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% Main local result:
%   At beta=1 and n=2, for every real coupling J,
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\title{\textbf{Oai227-A}\\
A Finite Stationary--Quench Certificate\\
for Classical SK Heat-Bath Autocorrelations\\[3pt]
\large A seven-stage audit with a fully proved two-spin instance}
\author{Ngo Cheung}
\date{14 October 2026}

\begin{document}
\maketitle

\begin{abstract}
We construct the finite certificate requested by the Family 227
audit plan without assuming its functional scaling-limit theorem.
For critical zero-field Sherrington--Kirkpatrick heat-bath dynamics
with rate-one clocks per spin, we freeze
\[
n=2,\qquad
(s,t)\in[4,5]\times[1,2],
\]
where physical waiting time and lag are \(2^{2/3}s\) and
\(2^{2/3}t\), and mean autocorrelations are multiplied by
\(2^{1/3}\). An exact four-state calculation proves, for every
real coupling, that the rescaled stationary autocorrelation
exceeds the rescaled uniform-quench autocorrelation by less than
\(1/65536\), uniformly on this compact. Consequently the same
certificate holds with probability one under both Gaussian and
Rademacher disorder, without truncating Gaussian couplings.

We distinguish this within-realization comparison from
finite-size universality: at two spins the Rademacher
autocorrelation functions are deterministic, whereas their
Gaussian counterparts are nondegenerate. We also provide
finite-state normalization checks, compact interpolation bounds,
sampling and clock-history allowances, and a rate-perturbation
ledger. The external stationary-edge, warming, entrance-selection,
and functional replacement assertions are audited but not
imported. Several ambiguities in the proposed plan are corrected,
including the difference between a common limiting law and
identical stationary and quench functions, and the existence of
an implicit semigroup representation of the proposed limit.
No thermodynamic-limit theorem, quantum SK theorem, QAOA
identification, Page curve, or gravitational conclusion is proved.
\end{abstract}

\tableofcontents

\section{Scope and the necessary debugging}
\label{sec:scope}

\subsection{The one completed next theorem}

The local submission claim is
Theorem~\ref{thm:finite-certificate}: one specified finite-size,
positive-time, uniform stationary--quench comparison.

The choice \(n=2\) is explicit. It is the smallest interacting
SK system, not a large-system approximation and not evidence
for the claimed critical exponent. The finite-size powers
\(n^{1/3}\) and \(n^{2/3}\) are retained exactly, but their use
at one size proves no scaling law.

The requested identifier Oai227-A does not assign authorship
to N.~Cheung, assert registration in another series, or claim
endorsement of this development.

\subsection{Four distinctions that the plan must preserve}

\paragraph{A common law is not equality of the two limiting functions.}
The external quench manuscript asserts convergence of a
\emph{pair} \((A_{n,J},B_{n,J})\) to a pair
\((A_g,B_g)\), with one common pair law for the two disorders.
It does not assert \(B_g(s,t)=A_g(t)\) at every finite waiting
time. Its relaxation assertion concerns \(s\to\infty\)
\cite[Section 1, \texttt{thm:main}]{OAI227Quench}.

\paragraph{Convergence in \(C_{\mathrm{loc}}\) is convergence in law.}
Uniform-on-compacts is the topology of the random-function
space in the external claim. Without a specified coupling,
it is not a pathwise supremum comparison between a Gaussian
sample and a Rademacher sample.

\paragraph{The proposed limit has an implicit representation.}
The inspected source writes \(B_g\) as a series involving an
edge measure, its semigroup, and an entrance density.
Thus ``the theorem does not write \(B_g\)'' is too strong.
The correct distinction is between such a representation and
an evaluated formula, a convergence rate, or a finite certificate
\cite{OAI227Background,OAI227Entrance}.

\paragraph{A finite certificate does not verify the infinite theorem.}
Producing the requested finite comparison neither proves nor
automatically opens verification of the external process limit.
A necessary milestone must not be read as a sufficient
verification criterion.

\subsection{What ``under both disorders'' means in the certificate}

We certify the same statement separately under each disorder:
for the same realization \(J\), stationary and quench
autocorrelations are close on the stated compact.
Our proof is stronger in this particular finite model:
it holds for every \(J\in\R\).

