\section{Setting and conventions}
\label{sec:setup}

Throughout, $\tau=x+iy$ lies in the upper half-plane $\HH$,
$q=e^{2\pi i\tau}$, $\bar q=e^{-2\pi i\bar\tau}$, and $\eta$ is the
Dedekind eta function. We write
\[
 c=1+12a,\qquad a\geq0 ,
\]
so that $a>0$ is the regime of interest and $a=0$ is the boundary
$c=1$. For small $a>0$ we put $L=\log(1/a)$. Constants called $C$, $c$,
$\kappa$ may change from line to line; their dependence is indicated when
it matters.

\subsection{The admissible class}

For $c>1$ the non-degenerate Virasoro characters and the vacuum
character are
\[
 \chi_h(\tau)=\frac{q^{h-a/2}}{\eta(\tau)}\quad(h>0),\qquad
 \chi_{\vac}(\tau)=\frac{q^{-a/2}(1-q)}{\eta(\tau)} .
\]

\begin{definition}[Admissible partition functions]\label{def:admissible}
Let $0<g\leq\frac12$ and $a>0$. An \emph{admissible partition function
of gap $g$ at $c=1+12a$} is a function on $\HH$ of the form
\begin{equation}\label{eq:admissible-Z}
 Z(\tau)=|\chi_{\vac}(\tau)|^2
 +\sum_{j\in\mathcal J}\chi_{h_j}(\tau)\,\overline{\chi_{\bar h_j}(\tau)},
\end{equation}
where $\mathcal J$ is a countable index set and the following hold.
\begin{enumerate}[(i)]
\item \emph{No currents, integral spin, gap.} For every $j$,
 $h_j>0$, $\bar h_j>0$, the spin $s_j=h_j-\bar h_j$ is an integer, and
 the dimension $\Delta_j=h_j+\bar h_j$ satisfies $\Delta_j\geq g$.
\item \emph{Convergence.} $\sum_j e^{-2\pi y\Delta_j}<\infty$ for every
 $y>0$.
\item \emph{Modular invariance.} $Z(-1/\tau)=Z(\tau)=Z(\tau+1)$ for all
 $\tau\in\HH$.
\item \emph{Parity.} The family $(h_j,\bar h_j)_{j\in\mathcal J}$ is
 invariant, with multiplicity, under $(h,\bar h)\mapsto(\bar h,h)$.
\end{enumerate}
We call the indices $j$ the (nonvacuum) \emph{states}, and
$(\Delta_j,s_j)_{j\in\mathcal J}$ the \emph{spectrum}.
\end{definition}

An index is repeated according to multiplicity, so the multiplicity of
each weight $(h,\bar h)$ is a positive integer; by (ii) it is finite, and
only finitely many states have dimension below any given bound. The
vacuum term is the unique shortened one and has coefficient one. The
inequalities $h_j,\bar h_j>0$ say that $\Delta_j>|s_j|$: there is no
nonvacuum state with $h=0$ or $\bar h=0$. Nothing is assumed about the
size of $\Delta_j-|s_j|$. The hypotheses concern the torus partition
function only; no operator product data enter.

\begin{remark}\label{rem:gap-range}
The restriction $g\leq\frac12$ is harmless: a partition function whose
states all have $\Delta_j\geq g>\frac12$ is admissible of gap $\frac12$.
\end{remark}

Put
\begin{equation}\label{eq:reduced-trace}
 H_a(\tau)=(1-q)(1-\bar q)+\sum_{j\in\mathcal J}q^{h_j}\bar q^{\bar h_j},
 \qquad
 \widehat Z_a(\tau)=\sqrt y\,|\eta(\tau)|^2Z(\tau)
 =\sqrt y\,e^{2\pi ay}H_a(\tau).
\end{equation}
Since $\sqrt y\,|\eta|^2$ is invariant under the modular group,
$\widehat Z_a(-1/\tau)=\widehat Z_a(\tau)=\widehat Z_a(\tau+1)$.
At $a=0$ the same formulas are used for the limiting objects of
Theorem~\ref{thm:compactness}, where $h_j,\bar h_j\geq0$ is allowed.

