enlarged so that $C\geq1$. It bounds
$|\delta_{3,a}|$ and $\varsigma_a\leq\beta^{1/2}$, and
$\beta^{1/2}H\leq a^{1/2}L+bH\leq a^{1/4}+Cb^{1/2}\leq C\theta_a^{25}$
by (K1) and the inequality used in \eqref{eq:col-flat}, so
$\varepsilon\leq\varrho_a^2$ for small $a$.
Lemma~\ref{lem:resolved-budget}(iii) with the width $\epsilon/\sqrt2$
gives
\begin{equation}\label{eq:col-W-budget}
 \sum_j\Phi^{(3)}_{t_a,\epsilon/\sqrt2}(X_j^\sigma)
 \leq C\theta_a^{-4}\Bigl(\bigl(a+\beta\,\theta_a^{-20}H^2\bigr)
 \theta_a^{-8}H^2+b^8\Bigr)\leq C\,\beta\,\theta_a^{-32}H^4 .
\end{equation}

\emph{Step 6: the row from above.} By
\eqref{eq:ker-row-inequality} with $t=t_a$ and $w=3$,
\eqref{eq:col-W-budget}, and
$\epsilon^2=\pi c_\varrho^2\theta_a^4/(16H)$,
\begin{equation}\label{eq:col-row-upper}
 \bigl|\langle\mu_a,\mathsf D_{t_a}\psi_\epsilon\rangle\bigr|
 \leq C\,\epsilon^3\beta\,\theta_a^{-32}H^4+C\,a\,\epsilon^{-1}
 \leq C\,\beta\,\theta_a^{-26}H^{5/2}.
\end{equation}

\emph{Step 7: the row from below.} By
Lemma~\ref{lem:row-preimages}(i),
$\langle\mu_a,\mathsf D_{t_a}\psi_\epsilon\rangle$ is the combination,
with coefficients $b_lb_kb_j$ of total modulus $64$, of the numbers
$\langle\mu_a,\psi_{\epsilon,x}\rangle$, $x=x_{l,k,j}(t_a)$. Each of
these is the sum of $\psi_{\epsilon,x}$ over the physical points of all
states, with weights $\frac14$, and over the points of $V$, with the
weights \eqref{eq:ker-vacuum-masses}. By
Lemma~\ref{lem:row-preimages}(ii) and \eqref{eq:ker-narrow-tail}, the
points at distance at least $\varrho_a$ from $x$ contribute at most
$(C_1+4)e^{-\pi\varrho_a^2/(2\epsilon^2)}=(C_1+4)b^8$. The points within
$\varrho_a$ of $x$ were determined in Step~0. For $(k,j)\neq(0,0)$
there are none. For $x_{1,0,0}=(1,1)$ they are the atom of $\nu_{\vac}$
at $(1,1)$, at which $\psi_{\epsilon,(1,1)}$ vanishes, and the point
$(\sqrt h,\sqrt h)$ of the long scalar, with weight $\frac14$; by
Lemma~\ref{lem:row-preimages}(iii),
\[
 \Bigl|\langle\mu_a,\psi_{\epsilon,(1,1)}\rangle-\frac{z_a}{16}\Bigr|
 \leq\frac{\varsigma_a^2}{32}+\frac{\varsigma_a^3}{2\epsilon^2}
 +(C_1+4)b^8\leq C\,\beta ,
\]
where $\varsigma_a^3\epsilon^{-2}\leq C\varsigma_a^2\,
\varsigma_aH\theta_a^{-4}\leq C\varsigma_a^2$ by \eqref{eq:col-flat};
the expansion of that lemma is valid for all $|z_a|\leq1$ and requires
no comparison between $\varsigma_a$ and $\epsilon$. For
$x_{0,0,0}=(2,0)$ they are the two points of $j_+$, each with weight
$\frac14$, and by Lemma~\ref{lem:row-preimages}(iv) and
\eqref{eq:col-axis-bound} they contribute at most
$\frac1{16}\delta_a^+\leq C\beta\theta_a^{-20}H^2$ in modulus. For
$x_{2,0,0}=(0,2)$ they are the two points of $j_-$, whose contributions
cancel by Lemma~\ref{lem:row-preimages}(iv). The coefficients of the
three preimages with $k=j=0$ are $1,-2,1$. Hence
\begin{equation}\label{eq:col-row-value}
 \Bigl|\langle\mu_a,\mathsf D_{t_a}\psi_\epsilon\rangle
 +\frac{z_a}8\Bigr|\leq C\,\beta\,\theta_a^{-20}H^2 .
\end{equation}
The pairing here is the physical one \eqref{eq:ker-physical-pairing},
in which the vacuum atom at $(1,1)$ is not shifted; the central shift
is contained in \eqref{eq:ker-row-inequality}.

