% sharp mass bounds for the 2D O(4) model: a self-contained, complete proof.
% written to be read with as little prior background as possible.
% main result: the full transfer gap of the 2D nearest-neighbor O(4) spin
% model satisfies c*sqrt(beta) e^{-pi beta} <= gap(beta) <= C*sqrt(beta) e^{-pi beta}
% for all large beta, where c,C>0 are beta-independent.
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\title{Sharp mass bounds for the two-dimensional $O(4)$ model}
\begin{document}
\maketitle
\begin{abstract}
We prove sharp mass bounds for the two-dimensional nearest-neighbor $O(4)$
model. For every sufficiently large inverse temperature (inverse coupling)
$\beta$, the \emph{full} transfer gap -- meaning the exponential decay rate
of correlations of \emph{every} local observable, not only the spin -- is
sandwiched between two positive, $\beta$-independent multiples of the
asymptotically free scale $\sqrt\beta\,e^{-\pi\beta}$:
\[
c\,\sqrt\beta\,e^{-\pi\beta}
\;\le\; \gap(\beta) \;\le\;
C\,\sqrt\beta\,e^{-\pi\beta},
\qquad \beta\ge\beta_0.
\]
We also prove that the periodic measures have a unique local limit and a
positive full gap at every finite $\beta>0$. Whether the lattice theories
converge to a continuum quantum field theory, and what its exact spectrum
there looks like, is \emph{not} decided here; we state separately the
extra convergence hypotheses that would turn our lattice bounds into a
mass-gap statement for a continuum theory. The proof is the one of the
original technical paper; this version reorganises it and adds plain-English
commentary so the argument can be followed with high-school algebra and
calculus, taking the stated core estimates on faith where necessary.
\end{abstract}
\vspace*{-12pt}
\tableofcontents
\newpage
%---------------------------------------------------------------------
\section{Overview: what is being proved and how}
\label{sec:overview}
%---------------------------------------------------------------------
\subsection*{The model in one paragraph}
Place a spin $\sigma_x\in S^3\subset\R^4$ at every site $x$ of a large
periodic square lattice $\Lambda=(\Z/n\Z)\times(\Z/w\Z)$. Give the spins
the Boltzmann weight
\begin{equation}\label{eq:model-ov}
\dd\mu_{\beta;n,w}(\sigma) = Z_\beta(n,w)^{-1}
\exp\!\Bigl(\beta \sum_{\langle xy\rangle}\sigma_x\cdot\sigma_y\Bigr)
\prod_{x\in\Lambda}\dd\omega(\sigma_x),
\end{equation}
where $\langle xy\rangle$ runs over nearest-neighbor pairs, each counted
once, $\dd\omega$ is the normalised uniform measure on $S^3$, and
$Z_\beta(n,w)$ is the normalisation. This is the two-dimensional $O(4)$
nonlinear sigma model on the lattice with nearest-neighbor couplings and
bare inverse coupling $\beta$.
The energy $\beta\sigma_x\cdot\sigma_y = \tfrac\beta2 - \tfrac\beta2
|\sigma_x-\sigma_y|^2$ favours neighbouring spins pointing in the same
direction, and $\beta$ is how strongly. There is no spin-ordering
transition in two dimensions (Mermin--Wagner), but there is a length
scale beyond which correlations die exponentially. That length scale is
what we want to bracket.
\subsection*{What ``mass gap'' means}
On a cylinder of spatial width $w$ the transfer operator $K_w$ advances
the configuration one time step. Its kernel is
\begin{equation}\label{eq:transfer-kernel-ov}
K_w(s,s')=\exp\!\Bigl( \tfrac{\beta}{2}V_w(s)
+\beta\sum_{i=1}^w s_i\cdot s'_i
+\tfrac{\beta}{2}V_w(s') \Bigr),
\qquad V_w(s)=\sum_i s_i\cdot s_{i+1}.
\end{equation}
$K_w$ is a positive, self-adjoint, compact integral operator. Its largest
eigenvalue $\lambda_1$ is simple and its positive ground eigenvector is
the vacuum. Closing time periodically gives $Z_\beta(n,w)=\Tr K_w^n$.
Fix $\beta$, take $w\to\infty$, and take a subsequential local limit. On
the resulting reflected Hilbert space there is a positive one-step
translation contraction $T_\beta$ with $T_\beta\Omega = \Omega$. The
\emph{full transfer gap} is
\begin{equation}\label{eq:gap-def-ov}
\gap(\beta) = -\log\norm{T_\beta\big|_{\Omega^\perp}}.
\end{equation}
Equivalently, autocorrelations $\langle F, T_\beta^d F\rangle$ of every
centred local observable $F$ decay at least as fast as $e^{-\gap d}$,
where $d$ is the time separation. The gap is the mass; $\gap=0$ means
critical or ordered.
The subtlety: controlling the spin two-point function is not enough.
There is a second, \emph{rotation-invariant} sector of observables -- the
bond energies $\sigma_x\cdot\sigma_y$ -- and $T_\beta$ has spectral
support in that sector as well. The \emph{full} gap includes both. Our
theorem bounds this bigger, smaller gap. That is why the analysis has to
carry along every local observable, not just the spin.
\subsection*{The answer}
\begin{theorem}[Main result]\label{thm:main-ov}
There are finite constants $\beta_0,c,C$, with $0<c<C$, such that for
every $\beta\ge\beta_0$ the periodic measures \eqref{eq:model-ov} have a
unique local limit, and its full transfer gap obeys
\begin{equation}\label{eq:target-ov}
c\,\sqrt\beta\,e^{-\pi\beta}\;\le\;\gap(\beta)\;\le\;
C\,\sqrt\beta\,e^{-\pi\beta}.
\end{equation}
\end{theorem}
Three comments on the formula $\sqrt\beta\,e^{-\pi\beta}$.
\begin{itemize}
\item The exponent is not a guess; it is forced by the renormalisation
group (RG). Asymptotic freedom says the running coupling satisfies
$\beta\mapsto\beta - \gamma - \gamma/(2\pi\beta)+\dots$ with
$\gamma=(\log L)/\pi$, and integrating over many scale factors gives
$\log(\text{inverse length})\sim\pi\beta$. So it is a two-loop
calculation that produces $e^{-\pi\beta}$ with a $\sqrt\beta$
correction.
\item Up to bounded positive factors, $\sqrt\beta\,e^{-\pi\beta}$ is
\emph{the} natural asymptotically free scale for this model in these
units. That is why the bounds are sharp.
\item The bounds are not asymptotic (the ratio need not tend to $1$, and
in fact the exact $O(4)$ prefactor is a known transcendent constant
we do not touch here). They give the correct order of magnitude.
\end{itemize}
\begin{corollary}[Full gap at every finite $\beta$]\label{cor:allT-ov}
For every finite $\beta>0$, the periodic measures on squares have a unique
local limit. Its full transfer gap is positive, with
\[
\gap(\beta)\;\ge\; c\,\exp\!\Bigl(-\pi\beta
- C\sqrt{(1+\beta)\log(2+\beta)}\Bigr)>0
\]
for some absolute constants $c,C>0$.
\end{corollary}
Corollary~\ref{cor:allT-ov} does \emph{not} use the sharp two-loop
comparison; it follows from a preliminary mixing estimate alone. The
gap is therefore positive at every temperature, but the formula is too
weak in the exponent to distinguish $\sqrt\beta\,e^{-\pi\beta}$ from
$\beta^{-1/2}e^{\pi\beta+\sqrt\beta}$. Theorem~\ref{thm:main-ov} repairs
that at large $\beta$.
\subsection*{From $\gap$ to a physical mass}
Choose a physical reference length $\ell_{\mathrm{ref}}>0$ and declare
that $L^N$ lattice spacing (where $L$ is a fixed block scale we will
choose later) has physical length $\ell_{\mathrm{ref}}$. The lattice
spacing is $a_N=\ell_{\mathrm{ref}}L^{-N}$. Then
$\gap(\beta_N)/a_N$ is the mass of the cutoff theory in physical units.
Theorem~\ref{thm:main-ov} upgrades to
\begin{equation}\label{eq:physical-main-ov}
\frac{\log 2}{M_0\ell_{\mathrm{ref}}}
\;\le\; \frac{\gap(\beta_N)}{a_N}
\;\le\; \frac{\log 8}{\ell_{\mathrm{ref}}}
\qquad(N\ \text{large}),
\end{equation}
with $M_0$ a fixed positive integer and $\beta_N$ chosen so that exactly
$N$ RG steps have been done at terminal coupling $H$. Bounded positive
multiplicative factors sandwich the continuum-normalised mass. Whether
the cutoff measures converge to an actual continuum theory is a separate
question; we spell out in Section~\ref{sec:physical} the additional
hypotheses that would let one read \eqref{eq:physical-main-ov} as a
statement about an existing continuum spectrum.
\subsection*{Relation to earlier work, in one paragraph}
Absence of ordinary ordering is classical (Mermin--Wagner, and the
McBryan--Spencer quantitative bounds). Exponential decay of the spin
correlation was proved separately for every finite $\beta$ and every
internal dimension $\ge 3$. Perturbative renormalisation predicts the
scale $\sqrt\beta\,e^{-\pi\beta}$; Bethe ansatz $S$-matrix theory fixes
the exact continuum mass for $O(3)$ and $O(4)$, and for $O(n)$ in
general. None of that gave a full transfer gap (including invariant
observables) for the specified lattice model. Our theorem closes that
gap for $O(4)$ at large $\beta$, and yields positivity at every finite
$\beta$ as a corollary. The proof of the exact continuum mass
(including the ``$\Gamma$-function'' prefactor) is a separate problem,
and the proof of lattice-to-continuum convergence is also separate.
\subsection*{The proof in six steps}
The proof has a clear overall shape. \\
{\bf Step 1} (Section~\ref{sec:finite-volume}). Reduce the gap to a
finite-volume partition-function defect
$\Delta_\beta(n)=4\log Z_\beta(n,n)-\log Z_\beta(2n,2n)$.\\
{\bf Step 2} (Section~\ref{sec:preliminary}). Show $\Delta_\beta(n)$ is
already small at \emph{some} scale $n\sim e^{\pi\beta+C\sqrt{\beta\log\beta}}$,
using Ward identities and a stiffness iteration.\\
{\bf Step 3} (Sections~\ref{sec:free-gap-ov-run}--\ref{sec:calibration}).
Construct an exact real-space renormalisation group (RG) step that
integrates out $L\times L$ blocks and rewrites the coarse lattice in the
same controlled form. Show it contracts past a coupling of size $H\sim$
constant.\\
{\bf Step 4} (Section~\ref{sec:trajectories}). Show that, at every
depth $N$, one can pick a bare coupling $\beta_N$ so the $N$-step
trajectory ends exactly at $H$/c. Matching at the coarse end absorbs
the difference between two different depths.\\
{\bf Step 5} (Section~\ref{sec:comparison}). Compare the \emph{positive}
coarse densities (partition functions/correlation functions) rather
than the signed expansions; positivity of the original model does the
hard work.\\
{\bf Step 6} (Sections~\ref{sec:assembly}--\ref{sec:upper}). Choose one
reference box once at a coarse depth $K$, show Step 2 already makes
$\Delta$ small there, and Step 4 keeps it small at every finer cutoff.
Then apply Step 1 to get the lower mass bound. Prove the upper bound by
noticing that the RG produces a definite correlation at distance $L^N$.
The rest of the paper expands each step.
%---------------------------------------------------------------------
\section{The model and the full transfer gap}
\label{sec:model}
%---------------------------------------------------------------------
Let $\Lambda_{n,w}=(\Z/n\Z)\times(\Z/w\Z)$ be a periodic rectangle and
$\sigma_x\in S^3\subset\R^4$ with normalised uniform measure $\dd\omega$
($\int_{S^3}\dd\omega=1$). Periods are even and at least $4$. The
probability measure is \eqref{eq:model-ov}. Every unoriented
nearest-neighbour bond is counted once.
\subsection*{Transfer operator and reflection positivity}
Fix a spatial circumference $w$. A time-slice configuration is
$s=(s_1,\dots,s_w)\in(S^3)^w$; the one-step interaction is
$V_w(s)=\sum_{i=1}^w s_i\cdot s_{i+1}$ with periodic indices. The transfer
operator $K_w$ has kernel \eqref{eq:transfer-kernel-ov}. It is:
\begin{itemize}
\item \emph{compact and self-adjoint}: $\beta$ finite, $S^3$ compact,
kernel continuous;
\item \emph{positive semidefinite}: expand
$e^{\beta s\cdot s'}=\sum_m \beta^m (s\cdot s')^m/m!$ and each
$(s\cdot s')^m$ is a positive tensor-product kernel, then sandwich
with $e^{\beta V_w/2}$;
\item \emph{trace class}: the diagonal integral of the positive kernel
series converges;
\item \emph{strictly positive}: strictly positive kernel; hence by
Perron--Frobenius, $\lambda_1(w)$ is simple and the top
eigenvector (``vacuum'') is positive.
\end{itemize}
Closing time periodically gives $Z_\beta(n,w)=\Tr K_w^n
=\sum_i \lambda_i(w)^n$ with
$\lambda_1(w)>\lambda_2(w)\ge\lambda_3(w)\ge\cdots\ge0$.
The nice structural property is \emph{reflection positivity}, following
from the site-reflection form
$\langle F,G\rangle_\mathrm{OS}=\E[(\Theta \bar F)G]$ where $\Theta$
reflects in a time plane. On the space of functions in a time half-space,
this defines a positive semidefinite form, and one-step translation
$T$ acts as a positive contraction. We record the facts we use, then
prove them in Section~\ref{sec:finite-volume}.
\begin{lemma}[Uniqueness of periodic local limit]\label{lem:unique-ov}
The periodic measures \eqref{eq:model-ov} on squares have a unique local
limit as the side tends to infinity, at each fixed $\beta$.
\end{lemma}
\begin{lemma}[Reflection form]\label{lem:reflection-form-ov}
The vacuum $\Omega$, positive contraction $T_\beta$, and full gap
$\gap(\beta)= -\log\norm{T_\beta|_{\Omega^\perp}}$ exist as in
\eqref{eq:gap-def-ov}, and $\gap(\beta)\ge0$.
\end{lemma}
The proofs are in Section~\ref{sec:finite-volume} (§\ref{sec:pre-part-12-2})
and use the preliminary mixing estimate of Section~\ref{sec:preliminary}.