\subsection{Momentum variables and the spectral measures}

We use real \emph{momentum} coordinates $X=(P,Q)\in\RR^2$, related to the
weights by $h=P^2$, $\bar h=Q^2$, and the \emph{light-cone} coordinates
\[
 u=P+Q,\qquad v=P-Q,\qquad
 \Delta=P^2+Q^2=\tfrac12(u^2+v^2),\qquad s=P^2-Q^2=uv .
\]
A function or distribution on $\RR^2$ is called \emph{even} if it is
invariant under each of the reflections $P\mapsto-P$ and $Q\mapsto-Q$
separately. For $h\geq0$ let $A_h$ be the even probability measure on
$\RR$ carried by $\{\pm\sqrt h\}$,
\[
 A_h=\tfrac12\bigl(\delta_{\sqrt h}+\delta_{-\sqrt h}\bigr)\quad(h>0),
 \qquad A_0=\delta_0 .
\]
A state of weights $(h,\bar h)$ is represented by $A_h\otimes A_{\bar h}$.
For an admissible partition function we define
\begin{equation}\label{eq:physical-measures}
 \rho_a=\sum_{j\in\mathcal J}A_{h_j}\otimes A_{\bar h_j},\qquad
 \nu_{\vac}=\sum_{k,l\in\{0,1\}}(-1)^{k+l}A_k\otimes A_l,\qquad
 \mu_a=\nu_{\vac}+\rho_a .
\end{equation}
Thus $\rho_a$ is a positive even measure with $|X|^2=\Delta$ on its
support, each state carries total mass one, and atoms of $\rho_a$ have
masses in $\frac14\ZZ_{>0}$. We call $\rho_a$ the \emph{long measure} and
$\mu_a$ the \emph{Fock measure} of the partition function; both are built
from the physical weights $(h_j,\bar h_j)$ and do not involve $a$
otherwise. The \emph{counting function} and the thermal sum are
\[
 N_a(E)=\rho_a\bigl(\{|X|^2\leq E\}\bigr)=\#\{j:\Delta_j\leq E\},
 \qquad
 P_a(y)=\int e^{-2\pi y|X|^2}\dd\rho_a(X)=\sum_je^{-2\pi y\Delta_j}.
\]
With the Gaussians
\begin{equation}\label{eq:G-tau}
 G_\tau(P,Q)=\sqrt y\,e^{2\pi i\tau P^2-2\pi i\bar\tau Q^2},
\end{equation}
equation \eqref{eq:reduced-trace} reads
$\widehat Z_a(\tau)=e^{2\pi ay}\langle\mu_a,G_\tau\rangle$.

\subsection{The Lorentzian Fourier transform and circle lattices}

Let
\[
 B(X,X')=2(PP'-QQ')=uv'+vu'
\]
be the Lorentzian pairing, and define for Schwartz functions
\begin{equation}\label{eq:S-transform}
 (\mathbb Sf)(X)=2\int_{\RR^2}f(X')\,e^{2\pi iB(X,X')}\dd P'\dd Q'
 =\int_{\RR^2}f(X')\,e^{2\pi i(uv'+vu')}\dd u'\dd v' .
\end{equation}
In light-cone coordinates $\mathbb S$ is the unitary Fourier transform
followed by the exchange of the two variables; it preserves evenness, and
$\mathbb S^2f(X)=f(-X)$. It extends to tempered distributions by
duality. The \emph{phase operator} is multiplication by a unimodular
function,
\[
 \mathcal CT=e^{2\pi i(P^2-Q^2)}\,T ,
\]
and the \emph{boost generator} is the vector field
\begin{equation}\label{eq:boost}
 \mathcal K=P\partial_Q+Q\partial_P=u\partial_u-v\partial_v .
\end{equation}
A direct computation gives
\begin{equation}\label{eq:G-modular}
 \mathbb SG_\tau=G_{-1/\tau},\qquad \mathcal CG_\tau=G_{\tau+1},
\end{equation}
and, for $b>0$, $z\in\CC^2$ and $J=\operatorname{diag}(1,-1)$,
\begin{equation}\label{eq:S-gaussian}
 \phi_{b,z}(X)=e^{-b|X|^2+z\cdot X},\qquad
 \mathbb S\phi_{b,z}(X)=\frac{2\pi}b
 \exp\!\Bigl(\frac{(z+4\pi iJX)\cdot(z+4\pi iJX)}{4b}\Bigr).
\end{equation}
Here and below, in formulas for entire functions $|X|^2$ denotes the
polynomial $P^2+Q^2$ and $z\cdot X$ the bilinear product, without complex
conjugation. In particular $\mathbb S\phi_{b,z}$ is a constant times
$\phi_{4\pi^2/b,\,2\pi iJz/b}$.