\emph{Step 8: the long scalar.} By \eqref{eq:col-row-upper} and
\eqref{eq:col-row-value},
\[
 \tfrac18\varsigma_a\leq C\,(a+\varsigma_a^2)\,\theta_a^{-26}H^{5/2}.
\]
By \eqref{eq:col-flat},
$C\varsigma_a\theta_a^{-26}H^{5/2}\leq C\theta_a^{4}\leq\frac1{16}$ for
small $a$, so the quadratic term is at most $\frac1{16}\varsigma_a$,
and with \eqref{eq:col-a-theta}
\begin{equation}\label{eq:col-scalar-bound}
 \varsigma_a\leq C\,a\,\theta_a^{-26}L^{5/2}\leq C\,a^{974/1000}L^{5/2}
 <a^{1/3}.
\end{equation}

\emph{Step 9: the final budgets.} We now use the width
\eqref{eq:col-width-L}. As in Step~2, with
$\varepsilon=\varsigma_a<a^{1/3}\leq\varrho_a^2$,
Lemma~\ref{lem:resolved-budget}(iii) gives
\[
 \sum_j\Phi^{(p)}_{t_a,\epsilon}(X_j^\sigma)
 \leq C\theta_a^{-4}\bigl((a+a^{2/3})\theta_a^{-8}L^2+a\bigr)
 \leq C\,a^{2/3}\theta_a^{-12}L^2 ,
\]
so that $\Phi^{(p)}_{t_a}(X_j^\sigma)\leq Ca^{2/3}\theta_a^{-12}L^2$ for
$\Delta_j\leq E_*$. For $k\in\{1,2\}$ we apply
Lemma~\ref{lem:col-anchor} at $X_0(t)=B_k(t)=Y_{p_k}(t)$ to
$\Phi^{(p)}_t\geq S_{C^\sharp}S_{B_0(t)}S_{B_0(t)^\sharp}$, with the
intended zero $S_{B_0(t)}$ and the two factors of
Table~\ref{tab:col-phases}:
\[
 |\delta_{p_k,a}|\leq C\,\theta_a^{-4}\bigl(a^{2/3}\theta_a^{-12}L^2
 \bigr)^{1/2}=C\,a^{1/3}\theta_a^{-10}L\qquad(k=1,2).
\]
Put $\varepsilon_a=C\,a^{1/3}\theta_a^{-10}L$ with $C\geq1$; it bounds
$\varsigma_a$ and $|\delta_{p_k,a}|$, and
$\varepsilon_a\leq\varrho_a^2$ for small $a$ by
\eqref{eq:col-a-theta}. For $k=0,1,2$ the centres of
$\Phi^{(p_k)}_{t_a}$ with negative coefficient and positive mass are
$\pm C$, $\pm C^\sharp$, $\pm B_k(t_a)$, $\pm B_k(t_a)^\sharp$, and
Lemma~\ref{lem:resolved-budget}(iii) with
$\varepsilon=\varepsilon_a$ gives
\begin{equation}\label{eq:col-three-budgets}
 \sum_j\Phi^{(p_k)}_{t_a,\epsilon}(X_j^\sigma)
 \leq C\theta_a^{-4}\Bigl(\bigl(a+a^{2/3}\theta_a^{-20}L^2\bigr)
 \theta_a^{-8}L^2+a\Bigr)\leq C\,a^{2/3}\theta_a^{-32}L^4 .
\end{equation}