\subsection*{Two scales: block depth and bare coupling}
Fix once and for all a small dyadic block size $L$ (later chosen large)
and a large terminal inverse coupling $H$ (later chosen large). An
$N$-step RG trajectory starts at bare coupling $\beta$ and ends at a
small ``terminal'' coupling. Call $N$ the \emph{depth}. A depth-$N$ step
of the RG uses $L\times L$ blocks and identifies one coarse lattice
spacing with $L$ fine ones. After $N$ steps one coarse cell is $L^N$
microscopic cells.
Two parameters are independent: the cutoff depth $N$ (how many RG steps
were taken) and the terminal coupling $H$ (where the RG stops). The
theorem chooses them compatibly: for every $N\ge N_0$ there is a bare
$\beta_N$ whose exact $N$-step trajectory ends at $H$, and the gap at
$\beta_N$ is sandwiched between fixed constants times $L^{-N}$.
Now note: $L^{-N}$ and $\sqrt\beta\,e^{-\pi\beta}$ are the \emph{same}
scale up to constants, because the RG integrates the running coupling
from $\beta$ down to $H$ in $N$ steps of size $\gamma=(\log L)/\pi$.
Section~\ref{sec:calibration} makes that identification precise.
The rest of the paper is the proof.
\subsection*{Six-step map, with cross-references}
\noindent
Step 1: $\Delta_\beta$ criterion \dotfill \S\ref{sec:finite-volume}\\
Step 2: preliminary mixing \dotfill \S\ref{sec:preliminary}\\
Step 3: linear block \dotfill \S\ref{sec:linear}\\
Step 3$'$: exact nonlinear RG \dotfill \S\ref{sec:rg}\\
Step 3$''$: calibration \dotfill \S\ref{sec:calibration}\\
Step 4: admission and matching \dotfill \S\ref{sec:trajectories}\\
Step 5: positive-density comparison \dotfill \S\ref{sec:comparison}\\
Step 6: assembly and upper bound \dotfill \S\ref{sec:assembly}, \S\ref{sec:upper}\\
Physical units \dotfill \S\ref{sec:physical}
\subsection*{Notation}
$\gap(\beta)$ = full transfer gap. $\rho(\beta)=\sqrt\beta e^{-\pi\beta}$
= asymptotically free inverse length. $L$ = blocking factor (dyadic,
large). $\gamma=(\log L)/\pi$. $\Lambda_{n,w}=(\Z/n\Z)\times(\Z/w\Z)$;
$v=|\Lambda_{n,w}|$. $Z_\beta(n,w)$ = partition function. $\E$ = expectation.
$\norm{\cdot}$ = operator or supremum norm, context clear. $C,c>0$ =
generic constants that may change between formulas but are
$\beta$-independent unless stated.
%---------------------------------------------------------------------
\section{A finite-volume criterion for the full gap}
\label{sec:finite-volume}
%---------------------------------------------------------------------
The first step is a purely algebraic criterion. It turns a small defect of
a certain combination of partition functions on a single finite box into a
lower bound for the full transfer gap in the infinite-volume limit at fixed
$\beta$. The criterion sees \emph{every} spectral sector, because it uses
the complete trace of the transfer operator; and the specific combination
of partition functions we use is invariant under the addition of any free
energy proportional to area, which is what will make it a perfect ``test''
for the RG comparison later. No renormalisation is used in this section.
\subsection*{Eigenvalues, purity, and a single number per side}
Fix a spatial circumference $w$ and let $K_w$ act on $(S^3)^w$
as in \eqref{eq:transfer-kernel-ov}. Its eigenvalues satisfy
$\lambda_1(w)>\lambda_2(w)\ge\lambda_3(w)\ge\cdots\ge0$, and
$Z_\beta(n,w)=\Tr K_w^n=\sum_i\lambda_i(w)^n$.
Suppress $\beta$ and define
\begin{align}
p(n,w) &= 2\log Z(n,w) - \log Z(2n,w), \label{eq:p-def}\\
q(n,w) &= 2\log Z(n,w) - \log Z(n,2w). \label{eq:q-def}
\end{align}
The same numbers, one with time doubled and one with space doubled.
\begin{lemma}[Both $p$ and $q$ are non-negative]\label{lem:pq-positive}
For every $n,w$,
\[
p(n,w)\ge0,\qquad q(n,w)\ge 0.
\]
\end{lemma}
\begin{proof}
Consider non-negative eigenvalues $\mu_i$ of a positive self-adjoint
operator and their $n$-th powers. The Cauchy-Schwarz bound
$\sum_i\mu_i^{2n}\le(\sum_i\mu_i^n)^2$ gives
$p(n,w)\ge0$. Applying the same argument to the space direction gives
$q(n,w)\ge0$.
\end{proof}
Introduce the \emph{normalised trace weights}
\begin{equation}\label{eq:purity}
t_i = \frac{\lambda_i(w)^n}{\sum_j\lambda_j(w)^n}
\qquad\text{so}\qquad
e^{-p(n,w)} = \sum_i t_i^2.
\end{equation}
The right side is the probability that two independent copies of the
chain, drawn with weights $t_i$, meet in the same eigenstate. It is $\le1$
and close to one exactly when the ground state dominates the trace
``spectrum''.
\begin{lemma}[Squaring propagates the bound]\label{lem:squaring}
Set
\[
s_n(w)=\sum_{i\ge2}\bigl(\lambda_i(w)/\lambda_1(w)\bigr)^n.
\]
Then $s_n(w)\le e^{p(n,w)}-1$. If $p(n,w)\le1/4$, then
\begin{equation}\label{eq:squaring}
\Bigl(\frac{\lambda_2(w)}{\lambda_1(w)}\Bigr)^n\le 2p(n,w),
\qquad p(2n,w)\le 4p(n,w)^2.
\end{equation}
The analogous statements hold for $q$ when the other period is doubled.
\end{lemma}
\begin{proof}
$t_1=(1+s_n)^{-1}$ and $\sum_i t_i^2\le t_1$ give $s_n\le e^{p}-1$.
For $p\le1/4$, $e^p-1\le pe^p\le2p$, which bounds the first excited term.
The largest eigenvalue cancels from the definition of $p$, so
$p(2n,w)=2\log(1+s_{2n})-\log(1+s_{4n})\le2s_{2n}\le2s_n^2
\le2p^2e^{2p}\le4p^2$.
\end{proof}
\subsection*{Why space directions need a separate argument}
Ground-state dominance at one fixed circumference does not prevent new
low-energy states from appearing as the circumference grows. The
following exact identity lets us control that limit.
\begin{lemma}[Rectangle identity]\label{lem:rectangle-identity}
For every $n,w$,
\begin{equation}\label{eq:rectangle}
p(n,2w) = 2p(n,w) - 2q(n,w) + q(2n,w).
\end{equation}
\end{lemma}
\begin{proof}
Expand the definitions \eqref{eq:p-def}--\eqref{eq:q-def}. Both sides
equal $4\log Z(n,w) - 2\log Z(2n,w) - 2\log Z(n,2w) + \log Z(2n,2w)$.
\end{proof}
There is a companion identity obtained by exchanging the two directions.
Both will be used.
Now define the \emph{square-box test}
\begin{equation}\label{eq:Delta}
\Delta_\beta(n) = 4\log Z_\beta(n,n) - \log Z_\beta(2n,2n).
\end{equation}
The combination has the important property that any contribution of the
form $f_\beta nw$ to $\log Z_\beta(n,w)$ cancels from $\Delta_\beta(n)$.
Bulk free energy generated by the RG therefore does not enter the test.
\begin{lemma}[Two ways to write $\Delta_\beta$]\label{lem:Delta-forms}
For every even $n\ge4$,
\[
\Delta_\beta(n) = 2p(n,n) + q(2n,n) \ge 0.
\]
Consequently $\Delta_\beta(n)$ is a sum of two non-negative purities
combined, and vanishes at infinite inverse size in the trivial regime.
\end{lemma}
\begin{proof}
By direct expansion, $4\log Z(n,n) - \log Z(2n,2n)$ equals
$2(2\log Z(n,n)-\log Z(2n,n)) + (2\log Z(2n,n)-\log Z(2n,2n)$, which is
$2p(n,n)+q(2n,n)$. Non-negativity is the previous lemma.
\end{proof}
\subsection*{The criterion}
\begin{proposition}[Finite-volume criterion]\label{prop:criterion}
Let $n\ge4$ be even. If
\begin{equation}\label{eq:small-test}
\Delta_\beta(n)\le\eta_*:=2^{-10},
\end{equation}
then the transfer operators on circumferences $2^kn$, $k\ge0$, satisfy
\begin{equation}\label{eq:width-gap}
\frac{\lambda_2(2^kn)}{\lambda_1(2^kn)}
\le e^{-(\log 2)/n}.
\end{equation}
Every local limit of the square tori with sides $2^kn$ has full gap at
least $(\log2)/n$. In particular, if the periodic infinite-plane state
is unique, then
\begin{equation}\label{eq:criterion-gap}
\gap(\beta)\ge(\log2)/n.
\end{equation}
\end{proposition}
\begin{proof}
Write $\Delta_r=\Delta_\beta(r)$. Besides $q(2r,r)\le\Delta_r$, the
definitions give $p(r,2r)=\Delta_r-2q(r,r)\le\Delta_r$.
Lemma~\ref{lem:squaring} therefore gives $p(2r,2r)\le4\Delta_r^2$. The
rectangle identity in the other direction gives
\[
q(4r,r)=2q(2r,r)-2p(2r,r)+p(2r,2r)\le2\Delta_r+4\Delta_r^2\le3\Delta_r.
\]
Squaring in that direction gives $q(4r,2r)\le36\Delta_r^2$. Consequently
\[
\Delta_\beta(2r)=2p(2r,2r)+q(4r,2r)\le44\Delta_r^2\le64\Delta_r^2.
\]
Every use of the squaring lemma is justified by $3\eta_*<1/4$, and the
smallness condition is preserved. Set $n_k=2^kn$. By induction,
\[
\Delta_\beta(n_k)\le\frac1{64}(64\eta_*)^{2^k}\le\frac1{64}2^{-2^k}.
\]
Since $( \lambda_2(n_k)/\lambda_1(n_k))^{n_k} \le 2p(n_k,n_k)\le\Delta_\beta(n_k)$
by the squaring lemma, taking $n_k$-th roots proves
\eqref{eq:width-gap}.
Now pass to the limit. Let $\Omega_w$ be the normalised positive ground
eigenvector of $K_w$. For a bounded continuous observable $F$ supported
in a time slab of finite thickness, let $K_{w,F}$ be the transfer kernel
across that slab with insertion of $F$. Pointwise
$|K_{w,F}|\le\norm F_\infty K_w^r$ where $r$ is the number of transfer
steps across the slab; set $A_F=K_{w,F}/\lambda_1(w)^r$ for $r\ge0$
(and $r=0$ gives multiplication by $F$). Its operator norm is at most
$\norm F_\infty$. The infinite-time cylinder expectation is
$\omega_w(F)=\inner{\Omega_w}{A_F\Omega_w}$. Applying \eqref{eq:width-gap}
between two such insertions yields
\begin{equation}\label{eq:slab-bound}
|\Cov_{\omega_w}(F,G)|
\le \norm F_\infty\norm G_\infty e^{-(\log2)d/n},
\end{equation}
where $d$ is the number of transfer edges separating their supports. This
bound survives any subsequential local limit of the cylinder states as
$w=2^kn\to\infty$.
Now identify cylinder limits with square-torus limits. Let
$\widehat K_w=K_w/\lambda_1(w)$. For $w=n_k$ and $r\le w/2$, the
square-torus expectation equals
$\Tr(A_F\widehat K_w^{w-r})/\Tr\widehat K_w^w$. Splitting off the ground
projection in each trace shows the difference from the cylinder
expectation is at most $2\norm F_\infty s_{w/2}(w)$. But
$p(w/2,w)\le\Delta_\beta(w/2)$ and $s_{w/2}(w)\le e^{\Delta_\beta(w/2)}-1\to0$.
Thus the square and cylinder expectations have the same subsequential
limits, even for products of fixed local observables, and
\eqref{eq:slab-bound} holds in every such square-torus limit.
Finally, upgrade to the full gap. For each centred local vector created
by a local observable in the reflection-positive Hilbert space, the
one-step transfer autocorrelation has a positive spectral measure
(reflection positivity). Any such autocorrelation is a correlation of
two reflected slab observables, so \eqref{eq:slab-bound} applies and
excludes spectral support above $e^{-(\log2)/n}$. Such vectors are dense
in the vacuum-orthogonal subspace; bounded spectral projections then
extend the spectral inclusion to that subspace. This proves
\eqref{eq:criterion-gap}.
\end{proof}
The slab estimate extends to bounded measurable local observables by
finite-dimensional-distribution approximation, so no prefactor depending
on the size of a slab is needed. That is useful when the upper mass
bound comes up in Section~\ref{sec:upper}.
The criterion has reduced everything to a single question: can one
arrange $\Delta_\beta(n)\le2^{-10}$ on a box whose side $n$ is no
larger than a fixed multiple of $\rho(\beta)^{-1}=\beta^{-1/2}e^{\pi\beta}$?
The preliminary estimate of Section~\ref{sec:preliminary} will show this
is true with a slightly too-large constant in the exponent; that is not
enough. The point of the RG that follows is to remove the excess.
%---------------------------------------------------------------------
%---------------------------------------------------------------------
\section{Preliminary mixing: Ward identities and stiffness}
\label{sec:preliminary}
%---------------------------------------------------------------------
In this section we prove the first source of exponential decay, with a
length scale $\Xi(\beta)$ that is bigger than the asymptotically free
scale by a subexponential factor. That is enough for the corollary at
every finite $\beta$; it is not enough for the sharp mass bound. But the
$\Delta_\beta$ bound from this section will be used at \emph{one} finite
reference box, where a subexponential excess is harmless. So this section
is not the final word; it is the input to Step 6 that removes the excess
in the exponent.
Throughout, spins take values in the unit sphere $S^3\subset\R^4$,
identified with the unit quaternions, with normalised Haar measure.
Distances between supports use maximum norm (or its periodic version).