For $r>0$ the \emph{circle lattice} of radius $r$ is
\begin{equation}\label{eq:circle-lattice}
 \Lambda_r=\Bigl\{(P,Q)=\Bigl(\frac{mr+n/r}2,\frac{mr-n/r}2\Bigr):
 m,n\in\ZZ\Bigr\}
 =\bigl\{(u,v)=(mr,\,n/r):m,n\in\ZZ\bigr\}.
\end{equation}
The reflection $Q\mapsto-Q$ exchanges $u$ and $v$ and maps $\Lambda_r$
onto $\Lambda_{1/r}$. On $\Lambda_r$ one has $B(\lambda,\lambda')=mn'+nm'$
and $s=mn$, so $\Lambda_r$ is an even unimodular lattice for $B$. Put
\begin{equation}\label{eq:circle-measures}
\begin{gathered}
 e_r=\sum_{\lambda\in\Lambda_r}\delta_\lambda,\qquad
 \rhoF_r=\tfrac12(e_r+e_{1/r}),\\
 \Theta_r(\tau)=\sum_{m,n\in\ZZ}
 q^{(mr+n/r)^2/4}\,\bar q^{(mr-n/r)^2/4},\qquad
 Z_r=\frac{\Theta_r}{|\eta|^2}.
\end{gathered}
\end{equation}
Poisson summation and integrality of the spin give
\begin{equation}\label{eq:circle-invariance}
 \mathbb Se_r=e_r,\qquad\mathcal Ce_r=e_r ,
\end{equation}
and $\rhoF_r$ is the even measure with
$\langle\rhoF_r,G_\tau\rangle=\sqrt y\,\Theta_r(\tau)$. The function
$Z_r=Z_{1/r}$ is the partition function of the compact boson at radius
$r$, with $r=1$ the self-dual radius; $\rhoF_r=\rhoF_{1/r}$ is its Fock
measure. A point of $\Lambda_r$ with charges $(m,n)$ has dimension
$\frac12(m^2r^2+n^2/r^2)$ and spin $mn$.

We call a radius $r$ a \emph{half-integer} if $2r\in\ZZ$; integers are
included. For radii $R,t\geq1$ we write
\begin{equation}\label{eq:sigma-R-t}
 \sigma(R,t)=\tfrac12\rhoF_R+\tfrac12\rhoF_t-\nu_{\vac}
\end{equation}
for the long measure of the average $\frac12(Z_R+Z_t)$, and
$\mathcal B_g$ for the set of all $\sigma(R,t)$ with
$1\leq R,t\leq(2g)^{-1/2}$ and at least one of $R,t$ a half-integer.
Theorem~\ref{thm:classification} shows that these are positive measures
and are exactly the possible limits.

\subsection{Analytic evaluations and the Gaussian identity}
\label{subsec:analytic-evaluation}

At $a>0$ the character $\chi_h$ is a Gaussian in the momentum
$\sqrt{h-a/2}$, which is imaginary for the vacuum and possibly for states
with $h<a/2$. We therefore evaluate entire test functions at shifted
squared momenta instead of introducing a measure on real momentum space.