We do \emph{not} identify Gaussian and Rademacher finite-size
autocorrelation laws. Proposition~\ref{prop:finite-laws}
shows that those laws are different. A cross-disorder
finite-law approximation would require its own chosen metric,
tolerance, and theorem.

\subsection{Source and formalization status}

The inspected external quench manuscript is attributed to
OpenAI and dated 5 October 2026. Its stationary companion has
the same date \cite{OAI227Quench,OAI227Stationary}.

The scope note \texttt{lean/docs/227.md} describes selected
high-temperature, critical-mixing, and barrier statements;
it does not describe a formalization of the functional
autocorrelation limit \cite{Lean227}.
Moreover, the inspected Comparator targets
\texttt{SKRatio.lean} and \texttt{CriticalSKMixing.lean}
contain theorem bodies ending in \texttt{sorry}
\cite{LeanSKRatio,LeanCriticalMixing}.
They are not presented here as completed executed proofs.
No conclusion is drawn about uninspected artifacts.

The supplied Cheung passages are documentary comparisons.
Their model-specific theorems are not assumptions of the
finite SK calculation.

\section{Stage 0: freeze the classical model and the external claim}
\label{sec:freeze}

\subsection{Energy, disorder, and Gibbs law}

For \(n\ge2\), put
\[
\Omega_n=\{-1,1\}^n.
\]
For a symmetric array \(J\), with \(J_{ii}=0\), define the
classical energy
\begin{equation}
\mathcal H_{n,J}(\sigma)
=
-\frac1{\sqrt n}\sum_{i<j}J_{ij}\sigma_i\sigma_j.
\label{eq:energy}
\end{equation}
This is a real-valued function on a finite configuration space.
It is not a quantum Hamiltonian.

At inverse temperature \(\beta\), the Gibbs law is
\[
\pi_{n,J,\beta}(\sigma)
=
\frac{\exp(-\beta\mathcal H_{n,J}(\sigma))}
{\sum_{\tau\in\Omega_n}\exp(-\beta\mathcal H_{n,J}(\tau))}.
\]
Throughout the local certificate,
\[
\beta=1,\qquad \pi=\pi_{n,J,1}.
\]

The two nominated disorder laws are
\[
\DGauss=N(0,1),
\qquad
\DRad=\tfrac12\delta_{-1}+\tfrac12\delta_1,
\]
independently on the edges \(i<j\).
The normalized interaction matrix has off-diagonal entries
\(W_{ij}=J_{ij}/\sqrt n\).

These conventions are the zero-field SK normalization used
in the external claim. The original SK model is classical
\cite{SK1975,OAI227Quench}.

\subsection{Heat-bath clocks and generator}

Write \(\sigma^{i,a}\) for the configuration obtained by
setting coordinate \(i\) to \(a\). At criticality the local field is
\[
h_i(\sigma)=\frac1{\sqrt n}\sum_{j\ne i}J_{ij}\sigma_j.
\]
The site heat-bath operator is
\begin{equation}
(K_i f)(\sigma)
=
\sum_{a=\pm1}
\frac{e^{a h_i(\sigma)}}{2\cosh h_i(\sigma)}
f(\sigma^{i,a}).
\label{eq:site}
\end{equation}
Every site has an independent rate-one Poisson clock.
At a ring its spin is sampled from \eqref{eq:site}, including
the possibility that it keeps its value.

The generator and semigroup are
\begin{equation}
L_J=\sum_{i=1}^n(K_i-I),
\qquad
P_u^J=e^{uL_J}.
\label{eq:generator}
\end{equation}
Thus one physical time unit contains \(n\) update attempts
on average.

Let \(U_n\) denote the uniform law on \(\Omega_n\), independent
of \(J\). This is the quench initialization. It is not replaced
by a Gibbs sample.

\begin{lemma}[Finite chain and normalization]
\label{lem:chain}
For every finite \(J\), \eqref{eq:generator} defines a
nonexplosive irreducible continuous-time Markov chain.
Its invariant law is \(\pi\), and it is reversible.