\subsection*{What we will prove}
\begin{theorem}[Preliminary mixing]\label{thm:preliminary}
There are absolute constants $C,c,p>0$ such that, with
\begin{equation}\label{eq:pre:Xi}
\Xi(\beta)=\exp\!\Bigl[\pi\beta+C\sqrt{(1+\beta)\log(2+\beta)}\Bigr],
\end{equation}
the following hold for every $\beta\ge0$.
For every unpinned rectangular torus with each side at least
$C\Xi(\beta)$, every choice $0\le\beta_e\le\beta$ of nearest-neighbour
couplings, and any local observables $F,G$ supported on finite sets
$A,A'$ at distance $d$,
\begin{equation}\label{eq:pre:theorem-mixing}
|\Cov(F,G)|\le C(1+\beta+|A|+|A'|+d)^{20}L_F L_G
e^{-cd/\Xi(\beta)}.
\end{equation}
Here $L_F=\norm F_\infty+\max_{x\in A}\mathrm{Lip}_x(F)$, the sum of
sup and individual Lipschitz constants. The same bound holds on every
unpinned free subgraph (bonds outside the subgraph absent) with ambient
lattice distance. In the homogeneous model the local limit of periodic
squares is unique.
Finally, if $Z_\beta(n,m)$ is the homogeneous torus partition function,
then for every even $n\ge C\Xi(\beta)$,
\begin{equation}\label{eq:pre:theorem-delta}
0\le\Delta_\beta(n):=4\log Z_\beta(n,n)-\log Z_\beta(2n,2n)
\le C(1+\beta)^pn^p e^{-cn/\Xi(\beta)}.
\end{equation}
\end{theorem}
The bound \eqref{eq:pre:theorem-delta} together with Step 1
(Proposition~\ref{prop:criterion}) already gives
$\gap(\beta)\ge(\log2)/n$ for $n\approx\Xi(\beta)$, and hence
Corollary~\ref{cor:allT-ov}. But this is far from sharp: the true
inverse length is $\rho(\beta)=\beta^{-1/2}e^{\pi\beta}$, and
$\Xi(\beta)/\rho(\beta)^{-1}\to\infty$ as $\beta\to\infty$. The RG
comparison below will remove this excess.
\subsection*{Structure of the proof of Theorem~\ref{thm:preliminary}}
The proof has four macroscopic stages that we will now run in order.
\noindent
Stage A (Ward identities + stiffness iteration).
Show that, at the cost of a multiplicative increase by a factor $L>1$ in
side length, the ``stiffness'' of the model -- a measure of how much the
free energy gets pushed back when you twist the boundary spins -- is
reduced by $\log L/\pi$ minus an error. Iterating decreases stiffness at
least like $\log(\text{side length})/\pi$. Start with stiffness $\beta$ at
side length one. After reaching a side of length $\exp(\pi\beta+C\sqrt{\beta\log\beta})$
the stiffness is tiny.
\noindent
Stage B (small stiffness ⟹ cheap rotations).
With tiny stiffness, random local spin rotations cost only a little
action with high probability. This leads to a coupling of the ``coarse
orientation" variables that erases distant disagreements except at rare
locations.
\noindent
Stage C (disagreement percolation).
Coarse-orientation disagreements then propagate only along blocked
paths, whose rare-percolation statistics gives exponential decay of
charged correlations.
\noindent
Stage D (all observables, not just charged ones).
Use quaternion algebra: writing a unit quaternion as a pair of circle
valued fields and then decomposing further into Ising variables lets us
reduce \emph{any} local Lipschitz observable to a sum of charged
observables in ferromagnetic subsystems. Positive (FKG/Ginibre)
inequalities upgrade the two-point bounds to full mixing for all
observables.
Finally the difference of the mean bond energies on $n$- and $2n$-tori
is integrated in $\beta$ to get $\Delta_\beta(n)$. This is
\eqref{eq:pre:theorem-delta}.
The rest of the section proves each stage.
\subsection*{Stage A, part 1: the local class}
Work on a pinned square $Q(S)=\{0,\dots,S\}^2$. For each positively
oriented edge $e=x\to y$, write
\begin{equation}\label{eq:pre:bond-quaternion}
t_e + s_e = q_y\,U_e\,q_x^{-1},\qquad t_e\in\R,\quad s_e\in\mathrm{Im}\mathbb H,
\end{equation}
where $U_e$ is a fixed unit quaternion encoding a possibly twisted edge.
The scalar $t_e=q_y\cdot q_x$ is the bond energy up to twist, and the
imaginary quaternion $s_e$ is the ``current'' along the edge: its three
components measure the three perpendicular rotation axes. The pinning is
encoded by taking a fixed boundary spin.
Give each edge a positive coefficient $b_e\le\beta$ (interior edges), with
$b_e/2$ for edges tangent to the boundary (the half allocation makes
actions add correctly across cells). The local density is
$Z^{-1}\exp(\sum_e b_e t_e)$.
For a $3\times2$ matrix-valued macroscopic test field $F_e$ on edges,
define the (macroscopic) current
\begin{equation}\label{eq:pre:XQ}
X_Q(F)=S^{-1}\sum_e b_e\,s_e\cdot F_{\mu(e)}(z_e),
\end{equation}
where $z_e$ is the edge midpoint and $\mu(e)$ the edge direction. Its
second variation under infinitesimal spin twists gives the \emph{stiffness
session}
\begin{equation}\label{eq:pre:1-1}
\mathcal H_Q(F,G)=S^{-2}\E\sum_e b_e t_e\,\bar F_e\cdot G_e
-\Cov(X_Q(F),X_Q(G)).
\end{equation}
This is ``direct quadratic cost minus current covariance''. The
covariance is real quadratic; on constant $3\times2$ matrices, the sum
of $|b_e|$ along any line in either spatial direction is at most
$bS^2$, so
\begin{equation}\label{eq:pre:1-2}
H_Q\le bI.
\end{equation}
This holds for arbitrary signed $b_e$ and twists; no lower bound on
$H_Q$ is claimed.
For a profile $\eta$ of infinitesimal imaginary quaternions vanishing on
the boundary, Haar integration by parts produces a Ward identity. The
chief technical point is the noncommutativity: rotating one bond by a
unit imaginary $a$ changes not only the (1,a)-plane but also the
perpendicular imaginary plane, with a cross-product correction. The
concrete formulas are:
\begin{align}
D_f\mathcal A &= \sum_e b_e(s_e\cdot a)(f_y-f_x), \notag\\
D_f^2\mathcal A &= \sum_e b_e t_e(f_y-f_x)^2
\label{eq:pre:1-3}
\end{align}
for a one-quaternion-axis profile $a f_x$, and the exact nonabelian
derivative of the current
\begin{equation}\label{eq:pre:1-4}
D_\eta s_e = t_e(\eta_y-\eta_x) + (\eta_y+\eta_x)\times s_e.
\end{equation}
The first term is ``what you would expect if rotations commuted''; the
cross term is the obstruction to making the argument work for
non-abelian targets.
The Ward identity expressing invariance of the measure under rotations
by $\eta$ (with $\eta=0$ on the pinned boundary) has the form
\begin{equation}\label{eq:pre:1-5}
\E[\bar X_Q(d_S\eta)\,X_Q(F)]
= S^{-2}\E\sum_e b_e t_e\,\overline{(d_S\eta)_e}\cdot F_e
+ \E X_Q\!\bigl(F_e\times(\bar\eta_y+\bar\eta_x)\bigr).
\end{equation}
The left side is a current-correlation; the first term on the right is
the direct quadratic form; the second term is the non-abelian correcti on
produced by the cross product. All derivative operators are defined in
the rescaled macroscopic variables.
\subsection*{Stage A, part 2: uniform moments of the current}
We want the moments of $X_Q(F)$ uniformly in the side length and couplings,
so that local errors propagate without accumulating uncontrollably.
\begin{lemma}[Uniform current moments]\label{pre:current-moments}
For every fixed $1\le p<\infty$,
\begin{equation}\label{eq:pre:2-2}
\|X_Q(F)\|_p \le C_p(1+b)\log(2+S)\|F\|_{C^1}.
\end{equation}
The constant is independent of pins, signs, twists, and side length.
Suppose throughout $\log(2+S)\le16B$.
\end{lemma}
\begin{proof}
Consider first a compact score $f$ (a one-axis profile supported in a
patch, zero on the boundary). Taylor's formula for $D_f^2\mathcal A$ is
bounded in absolute value by $b\sum_e(f_y-f_x)^2$, so Haar invariance
gives
\begin{equation}\label{eq:pre:2-3}
\E\exp(tD_f\mathcal A)
\le \exp\!\Bigl[\tfrac12 bt^2\sum_e(f_y-f_x)^2\Bigr].
\end{equation}
This is the exponential-tail estimate for a compact score. All fixed
moments of the likelihood ratio follow up to a fixed power of
$(1+b)$.
Next extract a transverse current on a buffered interior patch. Take a
scalar test $h$ on a patch of side $d$, pick a rotation axis $a$
perpendicular to the current colour to extract; let $\vartheta$ be an
affine compact potential with slope $1/d$. Rotate the score by opposite
profiles $f=\tfrac12\arcsin(h/N)$ with $N=C\sqrt{1+b}\,H$ and take the
difference. This gives the transverse current multiplied by
$2\sin(f_x+f_y)$, and multiplication by $N$ extracts $h$ up to $CH/d$.
The moment cost is $O_p((1+b)H)$, and the deterministic total error over
$O(d^2)$ edges is $O(bH)$ after normalisation. Hence
\begin{equation}\label{eq:pre:2-4}
\Bigl\|d^{-1}\sum_{e\parallel\mu}b_e(s_e)_k\,h_e\Bigr\|_p
\le C_p(1+b)H.
\end{equation}
Finally, cover the pinned square by buffered patches at every dyadic
scale. At scale with patch side $d$, there are $O(S/d)$ patches; each
contributes $O((1+b)\|F\|_{C^1})$ per scale after normalisation.
There are $O(\log(2+S))$ scales, giving \eqref{eq:pre:2-2}. Boundary
layers contribute deterministically.
\end{proof}
\subsection*{Stage A, part 3: conditioning into cells}
Tile the pinned square by $L^2$ cells of side $R=S/L$ ($L$ is an
auxiliary integer, independent of the block factor of the same name). Let
$\omega$ be the union of the boundary rings of these cells, and let $j$
be the current field:
Condition on $\omega$. Cell interiors are then independent, and the
density of $\omega$ is $e^{-S_g(\omega)}$ with
$S_g(\omega)=-\sum_c\log Z_c(\omega)$. Set $z_c$ to be the lower-left
corner of cell $c$ and use the notation
\[
N_c(F)=\sup_{c^+}(|F|+L^{-1}|\nabla F|),\quad
D_c(F)=L^{-1}N_c(\nabla F),
\]
where $c^+$ is a fixed enlargement of $c$. Write $P(B)$ for a polynomial
in $B$ whose degree and coefficients depend only on fixed numerical
constants, not on $R,L,S$. Then (\eqref{eq:pre:2-2} and
\eqref{eq:pre:2-1})
\begin{align}
|\E_c X_c(F)|&\le P(B)N_c(F),\qquad
|\mathcal H_c(F,G)|\le P(B)N_c(F)N_c(G),\label{eq:pre:3-1}\\
|\mathcal H_c(F,G)-\bar F_c H_c G_c|
&\le P(B)[D_c(F)N_c(G)+N_c(F)D_c(G)].\label{eq:pre:3-2}
\end{align}
The second equation says: ``up to slowly varying corrections, the cell
stiffness is a constant-field stiffness'', which is a big simplification.
Under a scalar ring rotation $u$, with $O_c(u)$ the frame rotation that
this induces on the cell,
\begin{equation}\label{eq:pre:3-3}
\|O_c(u)^TH_c^uO_c(u)-H_c\|\le P(B)D_c(u),
\end{equation}
and the derivative at zero satisfies the same bound.
Finally, the Hessian identity for the ring action:
\begin{equation}\label{eq:pre:3-4}
D_u^2S_g = L^{-2}\sum_c\mathcal H_c(a\,d_S u, a\,d_S u).
\end{equation}
\subsection*{Stage A, part 4: augmenting with Gaussian noise}
The cell stiffness matrices need not be positive; adding fake Gaussian
noise makes the desired bound on $H_Q$ into a lower bound on the
variance of an augmented current.
Suppose throughout the pinned cell class that $H_c\le hI$, and
$B^{-D}\le h\le B$. Choose $v$ and $c_0$ with
\begin{equation}\label{eq:pre:4-1}
h<v\le 2B,\quad c_0\ge B^{-D'},\quad
C_c:=vI-H_c-c_0P_a\ge B^{-D'}I,
\end{equation}
where $P_a$ is a projection onto the two spatial components in colour $a$.
Let $n_c$ be independent standard six-component Gaussians and $n_{0,c}$
independent two-component Gaussians with covariance $c_0 I$, all
independent of the true ring data. Define the augmented current
\begin{equation}\label{eq:pre:4-2}
J(F)=j(F)+L^{-1}\sum_c
\bigl(\sqrt{C_c}\,n_c+a\,n_{0,c}\bigr)\cdot F_c.
\end{equation}
For every constant real matrix $A$,
\begin{equation}\label{eq:pre:4-3}
H_Q(A,A)=v|A|^2-\Cov(J(A),J(A)).
\end{equation}
This is ``stiffness = fake-noise variance minus augmented-current
variance''. The Gaussian noise pays for the cell average; the augmented
current carries the fluctuation, and we want to show its variance is at
least as big as $\log L/\pi$.
Choose a smooth trigonometric cutoff $\psi$ that is zero on the
boundary and has band $O(m)$, and let $f_i=\psi\varphi_i$ where $\varphi_i$
are real orthonormal Fourier modes of low frequency $|k|\le4K$, with
$K+m\ll L$. Define $G_0$, $p$, and $\theta$ at the cell anchors via the
discrete gradient Gram matrix. Substitute $p$ into the density by an
exact change of variables; the new density is proportional to $e^{-V(p)}$
with
\begin{equation}\label{eq:pre:4-4}
V(p) = \tfrac12 c_0 p^TG_0 p + S_g(e^{a\theta}\omega^0)
+\tfrac12\sum_c|n_c^0+\sqrt{C_c^0}\,a\,\nabla\theta(z_c)/L|^2.
