\section{Classification of the limits at \texorpdfstring{$c=1$}{c=1}}
\label{sec:classification}

Theorem~\ref{thm:compactness} produces limits of the following kind.
Fix $0<g\leq\frac12$. A \emph{boundary partition function of gap $g$} is
a function $Z_0=H_0/|\eta|^2$ on $\HH$ with
$Z_0(-1/\tau)=Z_0(\tau)=Z_0(\tau+1)$ and
\begin{equation}\label{eq:cls-boundary-long}
 H_0=(1-q)(1-\bar q)+\sum_{j\in\mathcal J_0}q^{h_j}\bar q^{\bar h_j},
 \qquad h_j,\bar h_j\geq0,\quad h_j-\bar h_j\in\ZZ,\quad
 h_j+\bar h_j\geq g ,
\end{equation}
where $\mathcal J_0$ is countable and the sum converges absolutely on
$\HH$. Its long measure is
$\rho_0=\sum_{j\in\mathcal J_0}A_{h_j}\otimes A_{\bar h_j}$ and its Fock
measure is $\mu_0=\nu_{\vac}+\rho_0$. Parity invariance is not assumed in
this section.
Collecting equal exponents, $H_0=\sum d_{h,\bar h}\,q^h\bar q^{\bar h}$
with integers $d_{h,\bar h}$, the \emph{Fock coefficients}; this
expansion is unique. They are nonnegative except possibly at
$(h,\bar h)=(1,0)$ and $(0,1)$, where the vacuum contributes $-1$, and
$d_{0,0}=1$.

\subsection{The crystalline reduction}

The classification of all modular-invariant partition functions at
$c=1$ with discrete spectrum, under axioms that include positivity in
the irreducible $c=1$ characters, is due to Eberhardt and Levine
\cite{EL}. We use the harmonic-analytic part of their argument, in the
following form.

\begin{proposition}[{\cite[Theorem 3.1 and Sections 3.3--3.5]{EL}}]
\label{prop:cls-reduction}
Let $\mu$ be an even, real, purely atomic measure on $\RR^2$ whose atoms
have masses in $\frac14\ZZ$. Suppose that
\begin{enumerate}[(a)]
\item $|\mu|(\{|X|\leq\varrho\})\leq C\varrho^2$ for all $\varrho\geq1$;
\item $\mathbb S\mu=\mu$ as tempered distributions;
\item $P^2-Q^2\in\ZZ$ on $\supp\mu$.
\end{enumerate}
Then there are a finite set $\mathcal R\subset[1,\infty)$ and complex
numbers $c_r$ with $\mu=\sum_{r\in\mathcal R}c_r\,\rhoF_r$.
\end{proposition}

\begin{remark}\label{rem:cls-source}
We record how this statement is extracted from \cite{EL}, since it is
not displayed there in this form. Theorem~3.1 of \cite{EL}, applied to
$4\mu$, requires a tempered atomic measure with integer masses whose
transform $\mathbb S\mu$ is a measure of upper bounded density,
$|\mathbb S\mu|(\{|X|\leq\varrho\})\leq C\varrho^2$ for $\varrho\geq1$;
hypothesis (a) gives temperedness and, by (b), this bound. The subsequent
reduction \cite[(127)--(150)]{EL} of the resulting finite union of
lattice cosets to combinations of circle (Narain) combs uses, in
addition, only spin quantization (c), evenness, and (b). The authors
note that positivity enters their argument up to that point only
through the density bound, and that uniqueness of the vacuum is not
used. Their proof of Theorem~3.1 relies on Cohen's idempotent theorem
\cite{Cohen} and related results on Fourier quasicrystals, which we do
not reprove. Alternatively, a function of the form
\eqref{eq:cls-boundary-long} satisfies the full list of axioms of
\cite{EL}, so their final classification applies to it directly; the
additional solutions found there have a negative Fock coefficient at
$(h,\bar h)=(p^2,0)$ for large primes $p$ and are excluded by
\eqref{eq:cls-boundary-long}. Both routes lead to
Lemma~\ref{lem:cls-mixture} below.
\end{remark}

\begin{lemma}\label{lem:cls-growth}
The Fock measure $\mu_0$ of a boundary partition function satisfies the
hypotheses of Proposition~\ref{prop:cls-reduction}.
\end{lemma}
\begin{proof}
The measure is even and atomic, and its atoms have masses in
$\frac14\ZZ$ by construction; (c) holds because the spins are integers
and the points of the four vacuum terms have spin $0$ or $\pm1$.