For an entire function $f$ on $\CC^2$ let
$f^{\rm ev}(P,Q)=\frac14\sum_{\pm,\pm}f(\pm P,\pm Q)$ be its even part.
It is an entire function of $(P^2,Q^2)$: there is a unique entire
$\ev f$ on $\CC^2$ with
\begin{equation}\label{eq:natural}
 f^{\rm ev}(P,Q)=\ev f(P^2,Q^2).
\end{equation}
For $h,\bar h\geq0$, $\ev f(h,\bar h)=\langle A_h\otimes A_{\bar h},f\rangle$.
Given an admissible partition function at $c=1+12a$, define
\begin{equation}\label{eq:analytic-evaluation}
 \mathcal L_a(f)=\sum_{k,l\in\{0,1\}}(-1)^{k+l}
 \ev f\bigl(k-\tfrac a2,\,l-\tfrac a2\bigr)
 +\sum_{j\in\mathcal J}\ev f\bigl(h_j-\tfrac a2,\,\bar h_j-\tfrac a2\bigr)
\end{equation}
whenever the sum over $\mathcal J$ converges absolutely. The first sum
is the \emph{shifted vacuum}; the terms of the second are the
\emph{shifted} (or analytic) evaluations of the states, as opposed to
their \emph{physical} evaluations $\ev f(h_j,\bar h_j)$, whose sum with
the unshifted vacuum terms is $\langle\mu_a,f\rangle$. By construction
\begin{equation}\label{eq:L-a-of-G}
 \mathcal L_a(G_\tau)=\widehat Z_a(\tau),\qquad\tau\in\HH .
\end{equation}
The evaluation points of a state are $(\pm P,\pm Q)$ with
$P^2=h-\frac a2$, $Q^2=\bar h-\frac a2$. Hence $P^2+Q^2=\Delta-a$ is
real, $P^2-Q^2=s$ is the spin, at most one of $P,Q$ is imaginary when
$a<g$, and
\begin{equation}\label{eq:strip}
 |\operatorname{Im}P|,\ |\operatorname{Im}Q|\leq\sqrt{a/2},\qquad
 |\operatorname{Re}X|^2\geq\Delta-a .
\end{equation}

\begin{definition}[Gaussian mixtures]\label{def:gaussian-mixture}
For $b>0$ let $\mathcal G_b$ be the space of finite sums of functions
\[
 f(X)=\int_\Omega p(X,\omega)\,e^{-b|X|^2+z(\omega)\cdot X}\dd m(\omega),
\]
where $\Omega$ is a compact metric space, $m$ a finite complex Borel
measure on it, $z:\Omega\to\CC^2$ is continuous, and $p$ is a polynomial
in $X$ whose coefficients are continuous functions of $\omega$.
\end{definition}

Every $f\in\mathcal G_b$ is entire, and so are all its derivatives, with
$\partial^\alpha f\in\mathcal G_b$. Products of an element of
$\mathcal G_b$ with polynomials, with exponentials $e^{w\cdot X}$,
$w\in\CC^2$, and with functions such as
$\sinc(\ell(X))=\frac12\int_{-1}^1e^{i\sigma\ell(X)}\dd\sigma$ for a
real linear form $\ell$, remain in $\mathcal G_b$; so do real translates.
By \eqref{eq:S-gaussian}, $\mathbb S$ maps $\mathcal G_b$ onto
$\mathcal G_{4\pi^2/b}$. For every $0<b'<b$, every multi-index $\alpha$
and every $\varrho<\infty$ there is $C$ with
\begin{equation}\label{eq:mixture-envelope}
 |\partial^\alpha f(X)|\leq C\,e^{-b'|\operatorname{Re}X|^2}
 \qquad\text{for }|\operatorname{Im}X|\leq\varrho .
\end{equation}

\begin{lemma}[The exact Gaussian identity]\label{lem:gaussian-identity}
Let $Z$ be admissible at $c=1+12a$; or let $a=0$ and let
$Z_0=H_0/|\eta|^2$ be modular invariant with $H_0$ of the form
\eqref{eq:reduced-trace}, $h_j,\bar h_j\geq0$, convergent at every
$y>0$. For every $b>0$ and every
$f\in\mathcal G_b$ the sums defining $\mathcal L_a(f)$ and
$\mathcal L_a(\mathbb Sf)$ converge absolutely, and
\begin{equation}\label{eq:gaussian-identity}
 \mathcal L_a(f)=\mathcal L_a(\mathbb Sf).
\end{equation}
\end{lemma}
\begin{proof}
By \eqref{eq:strip} and \eqref{eq:mixture-envelope}, each term of the sum
over states is bounded by $Ce^{-b'(\Delta_j-a)}$, which is summable by
hypothesis (ii) of Definition~\ref{def:admissible}. The same bound holds
uniformly for $\phi_{b,z}$ with $(b,z)$ in a compact subset of
$\{\operatorname{Re}b>0\}\times\CC^2$, and for its derivatives in the
parameters. Consequently
$F(b,z)=\mathcal L_a(\phi_{b,z})-\mathcal L_a(\mathbb S\phi_{b,z})$ is
holomorphic on $\{\operatorname{Re}b>0\}\times\CC^2$, and one may
differentiate under $\mathcal L_a$.