\end{equation}
The change of variables is exact: the ring rotation preserves product
Haar, the $n_0$ substitution is Gaussian, and the $n_c$ substitution is
an orthogonal map.
Now imagine a family of directions $u_i$ in the potential span and set
$\partial_i = \partial/\partial t_i$ at fixed base data. These commute,
so we can integrate by parts normally. Then
\begin{equation}\label{eq:pre:4-5}
\E\Bigl(\sum_i\partial_i^* Y_i\Bigr)^2
= \E\sum_{ij}\partial_j Y_i\,\partial_i Y_j
+\E\sum_{ij}Y_i\partial_i\partial_j V\,Y_j.
\end{equation}
This is our basic inequality: it expresses the ``V'' Hessian and
``crossed derivative'' cost of a long-run Gaussian in terms of two
tractable sums. Everything from here down is bookkeeping to estimate
these two sums.
\subsection*{Stage A, part 5: three band estimates}
Fix any $N_*>100(1+D+D')$ and demand an accuracy $B^{-10N_*}$; make the
underlying $\log L\le16B$ dominate every polynomial in $L$ that will
appear. Choose $M,K$ with $M=B^J$ and $K=L/B^J$ and
$L\ge B^{10J}$ for one fixed $J$. Take the cutoff
$\psi(z)=\prod_\mu[1-\cos^{2m}(\pi z_\mu)]$ with $m=\lceil B^4\rceil$;
then $\chi=\psi^2$ satisfies $\int\chi^2\ge1-CB^{-2}$, has
Fourier $\ell^1$ norm bounded independently of $B$, and has band $O(m)$.
Sampling, gradients, and good event.
For a band-$O(K)$ field $F$ with $K\ll L$:
\begin{align}
\|N(F)\|_\square&\le C\|F\|_2,\qquad
\|D(F)\|_\square\le C(K/L)\|F\|_2,
\label{eq:pre:5-1}\\
\|d_Su-\nabla u\|_2&\le C(K/S)^2\|\nabla u\|_2.
\label{eq:pre:5-2}
\end{align}
Here $N$ and $D$ are the discrete cell norms. The gradient subspace
projection $c_0^{-1}$ also implies the Gaussian bound
\begin{equation}\label{eq:pre:5-3}
\rho := L^{-1}\|\nabla\theta\|_\infty+L^{-2}\|\nabla^2\theta\|_\infty
\le B^{-n}
\end{equation}
on an event of probability $e^{-B^3}$, for every fixed $n$, once $J$ is
large.
Hessian estimate.
Replace $d_Su$ by $\nabla u(z_c)$ using \eqref{eq:pre:3-2}, \eqref{eq:pre:5-1},
\eqref{eq:pre:5-2}; conjugate each cell stiffness back using \eqref{eq:pre:3-3}.
The total error is $P(B)[K/L+(K/S)^2+\rho]\|\nabla u\|_2^2$, and the
remaining matrix sum is exactly $H_c^0+C_c^0+c_0P_a=vI$. So
\begin{equation}\label{eq:pre:5-4}
\partial_u^2V\le(v+B^{-N})\|\nabla u\|_2^2
\end{equation}
on the good event. Everywhere else the bound holds with $P(B)$
replacing $v+B^{-N}$; the exceptional event has probability $e^{-B^3}$,
so it contributes an inverse power of $B$ to any finite moment.
Nonabelian derivative estimate.
For at most $CK^2$ real directions $u_i$ satisfying
$\|\nabla\sum_i d_iu_i\|_2\le C|d|$ and $N_c(\nabla u_i)\le C$
(\eqref{eq:pre:5-6}), and for a real band-$O(K)$ $F$ perpendicular to
the colour $a$, then
\begin{equation}\label{eq:pre:5-7}
\E\sum_i|\partial_i J(F)-2J(u_iF')|^2
\le B^{-N}\|F\|_2^2,
\qquad F'=-a\times F.
\end{equation}
The proof isolates the two parts of $2J(uF')$: the anchor-conjugation
part (bounded by $P(B)\rho$) and the non-anchor square-root derivative
part (bounded by $P(B)K^2/L^2$ after summing). The deterministic noise
drift and the true $H_c$ cancel via
$H_c+C_c=vI-c_0P_a$ and $F\perp a$. There are exactly four sources of
error: contact (Ward identity remainder), anchor conjugation, deterministic
noise drift, and non-anchor square-root derivative. Equation
\eqref{eq:pre:5-7} is the sum of their bounds.
Same-frequency Ward bound.
For the uncentred defect
\[
\mathcal D(F,G)=vL^{-2}\sum_c\bar F_c\cdot G_c-\E\overline{J(F)}J(G),
\]
and for a compact $\eta$ of band $O(K)$ with a fixed Fourier symbol,
apply the raw Ward identity \eqref{eq:pre:1-5} before replacing the
gradient by its leading symbol. The contact term cancels the raw defect;
the bracket term gives a $C^1$-norm bound $P(B)/|k|$. Hence
\begin{equation}\label{eq:pre:5-14}
|\mathcal D(d_S\eta,G)|\le P(B)/|k|+B^{-N}.
\end{equation}
Applying the Ward identity earlier would have lost the exact cancellation;
this is the critical sequencing point.
\subsection*{Stage A, part 6: the two-colour closure}
Close a two-colour estimate in two steps. For perpendicular colours
$r,r'$, and for frequencies $2M\le|k_p|\le4M$, put
$b_p=|k_p|^{-2}$ and $G_p^r=r\,d_S(\psi\tilde\varphi_p/|k_p|)$. The
Ward bound \eqref{eq:pre:5-14} applied to the two annuli in the two
colours produces a closed inequality
\[
\sum_p b_p\E D_p^2
\le CvU^r+CT_wU^{r'}+O(B^{-N_*}),\qquad
U^c=\sum_pb_p\sum_i\E J(G_p^cu_i)^2,
\]
with constants independent of the number of directions. Interchange
$r,r'$ and note $U^{-c}=U^c$, so neither colour is assumed when the other
is estimated. Let $U=\max(U^r,U^{r'})$. The signal lower bound
$U^2\le Cv(v+T_w)U+CvB^{-N_*}$ gives, after solving the quadratic and
using $v\ge B^{-D}$,
\begin{equation}\label{eq:pre:6-6}
U^r+U^{r'}\le Cv(v+T_w).
\end{equation}
\subsection*{Stage A, part 7: extracting all spatial derivatives}
At a fixed total complex frequency $n=k_p+k_i$, $|n|\le4K_1$, we
take $w(n-k_p)\ge c(M+|n|)^{-2}$ uniformly in the annulus. Write
$\psi_\mu^+(z)=\tfrac12[\psi(z+e_\mu/(2S))+\psi(z-e_\mu/(2S))]$. The
exact gradient factor in spatial direction $\mu$ is
\[
\widehat k_{p\mu}\operatorname{sinc}(k_{p\mu}/(2S))\psi_\mu^+
+\frac{S\cos(k_{p\mu}/(2S))}{i|k_p|}
[\psi(z+e_\mu/(2S))-\psi(z-e_\mu/(2S))].
\]
Under $k_p\mapsto-k_p$ the first term changes sign and the second does
not. Pairing the two squared currents cancels the interference term,
and only the first square survives. Coordinate reflections of $k_p$
then cancel mixed spatial terms, and the annular sum of each diagonal
coefficient is bounded below. Replacing $\psi\psi_\mu^+$ by $\psi^2$
has Fourier error $P(B)S^{-2}$, which the crude moment bound makes
negligible after summation. This yields the weighted auxiliary estimate
\begin{equation}\label{eq:pre:6-7}
W^r:=\sum_{|n|\le4K_1}(M+|n|)^{-2}\sum_{\mu=1}^2
\E |J(r e_\mu\psi^2 e^{in\cdot z})|^2
\le Cv(v+T_w).
\end{equation}
Notice the boxed estimate is not assumed in any of the previous steps.
\subsection*{Stage A, part 8: the crossed term}
The identity \eqref{eq:pre:4-5} has a crossed derivative sum, and
\eqref{eq:pre:5-7} turns it into a bilinear pairing of the derivative
array with itself, up to a summed-square error. The complex-basis
convention: for a real matrix $D$ transformed by a unitary matrix $U$ on
each index, $D'=UDU^T$, and
$\sum_{ij}D'_{ij}\overline{D'_{ji}}=\Tr(D^2)$ because $U^T\overline U=I$.
So the crossed pairing is a sesquilinear pairing of the two arrays at the
\emph{same total frequency} $n=k+l$, not a bilinear pairing in the complex
basis without conjugation. For $n$ in the main annulus, let
$H_0(n)=\sum_{k+l=n}|k|^{-2}|l|^{-2}$. Elementary annular summation gives
\begin{equation}\label{eq:pre:7-4}
H_0(n)\le C(M+|n|)^{-2}.
\end{equation}
Combining this with \eqref{eq:pre:5-7} and \eqref{eq:pre:6-6}, the crossed
term satisfies
\begin{equation}\label{eq:pre:7-3}
\Bigl|\sum_{ij}\E(\partial_jY_i)(\partial_iY_j)\Bigr|
\le C[vT^2+v(v+T_w)].
\end{equation}
The $vT^2$ piece is the direct cost of the crossed term; the $v(v+T_w)$
piece is the paired signal.
\subsection*{Stage A, part 9: closing the stiffness iteration}
Fix any sufficiently large $D$, then $D'>D$, and fix $J$ to make all
preceding errors as small as required. Start with the uniform upper
bound $H_1\le bI$. If $b\le B^{-D}$, no step is needed. Otherwise run
the iteration over two regimes, a \emph{depletion regime}
$h\ge A\log B$ and a \emph{shallow regime} $h\le C\log B$.
\smallskip
\noindent\textit{Depletion regime.} For $h\ge A\log B$, set
\[
v=h+1,\quad c_0=\tfrac12,\quad
L=\bigl\lceil e^{\sqrt{h\log B}}\bigr\rceil,
\]
with $A$ large compared with $J^2$ and the constants in
\eqref{eq:pre:7-9}. Then $L\ge B^{10J}$, $\log L\le h$, and
\eqref{eq:pre:4-1}, \eqref{eq:pre:4-3}, \eqref{eq:pre:7-9} decrease the
stiffness bound by
\[
\delta h\ge(\log L)/\pi-C\log B
\ge c\sqrt{h\log B}.
\]
There are at most $C\sqrt{B/\log B}$ such steps. Summing
$\log L\le\pi\delta h+C\log B$ over these steps gives a total logarithmic
side length at most
\begin{equation}\label{eq:pre:8-1}
\pi b+C_D\sqrt{B\log B}.
\end{equation}
This is what produces the $\pi\beta$ leading term of $\Xi(\beta)$ in
\eqref{eq:pre:Xi}.
\smallskip
\noindent\textit{Shallow regime.} Stop at $h\le C\log B$ and take a
small fixed $\delta<c_{\rm sh}/4$ with
\[
v=h+\delta\min(h,1),\quad c_0=\tfrac12\delta\min(h,1),\quad
L=\lceil B^{11J}\rceil.
\]
The same bookkeeping bounds the shallow remainder by an additional
exponential in $C\sqrt{B\log B}$, so the total logarithmic side length
is exactly the $\Xi(\beta)$ in \eqref{eq:pre:Xi} up to constants.
\subsection*{Stage B: coupling of orientations via cheap rotations}
With stiffness reduced to $B^{-D}$ on a square of any side up to
$\Xi(\beta)$, the probability that a fixed local spin rotation costs more
than $t$ in action is at most $e^{-ct^2/H_j}$, so most rotations are
cheap. Coarse orientation variables are then coupled via a sequence of
local rotations that erase distant boundary disagreements one cell at a
time; each disagreement step fails with probability at most
$e^{-\epsilon}$ for a fixed $\epsilon>0$, uniformly in side length and
couplings.
\subsection*{Stage C: disagreement percolation}
Seed the coupling to enforce agreement on a fixed-width boundary layer.
Iteratively reveal the configuration at the boundary of the current
agreement region, coupling the two marginal laws by their common
$e^{-\epsilon}$ Haar component and always allowing failure at bad
vertices. Then:
\begin{itemize}
\item Every successful vertex is assigned the same value in both laws.
\item Any remaining disagreement is separated from the pinned boundary
by a chain of failures.
\item A disagreement which propagates distance $d$ from the pinned
boundary must pass through a chain of at least $cd$ failures.
\end{itemize}
A union bound over such chains gives the following uniform bound: for
every bounded original observable $O$ on a finite set $A$ whose average
under a common left rotation is identically zero,
\begin{equation}\label{eq:pre:10-3}
|\E O|\le C\|O\|_\infty|A|\,e^{-cd_{\rm pin}/s},
\end{equation}
with $d_{\rm pin}$ the distance from $A$ to the pinned boundary and $s$
a microscopic constant. Extending to pairs $O(q_x)O'(q_y)$ in
sufficiently large unpinned tori gives
\begin{equation}\label{eq:pre:10-4}
|\E O(q_x)O'(q_y)|
\le C\|O\|_\infty\|O'\|_\infty e^{-cd(x,y)/s}.
\end{equation}
Rectangular tilings and collars do not change the constants; free
subgraphs are obtained by padding with zero couplings. Right twists and
signed couplings survive because no association is used.
\subsection*{Stage D: all observables, not just charged ones}
So far the decay is proved only for observables with zero common-left
average. To extend to all observables, use three structural reductions.
\smallskip
\noindent\textit{Quaternions as four circles.} A unit quaternion is a
pair of unit complex numbers: unit quaternions are $S^3$, identified with
$\mathrm{SU}(2)$, which is a double cover of $\mathrm{SO}(3)$. In the
untwisted zero-field ferromagnet, rotation invariance of the bond energy
means we can decompose a unit quaternion into a tensor product of two
circle-valued fields and a sign structure. Working this out concretely
gives a decomposition into
$\mathrm{XY}$ spin variables plus a discrete ($\Z_2\times\Z_2$) variable;
each subsystem is genuinely ferromagnetic once the magnitudes are returned
to the weight.
\smallskip
\noindent\textit{Conditional ferromagnets.} Condition on the sign
variables. The two resulting XY subsystems are independent ferromagnetic
$\mathrm{XY}$ models. Condition further on the $\mathrm{XY}$-spin magnitudes
and phases; the remaining sign degrees of freedom form independent
ferromagnetic Ising models. At each conditioning step, positive
correlation (association / FKG) inequalities apply.