For (a), let $P_0(y)=\sum_{j\in\mathcal J_0}e^{-2\pi y(h_j+\bar h_j)}$.
Since $|\nu_{\vac}|$ consists of the four vacuum terms with positive
signs,
\[
 \int e^{-2\pi y|X|^2}\dd|\mu_0|(X)
 \leq P_0(y)+(1+e^{-2\pi y})^2=H_0(iy)+4e^{-2\pi y}.
\]
Modular invariance of $\sqrt y\,H_0$ gives $H_0(iy)=y^{-1}H_0(i/y)
\leq y^{-1}(1+P_0(1))$ for $0<y\leq1$. Taking $y=\varrho^{-2}$ and using
$e^{-2\pi y|X|^2}\geq e^{-2\pi}$ on $\{|X|\leq\varrho\}$,
\[
 |\mu_0|(\{|X|\leq\varrho\})\leq e^{2\pi}\bigl((1+P_0(1))\varrho^2+4\bigr).
\]
In particular $\mu_0$ is a tempered distribution. For (b), note that
$\mathcal L_0(f)=\langle\mu_0,f\rangle$, so
Lemma~\ref{lem:gaussian-identity} gives
$\langle\mu_0,f\rangle=\langle\mu_0,\mathbb Sf\rangle$ for all
$f\in\mathcal G_{2\pi}$. This class contains the Hermite functions,
which span a dense subspace of the Schwartz space, and both sides are
continuous there. Hence $\mathbb S\mu_0=\mu_0$.
\end{proof}

The proof of (a) uses positivity only through the observation that the
total variation of $\mu_0$ exceeds its thermal trace by at most four
vacuum units; the general density estimate of \cite[Section~2]{EL} is not
needed.

\subsection{The mixture weights}

By Proposition~\ref{prop:cls-reduction} and Lemma~\ref{lem:cls-growth},
$\mu_0=\sum_{r\in\mathcal R}c_r\rhoF_r$ for a finite set
$\mathcal R\subset[1,\infty)$ of distinct radii, and pairing with
$G_\tau$ gives
\begin{equation}\label{eq:cls-theta-mixture}
 H_0=\sum_{r\in\mathcal R}c_r\,\Theta_r .
\end{equation}
Recall from \eqref{eq:circle-measures} that the monomials of $\Theta_t$
are indexed by the points $(u,v)=(mt,n/t)$ of $\Lambda_t$, and that two
points give the same monomial $q^h\bar q^{\bar h}$ exactly when they
are related by $(u,v)\mapsto\pm(u,v)$ or $\pm(v,u)$.

\begin{lemma}\label{lem:cls-mixture}
In \eqref{eq:cls-theta-mixture}, $c_r\in\frac12\NN$ for every
$r\in\mathcal R$, and $\sum_{r\in\mathcal R}c_r=1$.
\end{lemma}
\begin{proof}
Fix $r\in\mathcal R$ with $r>1$ and choose a prime $p$ larger than the
squares of all radii in $\mathcal R$. Consider the point
$(u,v)=(r,p/r)\in\Lambda_r$, of spin $p$, and let $(h,\bar h)$ be its
weights. A point $(mt,n/t)$ of $\Lambda_t$, $t>0$, has the same weights
if and only if it is one of $\pm(r,p/r)$, $\pm(p/r,r)$. In the first
case $mn=p$ and $t=r/m$, so $t\in\{r,r/p\}$; in the second $mn=p$ and
$t=p/(rm)$, so $t\in\{p/r,1/r\}$. For $t\in\mathcal R$, $t\neq r$, all
four values are impossible: $r/p<1$, $1/r<1$, and $p/r=t$ would give
$p=rt<p$. For $t=r$ the first case contributes the two points
$\pm(r,p/r)$, and the second would require $r^2=p/m\in\{1,p\}$, which is
excluded. Hence the coefficient of $q^h\bar q^{\bar h}$ in
\eqref{eq:cls-theta-mixture} is exactly $2c_r$. Since the spin $p$
exceeds one, no vacuum monomial contributes at these weights, so
$2c_r=d_{h,\bar h}\in\NN$.