\emph{Step 10: localization.} Let $j$ be a state with $\Delta_j\leq L$
and $X=[u,v]$ one of its physical points. As in Step~4 of the proof of
Proposition~\ref{prop:col-product}, the broad Gaussian at $X$ is at
least $e^{-1}$, so by \eqref{eq:col-three-budgets}
\[
 \Omega(X)\,S_{B_k(t_a)}(X)S_{B_k(t_a)^\sharp}(X)
 \leq\Gamma_a:=C\,a^{2/3}\theta_a^{-32}L^4\leq1\qquad(k=0,1,2),
\]
and $uv=s_j\in\ZZ$, $|u|,|v|\leq\Upsilon=\sqrt{2L}$. We apply
Lemma~\ref{lem:three-projection}(d) and (a) with $t=t_a$,
$t_-=\frac12$, $t_+=R_{\max}+1$, $\ell=1$ and
\[
 \zeta=\tfrac12\Gamma_a^{1/6}=C\,a^{1/9}\theta_a^{-16/3}L^{2/3},
 \qquad\zeta\,\Upsilon\leq C\,a^{1/9}\theta_a^{-16/3}L^{7/6}
 \leq C\,a^{119/1125}L^{7/6},
\]
by \eqref{eq:col-a-theta}. Since $\zeta\Upsilon\to0$, the smallness
conditions hold, and there is a real point $X_j'$, either in
$\Lambda_{t_a}\cup\Lambda_{1/t_a}$ or of the form $[m/2,n/2]$, whose
spin is exactly $s_j$, with
\begin{equation}\label{eq:col-sum-localization}
 \bigl|\Delta_j-|X_j'|^2\bigr|\leq C\,\zeta\,\Upsilon
 \leq C\,a^{1/9}\theta_a^{-16/3}L^{7/6}\leq C\,a^{119/1125}L^{7/6}.
\end{equation}
In the second case
$X_j'\in\Sigma_2=\Lambda_{1/2}\cup\Lambda_1\cup\Lambda_2$ by
\eqref{eq:ker-grid-circles}.

\emph{Step 11: contradiction.} As in Step~5 of the proof of
Proposition~\ref{prop:col-product}, the localization
\eqref{eq:col-sum-localization} of all states with $\Delta_j\leq L$
contradicts Theorem~\ref{thm:moving-localization}, applied with the
radii $\{\frac12,1,2\}$, the interval $I$, the radii $t_a$ and $u_*=1$.
\end{proof}

In Step~8 the quadratic term is absorbed because the comparison theorem
already gives $\varsigma_a\leq\theta_a^{60}$, which is much smaller
than the loss $\theta_a^{-26}$. The case $\varsigma_a=0$ needs no
separate treatment, since the scale $b=\varsigma_a+a$ is positive.

\subsection{Proof of Theorem~\ref{thm:main}}
\label{subsec:col-main-proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
The two formulations in the theorem are equivalent, and by
Remark~\ref{rem:gap-range} it suffices to consider
$0<g\leq\frac12$. Suppose that the theorem fails for such a $g$. Then
there are admissible partition functions of gap $g$ at central charges
$c=1+12a$ with $a$ arbitrarily small, hence an admissible sequence of
gap $g$ with $a\to0$. By Theorem~\ref{thm:compactness} a subsequence
converges, and by Theorem~\ref{thm:half-integer-limits} its limit is
$\sigma(R_1,R_2)$ with half-integers
$1\leq R_1\leq R_2\leq R_{\max}$. We show that a further subsequence,
with suitable radii, is a resolved sequence for collision data
$(R,T)$, $\{R,T\}=\{R_1,R_2\}$. This contradicts
Proposition~\ref{prop:col-product} if $R\notin\{1,2\}$ and
Proposition~\ref{prop:col-sum} if $R\in\{1,2\}$. Subsequences of
admissible sequences are admissible sequences, and all statements below
hold for sufficiently small $a$ in the sequence under consideration.