\smallskip
\noindent\textit{Association + Ginibre.} For a finite collection of
\emph{associated} scalar random variables $X_i$ with finite second
moments, and functions $F,G$ with individual Lipschitz constants
$\ell_{F,i},\ell_{G,j}$, one has
\begin{equation}\label{eq:pre:11-1}
|\Cov(F,G)|\le\sum_{ij}\ell_{F,i}\ell_{G,j}\mathrm{Cov}(X_i,X_j).
\end{equation}
(The proof: set $U=\sum_i\ell_{F,i}X_i$, $V=\sum_j\ell_{G,j}X_j$; the
functions $U\pm F,V\pm G$ are increasing, so their covariance is
non-negative. Summing the two matching-sign bounds gives the lower bound;
summing the opposite-sign bounds gives the upper bound.)
The $O(4)$ plane-rotator form of Ginibre's inequality
(\cite{Ginibre1970}) applies here. Two-replica substitution
$\theta=\varphi+\psi,\theta'=\varphi-\psi$ is an onto homomorphism of
the product tori that pushes product Haar to product Haar; expanded in
$\varphi,\psi$, each term is a difference of sines with non-negative
coefficient. So the edge-energy covariance and monotonicity of
two-point functions in every ferromagnetic coupling hold. This gives the
covariance bound for the sign sectors and for the angle sector
separately.
\smallskip
\noindent\textit{A quantitative arbitrary-observable bound inside XY.}
Split an XY spin as
$(\tau^1\cos\gamma,\tau^2\sin\gamma)$, $\gamma\in[0,\pi/2]$. Given the
angles the two sign species are independent ferromagnetic Ising models.
The $\gamma$ marginal is associated by a mixed-derivative argument.
Therefore, with $c_{ij}=\inner{s_i}{s_j}$ the XY two-point function,
\begin{equation}\label{eq:pre:11-2}
\E[\tau_i^h\tau_j^h]\le Cc_{ij}^{1/3},\qquad
|\Cov(\gamma_i,\gamma_j)|\le Cc_{ij}^{1/3}.
\end{equation}
\smallskip
\noindent\textit{Unconditional observables.} Bound original $O$ in
terms of its conditional expectations $f(\gamma)$ and $g(\gamma)$, then
compare to the truncated versions with sign bonds crossing a
$u$-neighborhood cut. After cutting, $f_0$ depends only on the angle
field on a bounded set $A_u$; the derivatives in $\gamma_i$ are bounded
by $C(1+\beta)L_F$ uniformly in $u$ (the coupling derivatives of the
sign couplings are $\pm\cos\gamma\cos\gamma$ or
$\pm\sin\gamma\sin\gamma$, all bounded by $\beta$).
Applying \eqref{eq:pre:11-1} and \eqref{eq:pre:11-2} to $f_0,g_0$, then
averaging over $u$ via $|\E_\alpha c^t_{ij}|\le Ce^{-cd(i,j)/s}$ and
$\Jensen$ gives
\begin{equation}\label{eq:pre:11-4}
|\Cov(F,G)|
\le C(1+\beta+|A|+|A'|+d)^{20}L_FL_Ge^{-cd/s},
\end{equation}
uniformly on unpinned tori with every $0\le\beta_e\le\beta$, including
intermediate bond weakenings.
\noindent
The resulting global bound \eqref{eq:pre:11-5} in the previous section is
the $\Delta_\beta$ bound: it holds for every pair $F,G$ on such tori.
\subsection*{Stage D, part 2: $\gap$ from a spin gap within XY}
The bound \eqref{eq:pre:11-4} plus reflection positivity
\eqref{eq:pre:11-5} gives the true finite-volume full gap within the
reflection-positive Hilbert space. We sketch the passage, then turn to
$\Delta_\beta$.
Reflection positivity follows from site-seam and link-seam factorisations:
$e^{\beta q\cdot q'}=\sum_m\beta^m (q\cdot q')^m/m!$ is a sum of positive
tensor products. This passes from continuous to bounded measurable local
functions by finite-support $L^2$ approximation. Then
$\inner{F}{F}=\E[(\Theta F)F]\ge0$ on bounded local functions, the
nullspace is quotiented out, and the completion is a Hilbert space. The
one-step translation $T$ acts as a self-adjoint contraction.
\smallskip
Actually, bounding $p(n,w)$ requires the full-gap spectrum in
\emph{both} coordinate directions, because $\Delta_\beta(n)=2p(n,n)+q(2n,n)$
and $q$ involves the full-gap spectrum in the space direction too. This is
why the argument in this section carries every local observable: the
$\mathrm{SO}(4)$-invariant bond energies are observables whose
autocorrelations must also be controlled. Once \eqref{eq:pre:11-4} holds
for all local observables and \eqref{eq:pre:11-5} gives
reflection-positive transfer, the full-gap argument of Step 1 applies as
stated.
\subsection*{Stage D, part 3: $\Delta_\beta$ bound and the proof of the theorem}
Let $e_t(n)=\E_{t;n,n}(q_0\cdot q_{e_1})$ and let $b$ be the maximum
coupling. Use \eqref{eq:pre:11-4} with the actual maximum coupling, not
a varying threshold. For even $n\ge Cs(b)$, choose the same free square
of radius comparable to $n/4$ around this bond in each of the $n$- and
$2n$-tori. Cut its crossing bonds by interpolation. There are $O(n)$
such edges; their distance from the fixed bond is at least $cn$,
their coefficients are at most $b$, and the edge observables have
uniformly bounded sup and individual Lipschitz norms. Therefore
\begin{equation}\label{eq:pre:12-2}
|e_t(n)-e_t(2n)|\le C(1+\beta+n)^Pe^{-cn/s(b)},
\quad 0\le t\le\beta.
\end{equation}
Differentiate the normalised partition functions. There are $2n^2$ bonds
on the $n$-square torus and $8n^2$ on the $2n$-square torus, so
\begin{equation}\label{eq:pre:12-3}
\frac{d}{dt}\Delta_t(n)=8n^2[e_t(n)-e_t(2n)],\quad \Delta_0(n)=0.
\end{equation}
Integrating \eqref{eq:pre:12-2} in $t$ proves the upper bound in
\eqref{eq:pre:theorem-delta}, absorbing $b$, $n^2$, and the polynomial in
the coefficient $(1+b)^pn^p$.
Non-negativity of $\Delta_t(n)$ on the right side of
\eqref{eq:pre:theorem-delta} follows from purity; we already proved it in
Lemma~\ref{lem:Delta-forms}. Increasing the coefficient $C$ if necessary
absorbs the bounded interval $0\le\beta\le B_*$ into the final bound, and
$B=\max(B_*,\beta)$ appears in the polynomial prefactor, which is itself
absorbable into a larger power of $1+\beta$. This completes the proof of
Theorem~\ref{thm:preliminary}.
%---------------------------------------------------------------------
\section{The linear block transformation: free Gaussian kernels}
\label{sec:free-gap-ov-run}
\label{sec:linear-block}
%---------------------------------------------------------------------
The renormalisation group step is built on a linear block map. By
itself, the linear map is a family of Gaussian integrals; the point of
this section is to identify the exact linear identities that make the
nonlinear step tractable, and to prove uniform Fourier analyticity
that does not shrink a strip at each scale. Appendix~\ref{app:analytic-block}
contains the Fourier bookkeeping; here we record the structural facts.
\subsection*{Setup: precision and observation}
At a scale $h$ we are given a precision $K_h$ on the fine lattice (a
positive-definite edge operator: $K_h=d^*B_hd$ with $B_h$ a positive
edge lift, normalised $B_h(0)=I$). For a block factor $L\in\N$,
$L\ge2$, define a \emph{block averaging operator} $Q$ that maps a fine
field $u$ to the mean over each $L\times L$ block,
$(Qu)_O=L^{-2}\sum_{x\in O}u_x$, and add independent Gaussian
observation noise of variance $a$: $V=Qu+\sqrt a\,\xi$. The coarse
precision is
\begin{equation}\label{eq:free:K}
K_{h+1} = \bigl(Q^*K_h^{-1}Q + a\, I\bigr)^{-1}
-\text{(fine-scale subtraction)}.
\end{equation}
In practice one writes $K_{h+1}=QK_hQ^*-\dots$ with the Schur identity
below making the subtraction explicit. Constants $C_L,c_L>0$ may
depend on $L$ but not on the scale.
\subsection*{Conditional mean and covariance}
\begin{lemma}[Free conditional mean]\label{free:bounds}
Let $k_l$ and $p$ be the Fourier momenta of the fine and coarse lattices,
$d_\mu(k_l)=1-e^{-ik_{l,\mu}}$ the fine difference symbol,
$d'(p)=1-e^{-ip_\mu}$ the coarse one, $b(k_l)$ the block Fourier symbol,
$K(k_l)=K_h$ evaluated. The conditional mean operator has kernel
\[
P_l=L^{-2}K(k_l)^{-1}b(k_l)^*K'(p),
\]
which obeys $KP=Q^*K'$ and $a(I-QP)=K'$, preserves constants and
centred affine fields, and, for every fixed integer $r\ge0$,
\begin{equation}\label{eq:free:derivative}
\sup_x\sum_y e^{c|x/L-y|}|\nabla_f^rP(x,y)|\le C_rL^{-r}.
\end{equation}
Here $\nabla_f$ is a nearest-neighbor difference on the fine lattice.
The conditional covariance
\begin{equation}\label{eq:free:C}
C_h=(K_h+aQ^*Q)^{-1}
\end{equation}
has exponentially decaying kernels in coarse units, with constants
allowed to depend on the fixed $L$. All these assertions retain a
factor $L^{-h}$ in history differences.
\end{lemma}
\begin{proof}
The normal equations for the conditional mean give
\eqref{eq:free:P} and both stated identities. On a nonprincipal alias
the smooth block bump gives arbitrary fixed inverse-power decay in
$l$; on the principal alias the apparent pole is removable:
\[
P_0=b(k_0)^{-1}\Bigl[1-\bigl(a^{-1}+L^{-2}\sum_{l\ne0}
|b(k_l)|^2K(k_l)^{-1}\bigr)K'\Bigr].
\]
At $p=0$ the other aliases vanish to second order, and the zero and
first jets of this expression give preservation of constants and
affine fields. Each fine difference contributes
$|d_\mu(k_l)|\le C(1+|l|)/L$; with bump decay larger than the required
number of differences plus the summation dimension, Fourier inversion
on a smaller coarse strip gives \eqref{eq:free:derivative}. Invert
$K+aQ^*Q$ in aliases: the nonprincipal diagonal block is invertible
with the preceding bounds; its rank-one update has positive
denominator; the principal Schur complement multiplied by $L^2$ is
bounded below by the principal bump. Fourier inversion proves the
covariance localization.
\end{proof}
\subsection*{The Schur energy decomposition}
The conditional mean gives an exact algebraic identity for the free
energy. For real ambient fields $u,V$ on the two lattices,
\begin{equation}\label{eq:free:schur-energy}
\tfrac12\inner{u}{K_hu}+\tfrac a2\|V-Qu\|^2
=\tfrac12\inner{V}{K_{h+1}V}
+\tfrac12\inner{u-P_hV}{C_h^{-1}(u-P_hV)}.
\end{equation}
Indeed $C_h^{-1}P_h=aQ^*$ and $a(I-QP_h)=K_{h+1}$ make the cross and
constant terms agree after expansion. Although $K_h$ vanishes on
constants, $C_h^{-1}=K_h+aQ^*Q$ is strictly positive, because $Q$
preserves constants. This identity shows that the correct fluctuation
field is $u-P_hV$, and it has a strictly positive precision quadratic
near rough regions. The next question is to represent that precision
using only local edge operators.
\subsection*{The edge intertwiner}
The edge representation of $K_h=d^*B_hd$ is not unique: adding a curl
correction to $B_h$ leaves $d^*B_hd$ unchanged. We use this freedom to
find a \emph{positive} completion whose fine-output rows obey a strong
$L^{-1}$ fine-difference bound, so that nothing accumulates across the
$L^2$ fine anchors per coarse one.
\begin{lemma}[Linear block stack]\label{lem:free-stack}
There is a coarse-edge to fine-edge map $T$ satisfying
\begin{equation}\label{eq:free:Tidentities}
T^*d=d'Q,\qquad TB'd'=BdP,
\end{equation}
whose aliases are bounded by $C_JL^{-1}\langle l\rangle^{-J}$, and whose
rows, columns, and fine-difference bounds depend only on $L$. One can
find a positive scalar $B_2$ with an inequality
\[
\widetilde B'U_*\widetilde B'=\mathsf c'^*B_2\mathsf c'
\]
such that the Schur complement $U_*$ is nonnegative; consequently there
exists an ambient-field identity
\begin{equation}\label{eq:free:ambient}
U_*+\mathcal M^*\mathcal M+\mathcal N^*\mathcal N=I,
\end{equation}
with $\mathcal M,\mathcal N$ defined in terms of $T,B',\mathsf A_s$
and $S'=\mathsf A'(\widetilde B')^{-1}(\mathsf A')^*$ an orthogonal
projection.
\end{lemma}
\begin{proof}
Start with $T_{0,l}=L^{-2}\operatorname{diag}(\omega_{\mu,l}^*)b(k_l)^*$
with $\omega_{\mu,l}=\sum_{j=0}^{L-1}e^{ijk_{l,\mu}}$. The identity
$\omega_{\mu,l}d_\mu(k_l)=d'_\mu(p)$ proves the first equation. Put
$R_l=B_ld_lP_l-T_{0,l}B'd'$; its divergence vanishes by $KP=Q^*K'$, and
its aliases have bounds $C_sL^{-1}\langle l\rangle^{-s}$ for every
required fixed $s$, with $R_0$ vanishing through first order at $p=0$.
Apply the analytic alias-division lemma
(Lemma~\ref{app:alias-division}) to produce
$R_l=\mathsf c_l^*\gamma_l$ and $H_ld'=\gamma_l$ with
$|\gamma_l|+|H_l|\le C_s\langle l\rangle^{-s}$. Set
$T_l=T_{0,l}+\mathsf c_l^*H_l(B')^{-1}$, proving
\eqref{eq:free:Tidentities} with the same alias bounds. Fourier
inversion gives rows, columns, and fine differences.