If $1\in\mathcal R$, choose a prime $p$ larger than every radius in
$\mathcal R$ and consider $(u,v)=(p,p)\in\Lambda_1$, of weights
$(p^2,0)$. A point $(mt,n/t)$ with these weights is $\pm(p,p)$, so
$mn=p^2$ and $t=p/m\in\{p,1,1/p\}$; only $t=1$ lies in $\mathcal R$, and
it contributes two points. Thus $2c_1=d_{p^2,0}\in\NN$, because
$(p^2,0)\neq(1,0)$.

Finally the coefficient of $q^0\bar q^0$ is one in $H_0$ and in every
$\Theta_r$, so $\sum_rc_r=1$.
\end{proof}

\subsection{The classification}

For $r\geq1$ put
\begin{equation}\label{eq:cls-m-r}
 m_r=\one_{\{2r\in\ZZ\}}+\one_{\{r\in\{1,2\}\}}\in\{0,1,2\}.
\end{equation}

\begin{theorem}[Classification of the limits]\label{thm:classification}
Let $0<g\leq\frac12$.
\begin{enumerate}[(i)]
\item Every boundary partition function of gap $g$, and in particular
 every limit in Theorem~\ref{thm:compactness}, has the form
\begin{equation}\label{eq:cls-two-circle}
 Z_0=\tfrac12(Z_R+Z_t),\qquad 1\leq R,t\leq(2g)^{-1/2},\qquad
 2R\in\ZZ\ \text{ or }\ 2t\in\ZZ ,
\end{equation}
 where $R=t$ is allowed. Equivalently $\rho_0=\sigma(R,t)\in\mathcal B_g$.
\item Conversely, for every pair $(R,t)$ as in \eqref{eq:cls-two-circle}
 the measure $\sigma(R,t)$ is positive, each of its weights
 $(h,\bar h)$ has a nonnegative integer multiplicity, all its states
 have integer spin and dimension at least $1/(2\max(R,t)^2)\geq g$, and
 $\frac12(Z_R+Z_t)$ is modular invariant.
\end{enumerate}
\end{theorem}
\begin{proof}
(i) By Lemma~\ref{lem:cls-mixture} either one radius occurs with weight
one or two radii occur with weight $\frac12$, so
$H_0=\frac12(\Theta_R+\Theta_t)$ with $R,t\geq1$, possibly equal. The
long part of $H_0$ is
\begin{equation}\label{eq:cls-long-numerator}
 \tfrac12(\Theta_R+\Theta_t)-(1-q)(1-\bar q)
 =\tfrac12(\Theta_R+\Theta_t)-1+q+\bar q-q\bar q ,
\end{equation}
and all its coefficients must be nonnegative. Only the coefficient at
$(h,\bar h)=(1,1)$ can fail to be. The points of $\Lambda_r$ with
weights $(1,1)$ are $(u,v)=(\pm2,0)$ and $(0,\pm2)$, which lie in
$\Lambda_r$ when $2/r\in\ZZ$, respectively $2r\in\ZZ$. For $r\geq1$ the
first condition holds only for $r\in\{1,2\}$, so the coefficient of
$q\bar q$ in $\Theta_r$ is $2m_r$, and the requirement is
\[
 m_R+m_t\geq1 ,
\]
that is, $2R\in\ZZ$ or $2t\in\ZZ$. The lowest nonzero dimension in
$\Theta_r$, $r\geq1$, is $1/(2r^2)$, attained at $(u,v)=(0,\pm1/r)$. The
corresponding scalar monomial has weights different from $(1,1)$, so it
survives in \eqref{eq:cls-long-numerator}, and the gap condition gives
$\max(R,t)\leq(2g)^{-1/2}$.

(ii) Every coefficient of $\Theta_r$ other than the constant term is
even, because $\lambda\mapsto-\lambda$ acts freely on
$\Lambda_r\setminus\{0\}$ and preserves the weights. Hence
$\frac12(\Theta_R+\Theta_t)$ has integer coefficients and constant term
one. In \eqref{eq:cls-long-numerator} the constant term cancels, the
coefficients at $(1,0)$ and $(0,1)$ are increased by one, and the
coefficient at $(1,1)$ is $m_R+m_t-1\geq0$. Thus all coefficients are
nonnegative integers. The remaining assertions follow from the
properties of $\Theta_r$ already used.
\end{proof}

\begin{remark}\label{rem:cls-averages}
The average of two different circle partition functions, for example
$\frac12(Z_1+Z_{\sqrt2})$ with gap $\frac14$, is not the partition
function of a conformal field theory: it is excluded in
\cite[Section~4.1]{EL} by an argument that uses current Ward identities
and nondegeneracy of two-point functions. Neither is available for a
limit of partition functions, so these averages must be retained here.
\end{remark}