If \(K=n^{-1}\sum_iK_i\), then
\begin{equation}
P_u^J
=
e^{-nu}\sum_{k=0}^\infty\frac{(nu)^k}{k!}K^k.
\label{eq:uniformization}
\end{equation}
In particular, \(K\) counts uniformly chosen single-site
attempts, whereas \(P_u^J\) uses rate one at each site.
\end{lemma}

\begin{proof}
Each \(K_i\) is a stochastic matrix. The exponential identity
follows from \(L_J=n(K-I)\). Its right-hand side is a
Poisson mixture of stochastic matrices, so it defines the
chain with rate-\(n\) attempts and uniformly chosen sites.
The number of attempts on a bounded interval is finite
almost surely.

Each conditional spin probability is strictly positive when
\(J\) is finite. Single-coordinate changes connect the cube,
which proves irreducibility. The operator \(K_i\) is conditional
expectation under \(\pi\), given all coordinates except \(i\).
Therefore it is self-adjoint on \(L^2(\pi)\), preserves
\(\pi\), and satisfies detailed balance. Summing gives
reversibility of \(L_J\) and \(P_u^J\).
\end{proof}

This is the usual finite-state heat-bath construction;
standard Markov-chain conventions are discussed in
\cite[Chapters 3 and 20]{LevinPeres}.

\subsection{The two autocorrelations and the averaging convention}

Let \(f_i(\sigma)=\sigma_i\). For physical waiting time
\(w\ge0\) and lag \(\ell\ge0\), define
\begin{align}
C_{n,J}^{\mathrm{stat}}(\ell)
&=
\frac1n\sum_{i=1}^n
\pi[f_iP_\ell^J f_i],
\notag\\
C_{n,J}^{\mathrm{q}}(w,\ell)
&=
\frac1n\sum_{i=1}^n
\E_{U_n}^J[
\sigma_i(w)\sigma_i(w+\ell)].
\label{eq:physical-correlation}
\end{align}
These are means of individual-spin autocorrelations, not
autocorrelations of the total magnetization.

Put
\[
v_n=n^{1/3},\qquad T_n=n^{2/3},
\]
and define
\begin{equation}
A_{n,J}(t)
=
v_n C_{n,J}^{\mathrm{stat}}(T_nt),
\qquad
B_{n,J}(s,t)
=
v_n C_{n,J}^{\mathrm q}(T_ns,T_nt).
\label{eq:scaled}
\end{equation}

Initial spins, clocks, and heat-bath resampling are averaged
in these functions. Disorder is not averaged.
Consequently \(A_{n,J}\) and \(B_{n,J}\) are random functions
only when \(J\) is sampled.

\subsection{The frozen external assertion}

\begin{claimrecord}[Family 227-01]
\label{record:main}
The external claim is that
\[
\Law_D(A_{n,J},B_{n,J})
\Longrightarrow
\Law_g(A_g,B_g),
\qquad D\in\{\DGauss,\DRad\},
\]
in
\[
C_{\mathrm{loc}}((0,\infty))
\times C_{\mathrm{loc}}((0,\infty)^2),
\]
with the same limiting pair law for the two disorders.
The stationary limiting law is not a point mass.

The quench starts from \(U_n\). Its selected entrance family
has positive-time probability densities \(h_s^g\), satisfies
a semigroup relation, and relaxes to equilibrium. In
particular, the stated conclusion includes
\[
\sup_{a\le t\le b}|B_g(s,t)-A_g(t)|
\longrightarrow0
\]
in probability over \(g\) as \(s\to\infty\), for
\(0<a<b<\infty\)
\cite{OAI227Quench,OAI227Stationary}.
\end{claimrecord}

This record is not a premise of any local theorem below.

\paragraph{Related records.}
The family also records a high-temperature gap and cutoff,
critical slowing, low-temperature barriers, and a critical
mixing exponent. Their statements concern different
quantities or temperature regimes \cite{Lean227,Catalogue}.
They are not premises of our finite certificate.
This does not assert that the external quench manuscript
has no dependencies on its companions: its own source
explicitly cites stationary, spectral, and mixing inputs
\cite{OAI227Background,OAI227Edge,OAI227Warming}.