First let $b=2\pi$. By \eqref{eq:G-modular}, \eqref{eq:L-a-of-G} and
modular invariance, $\mathcal L_a(G_\tau)=\mathcal L_a(\mathbb SG_\tau)$
for all $\tau$. The two sides are $\widehat Z_a(\tau)$ and
$\widehat Z_a(-1/\tau)$, whose defining series may be differentiated
termwise in $x$ and $y$, by hypothesis (ii) and $|s_j|\leq\Delta_j$; and
$\mathbb S$ commutes with $\partial_x$, $\partial_y$ applied to $G_\tau$.
Differentiating in $x$ and $y$ at $\tau=i$ produces, on
the left, $\mathcal L_a$ of polynomials in $P^2-Q^2$ and $P^2+Q^2$ times
$e^{-2\pi|X|^2}$; the factor $\sqrt y$ contributes only terms of lower
degree. By induction on the degree, the identity
$\mathcal L_a(\psi)=\mathcal L_a(\mathbb S\psi)$ holds for every
$\psi=P^{2i}Q^{2k}e^{-2\pi|X|^2}$. It holds trivially for monomials that
are odd in $P$ or in $Q$, because $\mathcal L_a$ annihilates functions
with vanishing even part and $\mathbb S$ preserves the parity in each
variable. These functions are the Taylor coefficients of
$z\mapsto\phi_{2\pi,z}$ at $z=0$, so the entire function $F(2\pi,\cdot)$
vanishes identically.

Next, $\partial_b\phi_{b,z}=-(\partial_{z_1}^2+\partial_{z_2}^2)\phi_{b,z}$,
and the same relation holds after applying $\mathbb S$. Therefore every
$b$-derivative of $F$ at $b=2\pi$ is a $z$-derivative of
$F(2\pi,\cdot)$ and vanishes, and $F\equiv0$ by the identity theorem.
Differentiating in $z$ and integrating over $\Omega$, which is justified
by the uniform bounds, gives \eqref{eq:gaussian-identity} for
$f\in\mathcal G_b$.
\end{proof}

The lemma never takes a Fourier transform of a distribution supported at
complex points. It applies to test functions that are not even: both
sides see only the even part.

\subsection{Derivatives in the squared momenta}

Taylor expansions of an evaluation $\ev f(h,\bar h)$ in the weights are
controlled by the following formula, which is regular at $h=0$.

\begin{lemma}\label{lem:sq-momentum-derivative}
Let $\phi$ be an entire function of one variable and let
$\varphi(h)=\frac12\bigl(\phi(\sqrt h)+\phi(-\sqrt h)\bigr)$, an entire
function of $h$. For every integer $k\geq1$ and every $h\in\CC$,
\begin{equation}\label{eq:sq-momentum-derivative}
 \varphi^{(k)}(h)=\frac1{4^k(k-1)!}
 \int_{-1}^1(1-t^2)^{k-1}\,\phi^{(2k)}(t\sqrt h)\dd t .
\end{equation}
For $h>0$, $\varphi^{(k)}(h)$ is also a linear combination of
$h^{-(2k-l)/2}\bigl(\phi^{(l)}(\sqrt h)+(-1)^l\phi^{(l)}(-\sqrt h)\bigr)$,
$1\leq l\leq k$, with coefficients depending only on $k$ and $l$.
\end{lemma}
\begin{proof}
For $k=1$ and $h>0$ the right side of \eqref{eq:sq-momentum-derivative}
is $\frac1{4\sqrt h}\bigl(\phi'(\sqrt h)-\phi'(-\sqrt h)\bigr)
=\varphi'(h)$. If the formula holds for $k$, differentiate it in $h$,
use $\partial_h\phi^{(2k)}(t\sqrt h)=\frac t{2\sqrt h}\,
\phi^{(2k+1)}(t\sqrt h)$ and
$t(1-t^2)^{k-1}=-\frac1{2k}\frac{d}{dt}(1-t^2)^k$, and integrate by
parts once in $t$; the boundary terms vanish and one obtains the formula
for $k+1$, the constant changing by the factor $1/(4k)$. Both sides are
entire in $h$
(the integrand is even in $t$), so the identity extends from $h>0$ to
$\CC$. The second statement follows by differentiating
$\phi(\pm\sqrt h)$ directly.
\end{proof}