We collect the elementary properties of the measures $\sigma(R,t)$ that
are used repeatedly.

\begin{lemma}\label{lem:boundary-clusters}
Let $\sigma(R,t)\in\mathcal B_g$ and $\mu_0=\frac12\rhoF_R+\frac12\rhoF_t$.
\begin{enumerate}[(i)]
\item $\mathbb S\mu_0=\mu_0$, $\mathcal C\mu_0=\mu_0$, and
 $\supp\mu_0\subset\Lambda_R\cup\Lambda_{1/R}\cup\Lambda_t\cup\Lambda_{1/t}$.
\item $\nu_{\vac}$ is supported on the nine points with
 $P,Q\in\{0,\pm1\}$, which lie in $\Lambda_1$; in light-cone coordinates
 they are $(0,0)$, $(\pm1,\pm1)$, $(\pm2,0)$ and $(0,\pm2)$.
\item Every cluster of $\sigma(R,t)$ consists of at most four states.
\item The scalar cluster of $\sigma(R,t)$ at dimension two consists of
 $m_R+m_t-1$ states, with $m_r$ as in \eqref{eq:cls-m-r}; the Fock
 measure $\mu_0$ has $m_R+m_t$ unit states there.
\item No state of $\sigma(R,t)$ has dimension below $1/(2\max(R,t)^2)$,
 and for every $E$ the number of its states of dimension at most $E$ is
 at most $A_g\max(E,1)$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i) follows from \eqref{eq:circle-invariance} and (ii) from
\eqref{eq:physical-measures}. For (iii), the coefficient of a monomial
in $\Theta_r$ is the number of points of $\Lambda_r$ among the at most
four points $\pm(u,v)$, $\pm(v,u)$, hence at most four, and so is the
coefficient in the average; the vacuum subtraction increases a
coefficient only at $(1,0)$ and $(0,1)$, where the coefficient of
$\Theta_r$ is $2\cdot\one_{\{r=1\}}$ and the number of states is at most
three. (iv) was shown in the proof of
Theorem~\ref{thm:classification}, and (v) there and in
Lemma~\ref{lem:thermal}.
\end{proof}

\subsection{A first consequence: emergent near-currents}

Although it is not needed in the sequel, the classification already
shows that an admissible partition function with $c$ close to one must
contain spin-one states close to the unitarity bound. For $r\geq1$ put
$D_r=\frac12(r^2+r^{-2})$ and let $B_g=g+\frac1{4g}=D_{(2g)^{-1/2}}$.

\begin{corollary}\label{cor:cls-near-currents}
Fix $0<g\leq\frac12$, $b>B_g$ and $\delta>0$. For all sufficiently
small $a>0$, every admissible partition function of gap $g$ at
$c=1+12a$ has exactly three pairs of states of spin $\pm1$ with
$\Delta\leq b$, and at least one of them has $1<\Delta<1+\delta$.
\end{corollary}
\begin{proof}
In $\Theta_r$ the monomials of spin $+1$ come from the two points
$(u,v)=\pm(r,1/r)$ of $\Lambda_r$, at dimension $D_r$; with weight
$\frac12$ they contribute one state to $\sigma(R,t)$, and the term $q$
of \eqref{eq:cls-long-numerator} contributes one state at dimension one.
Spin $-1$ is symmetric. Hence the spin-one states of
$\sigma(R,t)\in\mathcal B_g$ have dimensions $1$, $D_R$, $D_t$, all in
$[1,B_g]$. If the statement failed along a sequence $a_n\downarrow0$,
Theorems~\ref{thm:compactness} and~\ref{thm:classification} and
Corollary~\ref{cor:cluster-stabilization}, applied to
$U=\{|X|^2<b'\}$ and $U=\{|X|^2<1+\delta'\}$ with $b'\geq b$ and
$0<\delta'\leq\delta$ chosen so that $b'$ and $1+\delta'$ are not
dimensions of $\rho_0$, would give a contradiction. The inequality $\Delta>1$ is part of
Definition~\ref{def:admissible}.
\end{proof}