\emph{Step 1: the comparison measure.}
Theorem~\ref{thm:comparison} provides the indices $\iota_a$ and the
radii $t_a\geq1$. Passing to a subsequence we may assume that
$\iota_a$ is constant. Then $R=R_{\iota_a}$ and $T=R_{3-\iota_a}$ are
fixed half-integers, $\sigma(R,T)=\sigma(R_1,R_2)$, and $t_a\to T$. We
fix the collision data of Section~\ref{subsec:col-data} for $(R,T)$,
and with them the interval $I$, the integer $p$, the constants
$\varepsilon_0$, $c_0$ of Lemma~\ref{lem:quadratic-collisions}, and
$E_*$.

\emph{Step 2: the finite orders.} Put
\[
 m=20000,\qquad n=20000 .
\]
These integers are chosen after the data of Step~1, on which they do
not depend, and before any estimate involving $a$. By
Theorem~\ref{thm:four-scalar-slow} with the exponent $1/m$ and $k=1$,
\begin{equation}\label{eq:col-slow}
 a\leq\delta_a^{m}
\end{equation}
for all sufficiently small $a$. (The alternative
$\delta_a<a^{1/m}$ along a subsequence is the case excluded in the proof
of that theorem by Proposition~\ref{prop:four-scalar-budget} and
Theorem~\ref{thm:fixed-localization}.) Since $\delta_a\to0$, it follows
that
$\sqrt a\leq\delta_a$, hence
\[
 \delta_a\leq b_a\leq2\delta_a,\qquad H_a\leq\log(1/\delta_a),\qquad
 \bigl(a+b_a^{n+1}\bigr)^{1/84}
 \leq\bigl(\delta_a^{20000}+(2\delta_a)^{20001}\bigr)^{1/84}
 \leq2\,\delta_a^{20000/84}.
\]
As $20000/84>238$, for every pair of constants $C$, $K$ we have
\begin{equation}\label{eq:col-accuracy}
 C\,\bigl(a+b_a^{n+1}\bigr)^{1/84}(1+H_a)^{K}
 \leq2C\,\delta_a^{238}\bigl(1+\log(1/\delta_a)\bigr)^K
 \leq\delta_a^{200}
\end{equation}
for small $a$. Thus the bounds \eqref{eq:col-comparison-matching} and
\eqref{eq:col-matching}, for this $n$ and for the finitely many windows
and the number $E_*$ used below, are at most $\delta_a^{200}$.

\emph{Step 3: the collision scale.} Consider the window
$\mathcal W_{\eta_d}(E_d,0)$. A point of spin zero of
$\supp\sigma(R_1,R_2)\subset\Gamma$ has dimension in
$\frac1{2d^2}\ZZ$, and $3\eta_d<\frac1{2d^2}$, so the window is
isolated for $\mathfrak S(R_1,R_2)$. By
Lemma~\ref{lem:four-scalar-cluster} the states of $\rho_a$ in it have
the dimensions $E_d+z_{i,a}$, $1\leq i\leq4$, and
$\delta_a=\max_i|z_{i,a}|$. By \eqref{eq:ker-common-grid} the numbers
$d/R$, $dR$, $d/T$, $dT$ are integers. The classes with $mn=0$ and
$E_{m,n}(R)=E_d$ are those of $(d/R,0)$ and $(0,dR)$, neither of which
is the class of $(0,2R)$, since $d\geq4$; the classes with $mn=0$ and
$E_{m,n}(T)=E_d$ are those of $(d/T,0)$ and $(0,dT)$. By
Lemma~\ref{lem:col-windows}(i) the states of $\mathfrak S(R,t_a)$ in
the window have the dimensions
\[
 E_d,\qquad E_d,\qquad E_{d/T,0}(t_a)=E_d\,(t_a/T)^2,\qquad
 E_{0,dT}(t_a)=E_d\,(T/t_a)^2 .
\]
For two ordered quadruples the largest distances to $E_d$ differ by at
most the largest distance between corresponding entries. Hence, by
Theorem~\ref{thm:comparison}(iii) and \eqref{eq:col-accuracy},
\[
 \bigl|\delta_a-E_d\,\mathfrak d(t_a)\bigr|\leq\delta_a^{200},\qquad
 \mathfrak d(t)=\max\bigl\{|(t/T)^2-1|,\ |(T/t)^2-1|\bigr\}.
\]
For $|t-T|\leq\frac12T$ we have
$|t-T|/T\leq\mathfrak d(t)\leq10\,|t-T|/T$. Since
$\delta_a^{200}\leq\frac12\delta_a$, we get
$\frac12\delta_a\leq E_d\,\mathfrak d(t_a)\leq2\delta_a$ and
\begin{equation}\label{eq:col-theta-delta}
 \frac{T}{20E_d}\,\delta_a\leq|t_a-T|\leq\frac{2T}{E_d}\,\delta_a .
\end{equation}
In particular $t_a\neq T$, because $\delta_a>0$ by
\eqref{eq:col-slow}.