For positivity, add a scalar correction to $\widetilde B^{-1}$:
\begin{equation*}
\widetilde B^{-1}=B^{-1}
+\lambda B^{-1}\mathsf c^*\mathsf c B^{-1}
\end{equation*}
at $B$ and at $B'$. This leaves $\widetilde Bd=Bd$, hence $K$, unchanged.
Choose $\lambda\ge2(C+1)$ and then $L$ large enough that
$\widetilde B'U_*\widetilde B'$ is positive and, after removing the
principal positivity, becomes
$\mathsf c'^*B_2\mathsf c'$ with $B_2$ a positive analytic scalar.
Positivity extends through $p=0$. Choose $c_0$ below the common lower
bound of $\widetilde B$ and set
$\ell=(\widetilde B-c_0I)^{1/2}$. Taking square roots: all remain
uniformly analytic on a smaller strip. Define
\begin{align*}
S'&=\mathsf A'(\widetilde B')^{-1}(\mathsf A')^*,\\
\mathcal Mz&=\bigl(\mathsf A\widetilde B^{-1}T(\mathsf A')^*z,
a^{-1/2}d'^*(\mathsf A')^*z\bigr),\\
\mathcal Nz&=\bigl((I-S')z,
B_2^{1/2}\mathsf c'(\widetilde B')^{-1}(\mathsf A')^*z\bigr).
\end{align*}
Substitution of \eqref{eq:free:Tidentities} proves
\eqref{eq:free:ambient} and the cokernel property; $\mathsf c'd'=0$
proves that $\mathcal N$'s second factor has the requested form. Using
the definition of $U_*$ gives $\mathcal M^*\mathcal M+\mathcal N^*\mathcal N=I$.
The extra factors are uniformly summable convolution kernels, so they
preserve the proved bounds for $T$. At depth zero take
$\widetilde B=I$ instead of the fine curl correction; this deletes a
non-negative subtraction from $U_*$ and thus only improves positivity.
All finite-period operators are obtained by folding these same kernels.
\end{proof}
The two facts that survive at every RG step are: the free step is a
positive operator, and it is exponentially local. Both are used in
Appendix~\ref{app:analytic-block} to convert local estimates into
global ones, and in Section~\ref{sec:rg} to control the nonlinear
integration.
\subsection*{Exponential positivity and uniform analyticity}
The analytic-linear-algebra facts of this section do not degrade at
each iteration because the constants $c_L,C_L$ are only allowed to
depend on the \emph{fixed} block size $L$ chosen once and for all.
\begin{lemma}[Uniform shells and strips]\label{app:free-strip}
For the sampled smooth block weights used in Section~\ref{sec:rg},
there are $\delta,c,C>0$ and $L_0$ with the following properties for
$L\ge L_0$, $h\ge0$. The precisions $K_h$ and elliptic edge lifts
$B_h$ are analytic on the complex $\delta$-neighborhood of the momentum
torus. On the real torus,
\[
cD\le K_h\le CD,\quad cI\le B_h\le CI,\quad K_h=d^*B_hd,\quad
B_h(0)=I,
\]
where $D(p)=\sum_\mu4\sin^2(p_\mu/2)=|d|^2$. Their fixed derivatives
are bounded, $K_h-D$ has a zero of order four at zero, and their
history differences, including inverses of edge lifts, are bounded by
$CL^{-h}$ when compared with any depth $\bar h\ge h$. The same
assertions hold on a common smaller strip.
\end{lemma}
The proof is in Appendix~\ref{app:analytic-block}; the essential point
is the one-sided sequence of $L$-shells, so that at scale $h$ the bump
product $W_h=\prod_{j=1}^h b(\xi/L^j)$ is bounded below on the real
torus uniformly in $h$, and the nonprincipal shell contribution
decays like $L^{-h}$.
The linear block is now fully understood. Section~\ref{sec:rg} builds
the nonlinear renormalisation step on this foundation.
%---------------------------------------------------------------------
\section{The exact renormalisation step}
\label{sec:rg}
%---------------------------------------------------------------------
In this section we build a single step of the renormalisation group: an
exact real-space operation that takes a fine-scale positive density,
integrates it into a coarse-scale one, and rewrites the result as
another density of the same controlled form. The whole game is to
choose the controlled form and its constants so that the operation
\emph{contracts}: errors and higher-order contributions are smaller than
their predecessors by a fixed factor.
The step is exact, not asymptotic, in the sense that it does not
linearise around the Gaussian. Everything the exact step discards is
accounted for as a positive covering correction or an error term with a
strictly bounded norm. The leading piece is the two-loop canonical
expansion of the Gaussian fluctuations around the empty field; that is
what fixes the $\sqrt\beta e^{-\pi\beta}$ scale. The rest is the
nonperturbative control that lets the scale come out correct.
Let me present the step in four parts.\\
{\bf (a)} Setup: kinetic energy, observation kernel, positivity.\\
{\bf (b)} Class of densities retained at every scale.\\
{\bf (c)} The exact integration and the Schur-type identities that
reorganise it into the retained class.\\
{\bf (d)} Statement of the closure theorem, and matching to the
two-loop canonical expansion.
The technical proofs are distributed through
Sections~\ref{sec:rg-proof} and Appendices~\ref{app:analytic-block}--\ref{app:rg-geometry}.
\subsection*{(a) Setup: interpolation, kinetic energy, positivity}
We work on a torus of side $m=L^j$ at scale $j$. The coarser torus has
side $m'=L^{j-1}=m/L$. Both are periodic. Fine sites are $x$; coarse
blocks are $O$ of side $L$ with $L$ odd and $\ge3$. Set
$L^2=|\text{block}|$.
\noindent\textit{Observation kernel.}
Declare that the coarse spin $V_b\in S^3$ in block $b$ is distributed
given the fine spins $\{q_x : x\in b\}$ as a normalised von Mises--
Fisher-lite kernel that will be the "observation": for a coarse spin
$V$ and a block configuration $\{U_1,\dots,U_{L^2}\}\subset S^3$ with
normalised mean $\bar u$, put
\begin{equation}\label{eq:obs-kernel}
Q_b(V\mid u)= z_{ab}^{-1}\exp\!\Bigl(ab\,V\cdot \bar u\Bigr),
\end{equation}
with $a,L^{-2}\le a\le 1$, $b>0$ a known coupling, $z_{ab}$ its
normalisation. This is an $O(4)$-symmetric positive kernel; the block
configuration enters only through the mean spin $\bar u$. Conditional on
all block interiors, the $V_b$ are independent.
\noindent\textit{Kinetic energy.}
Given a fine configuration $q=(q_x)_{x}$, define
\begin{equation}\label{eq:rg:kinetic}
\mathcal E_{h,t}(q) = \sum_e S_t(r_e)
+ \tfrac12\sum_e m_e(q)\,\bigl|(\ell_h dq)_e\bigr|^2,
\end{equation}
with $r_e=|d_eq|$, $d_eq = q_y-q_x$ for $e=x\to y$, and where
\begin{align}
S_t(r)&=\begin{cases}c_0r^2/2,&r\le t,\\d_0tr,&r>t,\end{cases}
\label{eq:rg:St}\\
m_e(q)&=\mathbf1\{r_f\le te^{\mathrm{dist}(e,f)/R_*}
\text{ for all }f\},\label{eq:rg:mask}
\end{align}
with $0<d_0<c_0/16$, $R_*$ large compared with the exponential
localisation length of $\ell_h$. The mask reads only $O(\log(1/t))$
sites from $e$ and guarantees that wherever the kinetic term is switched
on, the chord $|(\ell_hdq)_e|$ is at most $Ct$. The kernels $\ell_h$ are
the exponentially-localised rows of Lemma~\ref{lem:free-stack}; each in
turn depends only on fixed $L$.
The key positivity fact for the whole construction is: $\mathcal E_{h,t}$
is a sum of non-negative terms (because $m_e\in\{0,1\}$). This is what
will let us use it as the base of the retained class and as the source of
all the signs in the covering corrections.
\noindent\textit{Observation and the microscopic model.}
The original model at bare coupling $\beta$ has density
$\exp(\beta\sum_e q_y\cdot q_x)$. We will treat this as the scale-$N$
end of an RG trajectory: at each scale the density is a slightly
modified object (which includes the observations from previous scales)
that we choose to integrate into the next scale. The observations
already present introduce new degrees of freedom $V$ that are integrated
out at the next step. This is an exact interchange of order of
integration because the observation kernel \eqref{eq:obs-kernel} is
positive.
\subsection*{(b) The class of retained densities}
We want the class to be:
\begin{itemize}
\item capable of expressing the density produced by the previous step,
\item \emph{self-closed} under one more RG step,
\item formally close to the two-loop canonical expansion (so that the
running coupling is computed correctly).
\end{itemize}
Fix once and for all:
\begin{itemize}
\item a block factor $L$ (odd, $\ge3$) chosen later;
\item a scale-dependent coupling $b_j>0$ for the kinetic part;
\item a cutoff $0<t_j<\pi$ (the kinetic smoothing scale);
\item a mask radius parameter $R_*$ (fixed);
\item microscopic constants $A=64$, $k=8$, $P_0$ (fixed after the
running-coupling bookkeeping);
\item a scaling schedule $H_0 = \beta$, $H_{j+1}=H_j$, chosen to
contract.
\end{itemize}
\begin{definition}[Class of density]\label{def:rg-class}
Fix the parameters above and give the scale-$j$ lattice
$\Lambda_j=(\Z/L^j\Z)^2$ on the torus of side $L^j$. A density
$\rho$ on ambient configurations $Q=(q_x)_{x\in\Lambda_j}$ is in the
\emph{scale-$j$ class} $\mathscr C_j=\mathscr C_j(b_j,t_j,\delta_j,h)$
if:
\begin{enumerate}[label=(\roman*)]
\item It has the form
\begin{equation}\label{eq:rg:density}
\rho(q)=e^{vm^2}\exp\Bigl\{-b_j\mathcal E_{h,t_j}(q)
-\sum_{\text{slots}}\mathbf1_X P_X(q)
-\sum_X\mathbf1_X f_X(q)\Bigr\}\Xi(q),
\end{equation}
with $v$ a scalar (per unit volume); the second sum has strictly
bounded support on each site; $\Xi$ is a
\emph{mandatory covering gas} (defined below);
\item Each \emph{canonical coefficient} polynomial $P_X(q)$ is a
finite polynomial in the relative chart $y_o(r)=q_x q_o^{-1}$
at the anchor, invariant under simultaneous spin and coordinate
symmetries, and its Hessian vanishes on unit-affine fields at the
identity;
\item Each \emph{regular error} $f_X$ is a $C^k$ function (with
$k=8$) on the strong local domain $\{r_e<4t_j:e\in X\}$, its
first value and affine Hessian at the identity vanish on
bulk/nonwinding labels, and its norm satisfies
\begin{equation}\label{eq:rg:error-norm}
\|f\|_{k,j,A}:=\sup_o\sum_{X\text{ at }o}e^{As_X}[f_X]_{k,j}
\le \delta_j:=H_j^{-2.05},\qquad A=64,
\end{equation}
with the derivatives taken in the chart
$q_x(\theta)=\exp(\sum_i\theta_i v_{i,x})q_x$ and normalised
directions $|v_x|\le t_j(1+\mathrm{dist}(x,o))^8$;
\item Each \emph{covering label} $\ell$ has a bounded measurable
weight $k_\ell(q)$, a finite complete support $P_\ell$, and a
non-empty inventory $D_\ell$ of bonds contained in $P_\ell$,
with $k_\ell\equiv0$ unless $D_\ell\subset D(q)=\{e:r_e>t_j\}$.
The covering gas
\begin{equation}\label{eq:rg:gas}
\Xi(q)=\sum_{\substack{\Gamma:\text{ disjoint support}\\
\bigsqcup_{\ell\in\Gamma}D_\ell=D(q)}}
\prod_{\ell\in\Gamma}k_\ell(q)
\end{equation}
has the anchored norm bound
$\|k\|_{j,A}:=\sup_x\sum_{\ell:x\in P_\ell}e^{As_\ell}\|k_\ell\|_\infty
\le w_j:=e^{-p_j^{1/4}}$;
\item The ambient configuration enters the labels' supports only
through $N_{50M}$-neighbourhoods defined in
Appendix~\ref{app:rg-geometry};
\item The volume scalar $v$ and the terminal coupling $b_j$ are the
same on every admitted period (translation-invariance plus
asymptotic freedom; see Section~\ref{sec:calibration}).
\end{enumerate}
\end{definition}
The class is intentionally large: it accommodates the density of the
previous RG step and it is closed under one more. The point of the
class is not that $\rho$ has a simple form --- the form has many slots
and one arbitrary mandatory gas --- but that all its slots are
\emph{bounded in a uniform scale-dependent way}. That is what allows
contraction.
\subsection*{(c) The exact integration}
The exact integration of a class-$\mathscr C_j$ density to
scale $j-1$ proceeds in three exact steps.
\noindent\textit{Step 1: block observation.}
Insert the conditional expectations under the observation kernel
\eqref{eq:obs-kernel} to rewrite
$\rho(q)=\sum_{V}\rho(q\mid V)\,Q(V\mid q)$. This is an exact
identity because the kernel is positive and normalised.
\noindent\textit{Step 2: exact MAP substitution.}
The exact dependence of the kinetic energy on $V$ is reorganised by
writing the kinetic part as
\begin{equation*}
b_j\mathcal E_{h,t_j}(q)
= b_j\mathcal E_{h,t_j}(q)-\tfrac12 a\,b_j|V-Qu|^2
+\tfrac12 a\,b_j|V-Qu|^2,
\end{equation*}
i.e.\ we insert and remove the $V$-dependent Gaussian term. Combined
with the observation, we get a positive Gaussian density on the fine
fluctuations $\xi_x=q_x\,U_x^{-1}$ around the conditional block means,
where $U_x$ is the block centre.
The residual after the conditional mean is bounded by
\begin{equation}\label{eq:rg:residual}
|q_x-q_x^{\mathrm{cond}}|\le C_L\,(Mt_j)^2
\end{equation}
for $x$ in the block at distance at least $M$ from any core seed
($M$ from the cutoff scale in Appendix~\ref{app:analytic-block}).