\section{Stage 1: finite identities and the five-stage audit}
\label{sec:audit}

\subsection{Locally proved identities that an audit may use}

\begin{lemma}[Dirichlet form and spin action]
\label{lem:dirichlet}
For real functions on the cube,
\[
-\ip{f}{L_J f}_\pi
=
\sum_{i=1}^n\pi[
\Var_\pi(f\mid \sigma_{-i})].
\]
Also,
\begin{equation}
L_J f_i=-f_i+\tanh h_i.
\label{eq:spin-action}
\end{equation}
The invariant law and both initialized processes are invariant
under global spin reversal; in particular their one-spin means
are zero.
\end{lemma}

\begin{proof}
Each \(K_i\) is an orthogonal conditional-expectation projection
on \(L^2(\pi)\). Hence
\[
\ip{f}{(I-K_i)f}_\pi
=
\pi[f^2]-\pi[(K_if)^2]
=
\pi[\Var_\pi(f\mid\sigma_{-i})].
\]
Only an update at site \(i\) changes \(f_i\), and its conditional
new mean is \(\tanh h_i\), proving \eqref{eq:spin-action}.
Both the energy and the update rule respect global reversal.
So do \(\pi\) and \(U_n\).
\end{proof}

\begin{lemma}[The exact spectral trace is a dynamical trace]
\label{lem:spectral}
Let \((\lambda_\alpha,\psi_\alpha)\) be an orthonormal
eigensystem of \(-L_J\) on \(L^2(\pi)\), with
\(\lambda_0=0\) and \(\psi_0=1\). Then
\begin{equation}
A_{n,J}(t)
=
n^{-2/3}
\sum_{\alpha\ge1}e^{-\lambda_\alpha T_nt}
\sum_{i=1}^n\ip{f_i}{\psi_\alpha}_\pi^2.
\label{eq:dynamical-trace}
\end{equation}
The weights are nonnegative. The eigenvalues here are those
of the heat-bath generator, not those of the \(n\)-by-\(n\)
interaction matrix.
\end{lemma}

\begin{proof}
Expand each \(f_i\) in the reversible eigenbasis.
Its constant coefficient vanishes by
Lemma~\ref{lem:dirichlet}. Apply \(P_{T_nt}^J\), take its
inner product with \(f_i\), and sum.
\end{proof}

\begin{lemma}[Replica-coordinate identity and finite entrance density]
\label{lem:replica}
For any orthonormal basis \(u_1,\ldots,u_n\) of \(\R^n\), set
\[
x_a(\sigma)=n^{-1/3}u_a\cdot\sigma.
\]
Then
\[
n^{-2/3}\sigma\cdot\tau
=
\sum_{a=1}^n x_a(\sigma)x_a(\tau).
\]
If
\[
w_s=\frac{d(U_nP_{sT_n}^J)}{d\pi},
\]
then
\begin{align}
A_{n,J}(t)&=\sum_a\pi[x_aP_{tT_n}^Jx_a],
\notag\\
B_{n,J}(s,t)&=\sum_a\pi[w_sx_aP_{tT_n}^Jx_a].
\label{eq:finite-entrance}
\end{align}
Moreover \(w_{s+t}=P_{tT_n}^Jw_s\).
\end{lemma}

\begin{proof}
The first identity is orthogonality. Apply it to the two
configurations in a two-time pair and condition on the first.
Since \(\pi\) has full support, the density exists. Reversibility
identifies the evolution of a density relative to \(\pi\)
with \(P_u^J\), proving the last assertion.
\end{proof}

This finite identity does not identify any limiting edge
measure or establish tightness.

\subsection{Audit 1: stationary edge input}

For Gaussian disorder the external route completes \(W\) by
an independent Gaussian diagonal. The added diagonal changes
the spin energy by a configuration-independent constant,
because \(\sigma_i^2=1\). This completion can supply Gaussian
orthogonal eigenframe structure without changing the finite
heat-bath dynamics \cite{OAI227Background}.

The audit must nevertheless keep three objects separate:
\begin{enumerate}
\item the spectral edge of \(W\);
\item the projected spin coordinates in
\eqref{eq:finite-entrance};
\item the spectrum and semigroup of the nonlinear heat-bath
generator in \eqref{eq:dynamical-trace}.
\end{enumerate}
Equation~\eqref{eq:spin-action} shows why diagonalizing \(W\)
does not generally diagonalize \(L_J\).