\emph{Step 4: anchoring the radius on an actual state.} By
Corollary~\ref{cor:col-matching} and \eqref{eq:col-accuracy} there is,
for small $a$, a map $j\mapsto j'$ from the states of $\rho_a$ with
$\Delta_j\leq E_*$ into $\mathfrak S(R,t_a)$ as described there, with
$|\Delta_j-\Delta'_{j'}|\leq\delta_a^{200}$. The moving state of the
class of $(1,p)$ has spin $p$ and dimension
$E_{1,p}(t_a)=E_p(t_a)\leq\varrho_1^2\leq E_*-1$, so it is the image of
a state $j_0$ of $\rho_a$, which has spin $p$ and
\begin{equation}\label{eq:col-anchor-state}
 |\Delta_{j_0}-E_p(t_a)|\leq\delta_a^{200}.
\end{equation}
This state does not depend on the choices made in
Corollary~\ref{cor:col-matching}: it is the only state of $\rho_a$ of
spin $p$ whose dimension is within $c\,\delta_a$ of $E_p(t_a)$, for a
suitable constant $c>0$. Indeed, the states of spin $p$ of
$\mathfrak S(R,t)$ are, by Lemma~\ref{lem:col-states}(ii), the moving
states of the classes of $(1,p)$ and $(p,1)$, of dimensions $E_p(t)$
and $E_p(1/t)$, and the fixed states of the same two classes, of
dimensions $E_p(R)$ and $E_p(1/R)$. The equation $E_p(x)=E_p(T)$,
$x>0$, holds only for $x=T$ and $x=p/T$, and $p/T>R_{\max}$. Hence the
dimensions other than $E_p(t)$ that tend to $E_p(T)$ as $t\to T$ are
$E_p(1/t)$ when $T=1$, and the constants $E_p(R)=E_p(T)$ when $R=T$
and $E_p(1/R)=E_p(T)$ when $R=T=1$. Since
$|E_p'|\geq4/(R_{\max}+1)$ on $I$ and
$E_p(1/t)-E_p(t)=\frac12(p^2-1)(t^2-t^{-2})$, each of them differs
from $E_p(t)$ by at least a constant multiple of $|t-T|$ near $T$; so
$E_p(t_a)$ is a simple node of the comparison and is separated from
the other nodes of spin $p$ by at least $c'|t_a-T|\geq2c\,\delta_a$,
by \eqref{eq:col-theta-delta}. As every state of $\rho_a$ of spin $p$
with dimension near $E_p(T)$ is within $\delta_a^{200}$ of the node to
which it is mapped, the claim follows.

Since $E_p$ is strictly decreasing on $I$, $T$ is interior to $I$ and
$\Delta_{j_0}\to E_p(T)$, there is a unique $\tilde t_a\in I$ with
\begin{equation}\label{eq:col-reanchor}
 E_p(\tilde t_a)=\Delta_{j_0},\qquad
 |\tilde t_a-t_a|\leq\tfrac14(R_{\max}+1)\,\delta_a^{200},
\end{equation}
by \eqref{eq:col-anchor-state} and the lower bound for $|E_p'|$. Let
$\tilde\theta_a=|\tilde t_a-T|$. By \eqref{eq:col-theta-delta} and
\eqref{eq:col-reanchor},
\begin{equation}\label{eq:col-theta-tilde}
 \frac{T}{40E_d}\,\delta_a\leq\tilde\theta_a\leq\frac{4T}{E_d}\,\delta_a .
\end{equation}