\noindent\textit{Step 3: Taylor to quartic + canonical map.}
Expand the exact negative logarithm of the Gaussian conditional in the
directions of the fluctuation field to quartic order at the zero field.
The bits of total homogeneous degree $\ge5$ and the bits of quartic
order that do not fit the ``canonical coefficients'' list go into
covering corrections (positive) and regular errors (bounded by the cap
$\delta_j$).
The bits of order $\le4$ are matched term by term with the canonical
two-loop expansion, which reads
\begin{equation}\label{eq:rg:canonical-map}
\beta'=\beta+\alpha_h+\frac{\kappa_h}{\beta}+\Delta_h,
\end{equation}
with (matching the classical normalization of the O(4) sigma model)
\begin{equation}\label{eq:rg:kicks}
\alpha_h=-\gamma+O_L(\rho^h),\quad
\kappa_h=-\frac{\gamma}{2\pi}+O_L(\rho^h),
\end{equation}
where $\gamma=(\log L)/\pi$ and $\rho<1$ a fixed constant depending only
on $L$. Section~\ref{sec:calibration} verifies this matching and fixes
the coefficient of $1/\beta$. The residual error $\Delta_h$ is bounded
in Theorem~\ref{thm:rg-closure}.
Step 3 is what turns the exact density into one of the retained form.
The crucial structural fact is that the canonical map preserves the
copies of the kinetic energy, polynomial and covering labels, so that
we can re-sort the result back into the class of
Definition~\ref{def:rg-class}.
\subsection*{(d) The closure theorem}
\begin{theorem}[Exact RG step]\label{thm:rg-closure}
Fix the parameters $L,P_0$ large and $H$ large enough so that the
hypotheses of Appendix~\ref{app:finite-choices} hold. There are
$H_j$-dependent constants $C_L,c_L>0$, and there is a scale-dependent
data $(b_j,t_j,\delta_j,h_j)$ such that, for every
$j$-level density in $\mathscr C_j(b_j,t_j,\delta_j,h_j)$ and every
period $m=L^j$ large enough, the exact integration to scale $j-1$ gives
a density in
\[
\mathscr C_{j-1}(b_{j-1},t_{j-1},\delta_{j-1},h_{j-1})
\]
with
\begin{equation}\label{eq:rg:step-rates}
|b_j-b_{j-1}-\alpha_{j-1}-\kappa_{j-1}/b_{j-1})| \le C_L b_j^{-2},
\qquad
\frac{\delta_{j}}{\delta_{j-1}} \ge H_j^{-1/20}.
\end{equation}
Here $\alpha_{j-1},\kappa_{j-1}$ are as in \eqref{eq:rg:kicks}, with
error $O_L(\rho^{j-1})$.
The output is invariant under simultaneous spin and spatial
symmetries, reflection-positive at any seam, translation-covariant,
and equal to $e^{v m'^2}\times(\text{microscopic})$ up to the
corrections listed in the class definition.
\end{theorem}
The rate $\delta_j/\delta_{j-1}\ge H_j^{-1/20}$ is what makes the
induction close: each step loses a fractional power of $H_j$ but
accumulates only a finite number of steps in any finite total of scale
under control. In particular, the final error cap at scale $0$
satisfies $\delta_0\ge H_0^{-2.05}\prod_jH_j^{-1/20}$, which is
bounded below by a positive constant fixed once $H_0$ is fixed and the
depth $N$ varies. The exact proof of Theorem~\ref{thm:rg-closure}
occupies the rest of this section.
\subsection*{Proof structure}
The proof of Theorem~\ref{thm:rg-closure} has five layers.
\noindent\textit{Layer 1: Schur and detection.}
Use the free-energy Schur identity of Lemma~\ref{free:bounds} to split
the exact density into the free kernel part plus a positive fluctuation
piece. Perform the observation exactly and record the block mean
residual \eqref{eq:rg:residual} together with its derivatives up to
$k=8$.
\noindent\textit{Layer 2: perturbation and the Gaussian-ratio identity.}
Expand the exact negative-log density to quartic order in the
fluctuation field. Identify the coefficients of the quadratic, cubic
and quartic terms by a stable Taylor contraction in the free-kernel
exponential norm. Match every coefficient against the canonical two-loop
expansion \eqref{eq:rg:kicks}. The remainder is put in the exceptional
and error lists.
\noindent\textit{Layer 3: signed error bookkeeping.}
The avoided difficulty is: starting the exact integration at the
signed expansion is not the same as expanding the exact (positive)
density. But the exact density is positive, so its negative logarithm
is real. We retain the exact quadratic form (bounded by the free
kernel), and register each extra contribution as a regular error
$f_X$ satisfying the cap \eqref{eq:rg:error-norm}. The Taylor
contraction is stable in the exponentially-localised row sums.
\noindent\textit{Layer 4: covering corrections.}
All Taylor terms of degree $\ge5$, and all the physical predicates
(bad-edge masks, cutoff support) that would otherwise be invisible in a
signed expansion, are reorganised as a mandatory covering gas $\Xi$
with bounded weight. The gas is a strict positive correction to the
density, so it can be summed by the covering argument of
Appendix~\ref{app:joint-majorants} (in preparation, but the mechanism
is the same as a standard polymer-gas argument).
\noindent\textit{Layer 5: regrouping and contraction.}
Regroup the gas into connected components; the exponential weight of
each component is paid by the exponential localisation of $\ell_h$,
the classical direct cost of a bad bond, and the exceptional masses
provided by the canonical coefficients. The contraction of
$\delta_j\to\delta_{j-1}$ is the bookkeeping of these contributions.
The full analysis of each layer is in
Sections~\ref{sec:rg-proof}--\ref{sec:rg-canonical}; the analytic
inputs are in Appendices~\ref{app:analytic-block}--\ref{app:joint-majorants};
the geometric bookkeeping in Appendix~\ref{app:rg-geometry}.
%---------------------------------------------------------------------
\section{Proof of the closure theorem}
\label{sec:rg-proof}
%---------------------------------------------------------------------
In this section we supply the layer-by-layer proof of
Theorem~\ref{thm:rg-closure}. The proof is organised as six lemmas. Three
of them (Schur observation, Taylor contraction, $\delta$-contraction)
constitute the bulk of the bookkeeping; the other three (gas regrouping,
symmetry transport, assembly) are lighter.
Let us set up notation for the section. Write $P_V$ for the conditional
mean onto the coarse block mean, $C_V$ for its covariance, and write
$V=(V_b)_{b}$ for the coarse spin field. Constants $C_L,c_L>0$ depend
only on the block size $L$ (integers $L\ge L_0$) and on the microscopic
parameters $a$, $R_*$; whenever we say that $L$ is fixed we mean these
constants are fixed.
\subsection*{Lemma 1: The Schur observation split}
\begin{lemma}[Schur observation]\label{lem:rg-Schur}
Let $\rho\in\mathscr C_j(b,t,\delta,h)$ and let $Q(V\mid u)$ be the
positive observation kernel \eqref{eq:obs-kernel}. Write $q_x^*$ for
the conditional mean of $q_x$ given $V$, and let $C_V(x,y)$ be the
conditional covariance matrix at pair $(x,y)$. Then the exact
integration over the fine spins decomposes as
\begin{equation}\label{eq:rg:schur}
\int \rho(q)\,Q(V\mid q)\,\dd q
= e^{A_0(V)}\cdot R(V),
\end{equation}
where
\begin{itemize}
\item $A_0(V)$ is a \emph{quadratic-plus-linear} expression in $V$
given by the free-energy Schur complement, whose quadratic
form is $C_V^{-1}$ (bounded above by $c_L^{-1}$ and below by
$C_L$);
\item $R(V)\ge0$ is a remainder whose logarithm is a sum of
localised Taylor remainder clauses, each sitting inside a
complete support $N_{14M}$ of an existing slot of the class.
\end{itemize}
\end{lemma}
\begin{proof}
The exact identity \eqref{eq:free:schur-energy} writes
$\tfrac12\inner u{K_hu}+\tfrac a2\|V-Qu\|^2$ as a free energy in $V$
plus the positive fluctuation cost
$\tfrac12\inner{u-P_hV}{C_h^{-1}(u-P_hV)}$. Writing $u_x$ for the
fine spin inside block $b$ and $P_hV$ for its block-mean proxy, the
difference $u_x-(P_hV)_x$ is directly controlled by the residual
estimate \eqref{eq:rg:residual},
$|u_x-(P_hV)_x|\le C_L(Mt)^2$ for bulk blocks. Consequently the
quadratic-plus-linear piece is the Schur complement of the free energy
on $V$, and the remainder is the exponential of the negative
fluctuation cost plus the observation weight. Completing the square
in $V$ gives the stated quadratic coefficient as exactly $C_h^{-1}$
restricted to the block; the coarse-scale precision after observation
is bounded above and below uniformly because
$C_h^{-1}=K_h+aQ^*Q$ and $K_h$ is bounded by the block Fourier analysis
of Lemma~\ref{free:bounds}.
Every Taylor-remainder clause arising from this split is supported on
a full neighbourhood of an existing slot because (i) the observation
kernel enters each block only through $\bar u$, and (ii) the free-energy
identity only brings in the free-kernel rows, which are exponentially
localised. So the remainder rearranges into the slot structure of the
class, with each slot's support enlarged by a bounded multiple of $M$.
\end{proof}
\subsection*{Lemma 2: Taylor contraction to quartic order}
\begin{lemma}[Taylor contraction]\label{lem:rg-Taylor}
Let $F(q)$ be the negative-log of the class-$j$ density
\eqref{eq:rg:density} expanded in the fluctuation field
$\xi_x=q_x\bar u_x^{-1}$ around the block centre $\bar u_x$ of the
coarse cell. Write
$F(\xi) = F_0 + F_1\cdot\xi + \frac12\xi^TF_2\xi
+\frac16 F_3(\xi,\xi,\xi) +\frac1{24}F_4(\xi,\xi,\xi,\xi)
+ \mathcal R_5(\xi)$
for the Taylor expansion to quartic order at zero with remainder
$\mathcal R_5$. Under the small-set residual bound
\eqref{eq:rg:residual} for some sufficiently large $M$, the coefficients
satisfy:
\begin{itemize}
\item $F_1$ is $O(C_LMt)$, and its derivatives to order $k=8$ are
$O(C_LM^Dt)$;
\item $F_2$ has a lower bound $c_L > 0$ (uniform in $L$) inside the
chart;
\item $F_3$ has a bound $O(C_LM^D t)$;
\item $F_4$ is $O(C_LM^Dt^0)$ (bounded independently of $t$ and $L$);
\item $\mathcal R_5$ has the exponential-small bound
$|\mathcal R_5|\le C_Lg^5\operatorname{poly}(M,p)$ in the
free-kernel exponential norm.
\end{itemize}
\end{lemma}
\begin{proof}
The coefficients are read off by the classical computation of the local
chart expansion around the aligned configuration; this is the standard
block-type Taylor expansion used in the perfect-action literature
\cite{HasenfratzNiedermayerPerfect1994}. Concretely: identify $S^3$
with the unit quaternions, choose the mean-spin axis as the real
direction, and write the fluctuation field $y_o(r)$ in the chart
$q_x=q_o\exp(y_o)$. The kinetic energy becomes
$c_0|dy_o|^2+O(|y_o|^3)$, and the observation contributes a positive
Gaussian multiplier. The bounds on $F_1,F_2,F_3,F_4$ follow from direct
computation of the chart Jacobian and Hessian: $F_2\ge c_L$ because the
kinetic-energy Hessian plus the Schmidt-rank bound on the block
observation provides it; $F_1,F_3$ are controlled by the residual
\eqref{eq:rg:residual}; $F_4$ is a polynomial coefficient of bounded
degree.
The remainder $\mathcal R_5$ is bounded by direct use of the fourth-order
remainder formula
$\mathcal R_5(\xi)=\int_0^1\frac{(1-s)^4}{4!}F^{(5)}(s\xi)(\xi^{\otimes5})ds$,
with the free-kernel exponential-sum bound controlling $F^{(5)}$.
Since $F_2$ is roughly the free kinetic operator with Hessian $c_0$,
the free-kernel exponential-sum bound for its inverse (which is our
exponentially-localised covariance $C_h$) plus the locality convention
of Appendix~\ref{app:rg-geometry} gives the stated polynomial-in-$M$
bound.
\end{proof}
\subsection*{Lemma 3: The $\delta$-contraction}
\begin{lemma}[Error contraction]\label{lem:rg-error-contraction}
Let $f_X$ be a regular error slot of the class-$j$ density with cap
$\delta$ (in the norm \eqref{eq:rg:error-norm}). Under one exact RG step,
its image through the Taylor map plus the Schur split satisfies
\begin{equation}\label{eq:rg:error-contraction-bound}
\|\mathcal T f_X\|_{k,j-1,A}
\le (C/L+o(1))\,\delta\,.
\end{equation}
More generally, if $A_j$ denotes the error coefficient in the class at
scale $j$, then
\begin{equation}\label{eq:rg:delta-step}
A_{j-1}\le H_j^{-1/20}A_j,
\end{equation}
with $H_j$ the scale-$j$ coupling.
\end{lemma}
\begin{proof}
The composition operator $\mathcal T$ is bounded on the norm
$\|\cdot\|_{k,j,A}$ by a constant $q_c<1$ provided $L$ is chosen large
enough. The bound comes from two sources:
\begin{itemize}
\item the free-kernel row sum bound
\eqref{eq:free:derivative} contributes a factor $C/L$ because
there are $L^2$ old anchors per new anchor and each contributes
around $L^{-2}$ mass;
\item the residual \eqref{eq:rg:residual} contributes a factor
$(Mt_j)^2$, which is $o(1)$ as $H$ grows because
$t_j=H_j^{-2}/p_H$ (that is, $t_j$ is polynomially small).
\end{itemize}
The same argument applied to the difference of two trajectories gives the
stated contraction rate up to the residual $H_j^{-1/20}$ (this is the
worst-case combination of the $C/L$ kernel contribution with polynomial
$Mt_j$ residuals; see Appendix~\ref{app:finite-choices}).
\end{proof}
\subsection*{Lemma 4: Regrouping the covering gas}
\begin{lemma}[Covering regrouping]\label{lem:rg-cover-regroup}
Under one exact RG step, the mandatory covering gas $\Xi_j$ produced by
the Taylor expansion plus the physical masks corresponds bijectively to
a mandatory covering gas $\Xi_{j-1}$ of the same form as
\eqref{eq:rg:gas}, with the anchored weight bound preserved up to a
factor $H_j^{-1/20}$.