The stationary edge law, its conditioned measure, convergence
of finite-coordinate integrals, and the limiting energy form
are substantive external inputs. Even if all of them are
supplied, the extra factor \(w_s\) in the quench trace still
requires control. Stationary convergence is not the quench
theorem.

\subsection{Audit 2: Gaussian estimates at the smallest edge scales}

The external Gaussian argument uses scales with
\(k_p=np^3\), capped resolvents, and a saddle equation.
A representative variance assertion is
\[
V_p(f)\le Cp^{-2}\mathcal E(f)+Ce^{-c k_p^\alpha},
\qquad 0\le f\le1,
\]
on stated disorder events and scale ranges
\cite[\texttt{prop:edge-estimates}]{OAI227Edge}.

A complete verification would have to check:
\begin{enumerate}
\item the saddle's existence, uniqueness, and positive denominators;
\item uniformity over every admitted field, function, and scale;
\item how a threshold depending on a requested disorder confidence
is chosen;
\item the integrability of accumulated relative errors across scales;
\item the transfer between spherical and cube quantities with
their stated reference measures.
\end{enumerate}

A threshold sent to infinity after a limsup in \(n\) is not
automatically a fixed-threshold probability tending to one.
Nor is a static resolvent estimate a heat-bath mixing theorem.
These Gaussian estimates are not reproved or assumed here.

\subsection{Audit 3: warming in the \(n^{2/3}\) window}

The external warming statement concerns positive-time Gibbs-relative
densities, entropy, and heat kernels at times \(sT_n\). Its constants
are controlled on disorder events chosen to meet a prescribed
confidence, and its stronger kernel bound begins beyond a
deterministic rescaled time tending to zero
\cite[\texttt{thm:warming}]{OAI227Warming}.

The audit must distinguish:
\[
\text{finite positive-time density bounds}
\quad\text{from}\quad
\text{distance to the equilibrium density }1.
\]
A bound by a finite function of \(s\) does not itself prove that
the distance to equilibrium tends to zero.

The time unit \(T_n=n^{2/3}\) is dynamical bookkeeping. No
positions, spatial propagation radius, or metric are specified
by it.

\subsection{Audit 4: selection of the quench entrance}

The source's entrance-selection condition is stronger than
merely requiring a zero-coordinate product-topology boundary.
It includes local positive-time density bounds, the edge
semigroup relation, and a sequential high-mode escape condition
inherited from the uniform initial states
\cite[\texttt{thm:entrance-selection}]{OAI227Entrance}.

The verification questions are:
\begin{enumerate}
\item Do all extracted Gaussian subsequences satisfy the stated
escape condition?
\item Does uniqueness hold in the exact density class used?
\item Is the selected family measurable in the edge environment?
\item Are the trace tails controlled when the density and
semigroup limits are inserted into the infinite sum?
\end{enumerate}
A possible semigroup entrance is not enough; it must be the
entrance selected by the finite uniform starts.

No conclusion at \(\beta\ne1\) is substituted for this selection.

\subsection{Audit 5: replacement and functional universality}

Gaussian and Rademacher couplings match their first three
moments. The fourth moments differ. In a fourth-order
replacement calculation, normalization contributes \(n^{-2}\)
while the number of edges is of order \(n^2\). These powers
alone give only order one, not an error tending to zero.

The source therefore uses clock-history cutoffs, projected
scores, derivative estimates including cutoff derivatives,
and a functional tightness argument
\cite{OAI227History,OAI227Scores,OAI227Replacement}.
The general replacement methodology has a separate literature
\cite{Chatterjee}.

A full audit must verify both removal of the cutoffs and
control between time-grid points. Comparison of finitely many
evaluations alone does not prove convergence in
\(C_{\mathrm{loc}}\).

Universality in the frozen statement concerns the two named
disorders. It does not assert universality for every
mean-zero disorder distribution.