\emph{Step 5: the sequence with the radii $\tilde t_a$ is resolved.}
(K1): by \eqref{eq:col-theta-tilde}, $0<\tilde\theta_a\to0$, and by
\eqref{eq:col-slow} and \eqref{eq:col-theta-tilde},
$a\leq\delta_a^{20000}\leq(40E_d\tilde\theta_a/T)^{20000}
\leq\tilde\theta_a^{1000}$ for small $a$. (K2): by the last assertion of
Corollary~\ref{cor:col-matching}, with
$e_a=\frac14(R_{\max}+1)\delta_a^{200}$, and by
\eqref{eq:col-accuracy}, there is a map $j\mapsto j'$ into
$\mathfrak S(R,\tilde t_a)$ with the required properties and
\[
 |\Delta_j-\Delta'_{j'}|\leq C\,\delta_a^{200}
 \leq C\,(40E_d\tilde\theta_a/T)^{200}\leq\tilde\theta_a^{60}.
\]
(K3): the state $j_0$ has spin $p$ and dimension
$E_p(\tilde t_a)$ by \eqref{eq:col-reanchor}. Hence the sequence,
with the radii $\tilde t_a$, is a resolved sequence for the collision
data $(R,T)$, which is excluded by
Propositions~\ref{prop:col-product} and~\ref{prop:col-sum}.
\end{proof}

\begin{remark}[The order of the choices]\label{rem:col-quantifiers}
The proof uses the following finite quantities, fixed in this order.
\begin{enumerate}[(1)]
\item The gap $g$ determines $d$, the grid $\Gamma$, the prime
 $\mathfrak p$ and the reconstruction operator $P_*$.
\item The limit of the sequence and the rule \eqref{eq:col-pin}, which
 involve no Taylor order, determine the pinned radius $R$ and the limit
 $T$ of the free radius. The pair $(R,T)$ determines the collision data
 and the constants of Lemma~\ref{lem:quadratic-collisions}. The
 collision exponent is two, the losses in
 Propositions~\ref{prop:col-product} and~\ref{prop:col-sum} are at most
 $\theta_a^{-36}$, and Definition~\ref{def:col-resolved} requires the
 accuracy $\theta_a^{60}$ and the bound $a\leq\theta_a^{1000}$. These
 exponents do not depend on $g$ or on the data.
\item The exponent $m=20000$ of the comparison between $a$ and
 $\delta_a$ and the Taylor order $n=20000$ are chosen so that
 $\min\{m,n+1\}/84>200\geq60$ and $m>1000$.
\item The order $n$ determines the local order $q=2n$, the tests
 $\mathcal F_q$ and the projection $\Pi_q$, then the number
 $D_0=4(n+2)/\pi$, then the window constant $M$ of
 \eqref{eq:col-M-condition}, as explained in the proof of
 Lemma~\ref{lem:normal-form}.
\item Finally $a$ is taken small. The certificates of
 Theorem~\ref{thm:moving-localization}, which are used for the
 exponents $\frac{29}{500}$ and $\frac{119}{1125}$ of
 \eqref{eq:col-product-localization} and
 \eqref{eq:col-sum-localization}, are fixed inside the proof of that
 theorem.
\end{enumerate}
No quantity depends on one that is chosen later, and no Taylor order
depends on $a$. With $a\leq\tilde\theta_a^{\,m'}$ for a larger exponent
$m'$ in place of $1000$, the same proofs give the localization exponents
$\frac1{16}-\frac9{2m'}$ and $\frac19-\frac{16}{3m'}$; the proof of
Theorem~\ref{thm:main} provides every $m'<20000$. The only input from
Theorem~\ref{thm:four-scalar-slow} is the inequality
\eqref{eq:col-slow}; no upper bound for $\delta_a$ by a power of $a$
is used, and none is available.
\end{remark}