\begin{lemma}[Uniform bounds for weight derivatives]
\label{lem:weight-derivative-bound}
Let $f$ be entire on $\CC^2$, let $k\geq0$ be an integer, and suppose
that for some $M<\infty$ and $\beta>0$
\[
 |\partial^\alpha f(X)|\leq M\,e^{-\beta|\operatorname{Re}X|^2}
 \qquad\text{for }|\alpha|\leq2k,\ \ |\operatorname{Im}P|,|\operatorname{Im}Q|\leq1 .
\]
Then for all real $h,\bar h\geq-1$ and all $i+l\leq k$,
\begin{equation}\label{eq:weight-derivative-bound}
 \bigl|\partial_h^i\partial_{\bar h}^l\,\ev f(h,\bar h)\bigr|
 \leq C_k\,M\,e^{4\beta}\,e^{-\beta(h+\bar h)},
\end{equation}
where $C_k$ depends only on $k$.
\end{lemma}
\begin{proof}
Treat the two variables separately. In a variable whose weight is at
most $2$, use \eqref{eq:sq-momentum-derivative} when a derivative falls
on it and the definition otherwise: the function is evaluated at momenta
$t\sqrt h$, $|t|\leq1$, whose imaginary parts are at most one and whose
real parts are bounded, and $1\leq e^{2\beta}e^{-\beta h}$. In a variable
whose weight exceeds $2$, use the endpoint form of
Lemma~\ref{lem:sq-momentum-derivative}: the momentum is $\pm\sqrt h$,
real, the coefficients $h^{-(2i-l)/2}$ are bounded, and the Gaussian
factor is $e^{-\beta h}$. The integral formula in one variable commutes
with either operation in the other.
\end{proof}

We shall use Lemma~\ref{lem:weight-derivative-bound} in two ways. First,
moving a state of weights $(h,\bar h)$ along its fixed-spin curve changes
both weights at the same rate, and for
$\Phi(d)=\ev f\bigl(h+\frac d2,\bar h+\frac d2\bigr)$ one has
$|\Phi^{(k)}(d)|\leq C_kMe^{4\beta}e^{-\beta(\Delta+d)}$. Secondly,
taking $k=1$ and integrating over the central shift gives the following.

\begin{corollary}[Central-shift replacement]\label{cor:central-shift}
Under the hypotheses of Lemma~\ref{lem:weight-derivative-bound} with
$k=1$, for $0\leq a\leq1$ and all $h,\bar h\geq0$,
\[
 \bigl|\ev f\bigl(h-\tfrac a2,\bar h-\tfrac a2\bigr)-\ev f(h,\bar h)\bigr|
 \leq C\,a\,M\,e^{5\beta}\,e^{-\beta(h+\bar h)} .
\]
\end{corollary}

\subsection{Thermal bounds}

\begin{lemma}[Thermal cap, counting bound, tails]\label{lem:thermal}
Let $0<g\leq\frac12$, put $C_g=2e^{\pi/2}g^{-2}$ and
$A_g=e^{2\pi}(1+C_g)$, and let $Z$ be admissible of gap $g$ at
$c=1+12a$ with $0<a\leq g^2/4$. Then
\begin{align}
 P_a(g^2)&\leq C_g,\label{eq:thermal-cap}\\
 P_a(y)&\leq(1+C_g)\,y^{-1}e^{2\pi a/y}\qquad(0<y\leq1),
 \label{eq:thermal-high}\\
 N_a(E)&\leq A_g\,E\,e^{2\pi aE}\qquad(E\geq1),\label{eq:count-bound}
\end{align}
and, for $\beta>2\pi a$ and $E_0\geq1$,
\begin{equation}\label{eq:thermal-tail}
 \sum_{\Delta_j>E_0}e^{-\beta\Delta_j}
 \leq A_g\,\beta\Bigl(\frac{E_0}{\beta-2\pi a}
 +\frac1{(\beta-2\pi a)^2}\Bigr)e^{-(\beta-2\pi a)E_0}.
\end{equation}
The same bounds hold with $a=0$ for every modular-invariant function
$Z_0=H_0/|\eta|^2$ in which $H_0$ has the form
\eqref{eq:reduced-trace} with $h_j,\bar h_j\geq0$, integer spins,
$\Delta_j\geq g$, and convergence at every $y>0$. This includes the
limits of Theorem~\ref{thm:compactness} and, by
Theorem~\ref{thm:classification}, every element of $\mathcal B_g$.
\end{lemma}
\begin{proof}
Let $V(y)=(1-e^{-2\pi y})^2$. On the imaginary axis
\eqref{eq:reduced-trace} is
$\widehat Z_a(iy)=\sqrt y\,e^{2\pi ay}\bigl(V(y)+P_a(y)\bigr)$, and
invariance under $\tau\mapsto-1/\tau$ gives
\begin{equation}\label{eq:thermal-S}
 V(1/y)+P_a(1/y)=y\,e^{2\pi a(y-1/y)}\bigl(V(y)+P_a(y)\bigr),
 \qquad y>0 .
\end{equation}
Take $y=Y=g^{-2}\geq4$ and put $C_Y=Ye^{2\pi a(Y-1/Y)}$. Then
\[
 P_a(1/Y)-C_YP_a(Y)=C_YV(Y)-V(1/Y)\leq C_Y ,
\]
and the contribution of a state to the left side is
$e^{-2\pi\Delta/Y}\bigl(1-Ye^{-2\pi(Y-1/Y)(\Delta-a)}\bigr)$. Since
$\Delta\geq g$, $a\leq g/2$ and $Y-1/Y\geq3/(4g^2)$, the subtracted
term is at most $g^{-2}e^{-3\pi/(4g)}\leq z^2e^{-2z}$ with
$z=1/g\geq2$, hence at most $4e^{-4}<\frac12$. Therefore
$\frac12P_a(g^2)\leq C_Y\leq g^{-2}e^{\pi/2}$, which is
\eqref{eq:thermal-cap}.