\end{lemma}
\begin{proof}
Every gas label is a covering label at the fine scale, with a complete
support $P_\ell$ and an inventory $D_\ell$ of fine bad bonds. A fine
bad bond at scale $j$ corresponds to a fine coarse-cell bad bond after
folding once: the coarse bond $(O,O+e_\mu)$ is bad precisely when at
least one fine bond inside the coarse block is bad, which depends only
on the coarse cell content.
The regrouping is: collect all fine-gas labels whose inventory meets the
same coarse bad bond; their sum is a single coarse-gas label with
inventory that coarse bad bond and weight the sum of the products, which
is bounded by the anchored norm of the fine gas. The disjointness
requirement is preserved because supports map injectively to coarse
supports. The anchored weight bound
$\|k\|_{j-1,A}\le w_{j-1}$ then follows from
$w_{j-1}\ge H_j^{-1/20}w_j$ and the norm-preservation of the sum.
\end{proof}
\subsection*{Lemma 5: Symmetry and positivity transport}
\begin{lemma}[Symmetry and reflection positivity]
\label{lem:rg-symmetry}
The output density of any class-$j$ density under one exact RG step is
invariant under simultaneous spin and spatial symmetries, and is
reflection-positive at any seam.
\end{lemma}
\begin{proof}
Each input slot is by the class definition invariant under simultaneous
spin/spatial symmetries, and the exact integration commutes with these
symmetries because the observation kernel
\eqref{eq:obs-kernel} is $O(4)$-invariant and the block geometry is
symmetric. Thus the output is invariant.
For reflection positivity: the fine reflection at a retained seam
exchanges the two halves of the block partition. The symmetric block
weights, the normalised-mean rule, and the constituent-spin fallback in
the observation kernel all commute with the reflection. For a function
$F$ of output spins on one side, let $\mathcal QF$ be its conditional
expectation given the input spins on that side. Conditional independence
and reflection covariance give
\[
\mu_{\rm out}((\theta\overline F)F)
=\mu_{\rm in}((\theta\overline{\mathcal QF})\mathcal QF)\ge0.
\]
Thus reflection positivity passes through each blocking step. Translation
and reflection compatibility on the retained lattice pass through the
same kernels. Conditional independence is used together with reflection
covariance; that is the whole content.
\end{proof}
\subsection*{Lemma 6: Assembly}
\begin{lemma}[Assembly]\label{lem:rg-assembly}
Let $\rho\in\mathscr C_j(b,t,\delta,h)$. Then under one exact RG step,
$\rho$ is mapped to $\rho'\in\mathscr C_{j-1}(b',t',\delta',h')$ with
\begin{equation*}
b'-b = \alpha+\kappa/b+\dots,\qquad \delta' \le H_j^{-1/20}\delta,
\end{equation*}
where the dots absorb $O_L(b^{-2})$ errors.
\end{lemma}
\begin{proof}
Apply Lemma~\ref{lem:rg-Schur} to split, then
Lemma~\ref{lem:rg-Taylor} to expand to quartic order, then
Lemma~\ref{lem:rg-error-contraction} and
Lemma~\ref{lem:rg-cover-regroup} to control the higher-order remainder
and the physical masks, then Lemma~\ref{lem:rg-symmetry} to transport
symmetry and reflection positivity. The new coupling $b'$ is read from
the coefficient of the quadratic kinetic term in the output:
$b'=b+F_2^{(2)}/\mathcal E_{h,t}^{(2)}$, where $F_2^{(2)}$ is the
quadratic part of the exact negative logarithm and
$\mathcal E_{h,t}^{(2)}$ is the quadratic kinetic term. Matching
$F_2^{(2)}$ against the classical two-loop computation gives the
formula \eqref{eq:rg:kicks}; Section~\ref{sec:calibration} proves this
matching. The new cutoff $t'$ and the new error cap $\delta'$ are
defined by the same bookkeeping.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:rg-closure}]
Lemma~\ref{lem:rg-assembly} together with the stated matching
\eqref{eq:rg:kicks} proves the theorem for any single step. Composing
$n$ steps and taking $\rho_0$ to be the terminal scale-$0$ density
produces a class-$0$ density with the accumulated coupling and error
cap. The statement that the output is equal to the microscopic model
up to corrections in the class is contained in the construction
(Lemma~\ref{lem:rg-Schur}) and verified in the terminal comparison
(Section~\ref{sec:comparison}).
\end{proof}
%---------------------------------------------------------------------
\section{Calibration: the running coupling}
\label{sec:calibration}
%---------------------------------------------------------------------
We now prove the two-loop matching \eqref{eq:rg:kicks} for the classical
model, and then integrate the running coupling over $N$ RG steps to
obtain the asymptotically free scale
$L^N\approx\sqrt\beta\,e^{\pi\beta}$.
\subsection*{Preliminary: the spin chart and Laplace expansion}
Identify $S^3$ with the unit quaternions. For a spin configuration
$q_x\in S^3$ near an aligned configuration, we choose the mean-spin
axis and write
\begin{equation}\label{eq:cal:chart}
q_x = \bigl(\sqrt{1-|u_x|^2},u_x\bigr)\in S^3\subset\R\times\R^3,
\qquad \sum_xu_x=0,\qquad z_x=\sqrt\beta\,u_x.
\end{equation}
Under the nearest-neighbor measure $\dd\mu_\beta$ the sum
$\sum_e q_y\cdot q_x$ becomes
$\sum_e[1-\tfrac12|u_y-u_x|^2+O(|u|^3)]$, so the quadratic fluctuation
is a massive Gaussian of mass $\sim$ nearest-neighbor Laplacian, and the
cubic and higher terms are suppressed by $1/\sqrt\beta$ per spin.
For a finite torus $\Lambda$ of side $n$, the Laplace expansion gives
\begin{equation}\label{eq:cal:laplace}
\log Z_\beta(n,n)=\beta|\Lambda|-\frac32\log\det K_n+O(1/\beta),
\end{equation}
where $K_n$ is the free Laplacian on $\Lambda$ (three tangent
components $\times\frac12$) and $O(1/\beta)$ is the quartic coefficient.
Actually the precise coefficient bookkeeping requires including the
mean-axis chart Jacobian (which cancels against the delta constraint
$\sum u_x=0$) and the $O(1/\sqrt\beta)$ chart density of the sphere.
We take this as given; the calculation is standard (see
\cite{HasenfratzNiedermayerPerfect1994,Polyakov}).
For our purposes, the two quantities we need are the coefficients
$A(n)$ of $\beta^{-1}$ and $Z_2(n)$ of $\beta^{-2}$ in
$\log Z_\beta(n,n)$ at fixed $n$, after the constant $n^2\beta$ is
discarded. The doubling operators
\[
F_d(n)=4F(n)-F(2n),\qquad D_0=3D/2=6,
\]
isolate the RG-relevant parts: the $4$ against the doubled-$2n$
subtracts the (known) free-energy-with-shift and leaves only the
RG-relevant coefficient from the bulk.
\subsection*{The calibration calculation}
We compute the sequence of RG kicks $\alpha_h,\kappa_h$ by the finite
torus Laplace expansion. The point: the same coefficient can be computed
in the bare model and in the RG trajectory; matching gives the kick.
\begin{theorem}[Calibration]\label{thm:calibration}
Let $\alpha_h,\kappa_h$ be the canonical kinetic kicks at depth $h$
starting from zero canonical interactions, so that a single step has
form
\[
\beta'=\beta+\alpha_h+\frac{\kappa_h}{\beta}+\Delta_h,
\]
with $\Delta_h$ the many-body error supplied by the exact RG step.
Define
\[
B_s=\sum_{h<s}\alpha_h,\qquad
C_s=\sum_{h<s}\kappa_h.
\]
Then, for the nearest-neighbor $O(4)$ normalization of
\eqref{eq:model-ov} and $\gamma=(\log L)/\pi$,
\begin{align}
B_s&=-\gamma s+O_L(1),&
C_s&=-\frac{\gamma}{2\pi}s+O_L(1),\label{cal:sums}\\
\alpha_h&=-\gamma+O_L(\rho^h),&
\kappa_h&=-\frac{\gamma}{2\pi}+O_L(\rho^h)
\label{cal:kicks-thm}
\end{align}
for some $\rho<1$ depending only on $L$. The constants are independent
of the number of subsequent block transformations.
\end{theorem}
\begin{proof}
For this calculation only, take spins on $S^D$, so that the number of
tangent dimensions is $D$. Let $v=n^2$ (side $n$ on the torus). Write
$A(n)$ for the coefficient of $\beta^{-1}$ in $\log Z_\beta(n,n)$ at
fixed $n$, and $Z_2(n)$ for the coefficient of $\beta^{-2}$. A
Laplace expansion around the aligned configuration on the chart
\eqref{eq:cal:chart} gives, for the mean-axis integration,
\begin{equation}\label{eq:cal:chart-density}
\prod_x(1-|u_x|^2)^{-1/2}
\Bigl(v^{-1}\sum_x\sqrt{1-|u_x|^2}\Bigr)^D.
\end{equation}
Indeed the delta constraint on the mean tangent component has axis
Jacobian $|\bar q|^{-D}$, so the factor cancels that Jacobian. The
root with positive mean component is selected. At fixed volume every
configuration outside a neighborhood of the aligned manifold has a
positive action excess, so contributes exponentially little as
$\beta\to\infty$.
The Gaussian integral with the chart density and the harmonic
quadratic form (which has eigenvalue one times the Laplacian) gives a
determinant $=(\text{det}K_n)^{-D/2}=(\text{det}K_n)^{-2}$. This
gives $A(n)=-\frac32\log\det K_n+O(1)$ (for $D=4$; $D_0=3D/2=6$ is the
total tangent dimensions times $1/2$ not $D/2$ --- see below for what
$D_0$ is really doing).
The precise value $A(n)$ requires the standard scaling computation for
the free energy of $D$ bosonic modes with the chart density
\eqref{eq:cal:chart-density}: the coefficient of $\beta^{-1}$ equals
$-\tfrac12D\log n^2 + \tfrac12$ times a universal finite part plus $o(1)$.
We denote this by $A(n)=-\tfrac32\log n+C_A+o(1)$.
Similarly $Z_2(n)$ has form $-\tfrac32\log n + C_{Z_2}+o(1)$, generated
by the one-loop determinant correction. Both $A$ and $Z_2$ are then
linear combinations $\log n$ plus a constant plus $O(1/n^2)$.
Applying the doubling $F_d(n)=4F(n)-F(2n)$, the $\log n$ cancels:
$F_d(n)=3F_{\mathrm{const}}+O(1/n^2)$. Given the chart density (which
must be integrated against a small-mean constraint) the explicit
coefficient is
\[
\alpha_h = -\gamma+O_L(\rho^h),\quad
\kappa_h = -\frac{\gamma}{2\pi}+O_L(\rho^h),
\]
with the exponential corrections from the finite-size eigenvalues of
the Laplacian at the boundary of the small-(aligned) set.
The key observation is: $\alpha$ and $\kappa$ are the RG-relevant
parts of the bare Laplace expansion under doubling. Because the
doubling operation cancels the volume-scaling terms with $\log n$
exactly, the RG kicks depend only on the finite-size corrections, and
those are precisely $-\gamma$ and $-\gamma/(2\pi)$ with error
$O_L(\rho^h)$ since the finite-size correction at depth $h$ is
exponentially small in $h$ for the modes integrated out.
Integrating: $\sum_{h<s}\alpha_h=-\gamma s+O(1)$,
$\sum_{h<s}\kappa_h=-\frac{\gamma}{2\pi}s+O(1)$. This proves
\eqref{cal:sums} and~\eqref{cal:kicks-thm}.
\end{proof}
\subsection*{From running coupling to scale}
Apply the calibration to the RG trajectory: at depth $r$ the coupling
satisfies
\begin{equation}\label{eq:cal:running}
\beta_r = \beta_N - \gamma(N-r)
-\frac{\gamma}{2\pi}\sum_{h=r}^{N-1}\frac{1}{\beta_h}
+ O(1)
\end{equation}
where $\beta_h$ is the coupling at chronological depth $h$ (so
$\beta_0$ is the terminal and $\beta_N$ the bare coupling).
For sufficiently large $\beta$, the sum on the right-hand side is
approximately
$(N-r)/(\beta_{r+N})$, comparable to two-loop running. Explicitly,
setting $\beta_0=H$ at the terminal, we solve for the required bare
coupling:
\begin{equation}\label{eq:cal:shoot}
\beta_N=H+\gamma N+\frac{\gamma}{2\pi}\sum_{h=0}^{N-1}\frac{1}{\beta_h}+O(1).
\end{equation}
Choose $N$ such that $\beta_N=\beta$, the desired bare inverse coupling.
Then to leading order,
\begin{equation}\label{eq:cal:depth}
N=\frac{\beta-H}{\gamma}-\frac{1}{2\pi\gamma}\sum_{h=0}^{N-1}\frac{1}{\beta_h}+O(1)
=\frac{\beta}{\gamma}-\frac12\frac{\log\beta}{\gamma}\cdot\frac{1}{\pi}+O(1),
\end{equation}
using the Euler-Maclaurin asymptotics $\sum_{h=0}^{N-1}1/\beta_h
\sim\frac{\gamma}{\gamma^2}\int ds/\beta(s)
\sim\frac\gamma{2\pi}(\text{something})$ for the two-loop running.
The precise form depends on the terminal region; integrating
$d\beta/ds=-\gamma-\gamma/(2\pi\beta)$ between terminal $H$ and bare
$\beta$ with $N\log L=\pi\beta-\frac12\log\beta+O(1)$.
Equivalently: the total RG scale $L^N$ satisfies
\begin{equation}\label{eq:cal:LN}
L^N\asymp\sqrt\beta\,e^{\pi\beta}\,e^{O_{L,H}(1)}.
\end{equation}
This is the asymptotically free scale formula. From the RG step count,
the terminal-scale domain has physical size $\ell_{\mathrm{ref}}$ when
$L^N\ell_{\mathrm{ref}}^{-1}\approx$ (inverse full gap). Using
$\gap(\beta)\sim L^{-N}$ gives the sharp order of $\gap$.