\subsection{A mixing bound does not close these gates}

\begin{lemma}[Mixing-to-correlation implication, with its scale]
\label{lem:mixing-comparison}
For fixed \(n,J\), define
\[
G_\ell(\sigma)=\frac1n\sum_i f_i(\sigma)(P_\ell^Jf_i)(\sigma).
\]
Then \(\norm{G_\ell}_\infty\le1\), and
\begin{equation}
|B_{n,J}(s,t)-A_{n,J}(t)|
\le
2v_n\,\TV(U_nP_{T_ns}^J,\pi).
\label{eq:tv-comparison}
\end{equation}
\end{lemma}

\begin{proof}
The semigroup sends functions bounded by one to functions
bounded by one. The two autocorrelations are \(v_n\) times
the expectation of \(G_{T_nt}\) under the two displayed laws.
The total-variation expectation bound gives the result.
\end{proof}

At a fixed total-variation threshold such as \(1/4\), the
right-hand side of \eqref{eq:tv-comparison} grows as
\(n^{1/3}\). Thus a mixing-time bound at that threshold does
not supply a rescaled correlation process limit. It also
does not identify a nondegenerate limiting law.

\section{Stage 2: price only specified finite operations}
\label{sec:pricing}

\subsection{Uniform finite-size interpolation}

\begin{lemma}[Deterministic compact moduli]
\label{lem:modulus}
For every \(n,J\), the finite autocorrelation functions are
smooth. Uniformly over their nonnegative time arguments,
\begin{align}
|\partial_t A_{n,J}(t)|&\le2n,
&
|\partial_t B_{n,J}(s,t)|&\le2n,
\notag\\
|\partial_s B_{n,J}(s,t)|&\le2n^2.
\label{eq:modulus}
\end{align}
For
\(\Delta_{n,J}(s,t)=A_{n,J}(t)-B_{n,J}(s,t)\),
\[
|\partial_s\Delta_{n,J}|\le2n^2,
\qquad
|\partial_t\Delta_{n,J}|\le4n.
\]
\end{lemma}

\begin{proof}
Finite matrix exponentials are smooth.
By \eqref{eq:spin-action},
\(\norm{L_Jf_i}_\infty\le2\). Since the semigroup is a
supremum-norm contraction and commutes with its generator,
differentiating a lag contributes at most
\[
n^{-2/3}\,n\,T_n\,2=2n.
\]
For the waiting derivative use
\[
B_{n,J}(s,t)=v_nU_nP_{T_ns}^JG_{T_nt}.
\]
Every \(G_\ell\) is bounded by one, and
\(\norm{L_JG_\ell}_\infty\le2n\). Thus the derivative is at
most \(v_nT_n2n=2n^2\).
Subtracting the two lag derivatives proves the final bounds.
\end{proof}

This is a finite-\(n\) modulus. It grows with \(n\), so it
does not by itself prove asymptotic equicontinuity.

\subsection{Complete-record sampling and finite coverage}

\begin{specification}[Independent trajectory repetitions]
\label{spec:sampling}
Fix \(n,J\), a finite time grid, and the intended initialization.
At each grid point use independent repeated trajectories.
For stationary data the initial law is \(\pi\); for quench
data it is \(U_n\). Every attempted repetition is retained.
Any implementation failure receives its predeclared score
and flag. No successful-run conditioning is substituted.
\end{specification}

For a two-time trajectory pair, its scaled score is
\[
Y=n^{-2/3}\sum_i\sigma_i(w)\sigma_i(w+\ell),
\qquad -v_n\le Y\le v_n.
\]

\begin{proposition}[Finite-grid coverage and compact extension]
\label{prop:coverage}
Suppose a compact rectangle has a finite grid \(\Gamma\) of
\(M\) points such that each point of the rectangle has a grid
point within \(h\) in each coordinate. At each grid point
estimate the stationary and quench quantities with \(N\)
repetitions as in Specification~\ref{spec:sampling}.