For $0<y\leq1$, \eqref{eq:thermal-S} with $y$ replaced by $1/y$,
monotonicity of $P_a$, $0\leq V\leq1$ and $P_a(1)\leq P_a(g^2)$ give
\eqref{eq:thermal-high}. Then $N_a(E)\leq e^{2\pi yE}P_a(y)$ with
$y=1/E$ gives \eqref{eq:count-bound}. Finally, integrating by parts
against $e^{-\beta E}$ and discarding the nonpositive boundary term at
$E_0$,
\[
 \sum_{\Delta_j>E_0}e^{-\beta\Delta_j}
 \leq\beta\int_{E_0}^\infty N_a(E)e^{-\beta E}\dd E
 \leq A_g\beta\int_{E_0}^\infty E\,e^{-(\beta-2\pi a)E}\dd E ,
\]
which is \eqref{eq:thermal-tail}. For $a=0$ the argument is unchanged.
\end{proof}

In particular, for $0<\beta\leq1$ and $a\leq\min\{g^2/4,\beta/(4\pi)\}$,
\begin{equation}\label{eq:thermal-moment}
 \sum_je^{-\beta\Delta_j}\leq C_g'\,\beta^{-1},
\end{equation}
by \eqref{eq:thermal-high} with $y=\beta/(2\pi)$, and
$N_a(ML)\leq A_gMLe^{2\pi aML}=O_M(L)$ for every fixed $M$: a
logarithmic energy window contains $O(L)$ states.

\subsection{Compactness}

\begin{theorem}[Compactness]\label{thm:compactness}
Fix $0<g\leq\frac12$. Let $Z_n$ be admissible of gap $g$ at
$c_n=1+12a_n$ with $a_n\downarrow0$, with states
$(h^{(n)}_j,\bar h^{(n)}_j)_{j\in\mathcal J_n}$. After passing to a
subsequence there is a countable family
$(h_j,\bar h_j)_{j\in\mathcal J_0}$ with
\[
 h_j,\bar h_j\geq0,\qquad h_j-\bar h_j\in\ZZ,\qquad h_j+\bar h_j\geq g,
 \qquad \sum_{j\in\mathcal J_0}e^{-2\pi y(h_j+\bar h_j)}<\infty\ \ (y>0),
\]
such that:
\begin{enumerate}[(a)]
\item the long measures $\rho_{a_n}$ converge vaguely on $\RR^2$ to
 $\rho_0=\sum_{j\in\mathcal J_0}A_{h_j}\otimes A_{\bar h_j}$;
\item $H_{a_n}\to H_0=(1-q)(1-\bar q)
 +\sum_{j\in\mathcal J_0}q^{h_j}\bar q^{\bar h_j}$ locally uniformly on
 $\HH$, together with all derivatives in $x$ and $y$;
\item $Z_0=H_0/|\eta|^2$ satisfies $Z_0(-1/\tau)=Z_0(\tau)=Z_0(\tau+1)$,
 and the family $\mathcal J_0$ is parity invariant.
\end{enumerate}
\end{theorem}

The limit may contain states with $h_j=0$ or $\bar h_j=0$; these are
terms $q^{h}\bar q^{\bar h}/|\eta|^2$ and not shortened characters. The
vacuum term of $H_0$ is still $(1-q)(1-\bar q)$ with coefficient one.

\begin{proof}
By \eqref{eq:thermal-cap}, the number of states of $Z_n$ with
$\Delta\leq E$ is at most $C_ge^{2\pi g^2E}$, uniformly in $n$ once
$a_n\leq g^2/4$, and their spins are integers with $|s|\leq E$. A
diagonal argument over $E\in\NN$ yields a subsequence along which, for
every $E$ that is not a limiting dimension, the finitely many states
below $E$ converge together with their spins. The limits form a locally
finite family $\mathcal J_0$ in which coinciding limits are repeated;
the closed conditions $\Delta\geq g$ and $h,\bar h\geq0$ persist, spins
stay integral, and parity invariance is preserved. This is (a).

For (b) we need uniform tails. By \eqref{eq:thermal-cap} and
\eqref{eq:thermal-high}, $P_{a_n}(y)$ is bounded uniformly in $n$ for
each fixed $y>0$. Hence on a compact subset
of $\HH$ with $y\geq y_0$, the contribution to $H_{a_n}$ of states with
$\Delta>E$ is at most $P_{a_n}(y_0/2)\,e^{-\pi y_0E}$, uniformly in $n$;
a derivative of order $k$ in $(x,y)$ costs a factor at most
$(2\pi\Delta)^k$ because $|s|\leq\Delta$, and is absorbed by replacing
$y_0/2$ with $y_0/4$. Together with (a) this gives (b), and the
convergence of the sum for $\mathcal J_0$ at every $y>0$ follows from
Fatou's lemma. The modular identities for
$\sqrt y\,e^{2\pi a_ny}H_{a_n}$ pass to the limit, which is (c). Since
$\Delta\geq g$ on $\mathcal J_0$, no state of the limit has weights
$(0,0)$, so the coefficient of $q^0\bar q^0$ in $H_0$ is one.
\end{proof}

\begin{definition}\label{def:converges-to}
In the situation of Theorem~\ref{thm:compactness} we say that the
sequence \emph{converges to} $\rho_0$ if (a) holds for the sequence
itself. An \emph{admissible sequence of gap
$g$} is a sequence $Z_n$ as in that theorem; we often index its objects
by $a=a_n$ rather than by $n$, writing $\rho_a$, $\mu_a$,
$\mathcal L_a$.
\end{definition}

We shall use vague convergence in the following form. Atoms of all the
measures $\rho_a$ have masses in $\frac14\ZZ_{>0}$, and each state has
total mass one. A \emph{cluster} of $\rho_0$ is the set of its states
with a given dimension and signed spin.

\begin{corollary}[Stabilization of clusters]\label{cor:cluster-stabilization}
Let an admissible sequence converge to $\rho_0$, and let $U\subset\RR^2$
be a bounded open set, invariant under the two reflections, whose
boundary does not meet $\supp\rho_0$. For all sufficiently small $a$ in
the sequence, the number of states of $\rho_a$ with support in $U$
equals that of $\rho_0$, no state of $\rho_a$ meets $\partial U$, and the
dimensions of the states in $U$ converge, with multiplicity and with
their spins, to those of $\rho_0$ in $U$.
\end{corollary}
\begin{proof}
Choose $E>\sup_U|X|^2$ that is not a dimension of $\rho_0$. By vague
convergence and integrality of the masses, for small $a$ the states of
$\rho_a$ with $\Delta<E$ are in bijection with those of $\rho_0$, with
equal spins and converging weights. Since $U$ is invariant under the two
reflections, the support of a state lies in $U$ or is disjoint from it.
A state of $\rho_0$ has its support in the open set $U$ or in the open
set $\RR^2\setminus\overline U$, and so, for small $a$, has the
corresponding state of $\rho_a$. States with $\Delta\geq E$ do not meet
$\overline U$.
\end{proof}

Statements that hold ``for all sufficiently small admissible $a$'' are
always obtained from statements about sequences by contradiction, and
the resulting thresholds are not effective.