With probability at least
\[
1-4M\exp\!\left(-\frac{Nr^2}{2v_n^2}\right),
\]
all the two types of mean estimates have errors at most \(r\).
On this event,
\begin{equation}
\sup|\Delta_{n,J}|
\le
\max_{\Gamma}|\widehat\Delta_{n,J}|
+2r+(2n^2+4n)h.
\label{eq:grid-certificate}
\end{equation}
A certified score-expectation bias \(b\) for each of the two
protocols adds \(2b\). A calibration-failure probability
\(\eta\) adds \(\eta\) to the total failure probability.
\end{proposition}

\begin{proof}
A score has range length \(2v_n\). Hoeffding's inequality
gives
\[
\Prob\{|\overline Y-\E Y|>r\}
\le2\exp(-Nr^2/(2v_n^2)).
\]
Apply a union bound to \(2M\) estimates.
Their difference has error at most \(2r\).
Lemma~\ref{lem:modulus} extends the bound from the grid to
the rectangle. Add the deterministic biases and calibration
failure by the triangle inequality and a union bound
\cite{Hoeffding}.
\end{proof}

The independence is across repetitions, not across the spins
within one trajectory. Correlated spin coordinates have not
been counted as independent samples.

This is the legitimate comparison with Cheung's bounded,
complete-record statistical certificates
\cite{CheungEn01B,CheungEn01C}. A confidence interval alone
does not give functional convergence. Equation~\eqref{eq:grid-certificate}
obtains finite-compact coverage only because a deterministic
modulus has also been proved.

\subsection{Clock histories and their actual time budget}

For an observation up to physical time \(u\), the attempted
update count is Poisson with mean \(nu\).
For the quench pair \((s,t)\), that mean is
\[
nT_n(s+t)=n^{5/3}(s+t).
\]
Clock attempts include updates that leave the spin unchanged.
Counting flips instead would change the history law.

In the final certificate, \(n=2\), \(s+t\le7\), and
\[
T=2^{2/3}<\frac85.
\]
Hence the maximum quench history has mean less than
\[
2\cdot7\cdot\frac85=\frac{112}{5}.
\]

\begin{lemma}[A finite clock-history allowance]
\label{lem:clock-tail}
For the frozen compact used in
Theorem~\ref{thm:finite-certificate}, the probability that
a trajectory needs at least \(144\) update attempts is less
than \(2^{-144}\), uniformly over every coupling.

If such a history is flagged and assigned score zero, rather
than deleted, the absolute bias of either scaled mean score
is at most \(2^{1/3}2^{-144}\).
\end{lemma}

\begin{proof}
If \(N\) is Poisson with mean \(\mu\), then
\[
\Prob\{N\ge m\}
\le \E\binom Nm=\frac{\mu^m}{m!}.
\]
Also
\[
\log(m!)\ge\int_1^m\log x\dd x
=m\log m-m+1,
\]
so \(m!\ge(m/e)^m\).
For \(m=144\), \(\mu<112/5\), and \(e<3\),
\[
\frac{e\mu}{144}
<
\frac{3(112/5)}{144}
=\frac7{15}<\frac12.
\]
This proves the tail bound. On the flagged event, changing a
score to zero changes it by at most \(2^{1/3}\). Take expectations.
\end{proof}

The proof is a clock ledger, not a sample execution.
Exact draws from a real Gaussian law are mathematical inputs;
no finite-bit exact Gaussian sampler or machine running time
is claimed.

\subsection{A generator allowance, not an imported diamond budget}

\begin{proposition}[Finite generator perturbation]
\label{prop:perturbation}
Let \(Q,\widehat Q\) be Markov generators on the same finite
state space, and suppose
\[
\delta=
\max_x\frac12\sum_y|Q(x,y)-\widehat Q(x,y)|.
\]
If initial laws have total-variation distance at most
\(\eta\), their one-time laws at time \(u\) differ by at most
\(\eta+u\delta\).

Their joint laws at times \(w,w+\ell\) differ by at most
\[
\eta+(w+\ell)\delta.
\]
A score in \([-v_n,v_n]\) consequently has expectation
difference at most
\[
2v_n\bigl(\eta+(w+\ell)\delta\bigr),
\]
or the trivial bound \(2v_n\), whichever is smaller.
\end{proposition}

\begin{proof}
Duhamel's formula expresses the semigroup difference as the
integral of
\[
e^{rQ}(Q-\widehat Q)e^{(u-r)\widehat Q}.
\]
Stochastic multiplication contracts the total variation of
signed measures, while multiplication by \(Q-\widehat Q\)
costs at most \(\delta\) for a probability input.
Integration and the initial-law comparison give the first bound.

For a two-time joint law, first compare the law at time \(w\),
then the conditional transition over the remaining lag.
The two allowances add. The bounded-score expectation
inequality gives the final assertion.
\end{proof}

A nominated quantum-channel distance budget cannot specify
the classical generator by itself. Proposition~\ref{prop:perturbation}
states the actual classical comparison interface needed here.

\subsection{Disorder randomness is not trajectory sampling error}

\begin{lemma}[Two sources of randomness]
\label{lem:variance}
Sample \(J\) once, and conditionally on \(J\) sample \(N\)
independent scores \(Y_1,\ldots,Y_N\) with conditional mean
\(m(J)\). Then
\[
\Var\!\left(\frac1N\sum_{k=1}^NY_k\right)
=
\frac1N\E[\Var(Y_1\mid J)]
+\Var(m(J)).
\]
\end{lemma}

\begin{proof}
Apply conditional independence and the law of total variance.
\end{proof}

Increasing the number of trajectories removes the first
contribution, not sample-to-sample disorder variation.
Resampling disorder on each repetition instead estimates an
annealed quantity and changes the experiment.

\subsection{Debugged template ledger}

\begin{longtable}{@{}p{.24\linewidth}p{.34\linewidth}p{.34\linewidth}@{}}
\toprule
Template & Legitimate comparison & Boundary or correction\\
\midrule
\endfirsthead
\toprule
Template & Legitimate comparison & Boundary or correction\\
\midrule
\endhead
\bottomrule
\endfoot
Em04-A,
\texttt{thm:reduction} &
A specified XY-chain evolution with vacuum and one-excitation
blocks. &
Its assumed quantum dissipative generator is not
\eqref{eq:generator}.\\[5pt]

Hawk02-D / Hawk02-E &
Complete residual recovery in specified fermionic or regulated
interacting models. &
Neither gives a classical SK heat-bath reduction.
Hawk02-E explicitly warns against this transfer.\\[5pt]

En07-C &
Thermal input and complete-complement distinctions. &
The supplied \texttt{thm:exact} is a quantum recovery
criterion, not an SK-temperature theorem.\\[5pt]

Landmark I; VIII-B &
Destructive quantum readout semantics and record-only
quantum recovery limitations. &
The supplied quantitative
\texttt{prop:classical-recovery} passage is in VIII-B,
not a proof of an SK autocorrelation assertion.\\[5pt]

En01-B / En01-C &
Bounded scores, complete records, and calibration/sample
allowances. &
Not asymptotic tightness or functional convergence.\\[5pt]

Em03-G,
\texttt{thm:lower} &
A positive recovery-error lower bound at every fixed finite
source parameter. &
The possibility of vanishing error with growing resources
uses separate results, not this lower bound alone.\\[5pt]

Oai265-A,
\texttt{thm:separation} &
Static records do not determine channels or histories. &
Not a dynamical SK process.\\[5pt]

Oai272-A &
A nominated secret-bit comparison in the supplied dossier. &
No secret-key result is imported as a spin-memory theorem.\\[5pt]

Oai283-A &
A nominated implementation-distance ledger. &
No unsupplied \texttt{thm:assembly} is reconstructed from its
label; no distance budget constructs the heat-bath rates.\\
\end{longtable}

The Cheung citations support these scoped comparisons, not
premise transfers
\cite{CheungEm04A,CheungHawk02D,CheungHawk02E,
CheungEn07C,CheungI,CheungVIIIB,CheungEm03G,CheungOai265A}.
The Oai272-A and Oai283-A nominations remain dossier-level
comparisons \cite{Dossier}.

\section{Stage 3: an implicit limit is not a finite evaluation}
\label{sec:formula}

The inspected source describes a conditioned edge measure
\(\Pi_g\), a Markov semigroup \(S_t^g\), and entrance densities
\(h_s^g\). In that notation it represents the proposed quench
limit by
\[
B_g(s,t)
=
\sum_{a\ge1}\int
h_s^g(x)x_a(S_t^gx_a)(x)\dd\Pi_g(x),
\]
and the stationary function by the corresponding expression
with density one
\cite{OAI227Background,OAI227Entrance}.

This is a substantive representation, not an absent formula.
However, evaluating it requires the edge environment, the
conditioned measure, the semigroup, the selected entrance,
and control of its infinite tail.
A representation of this kind is not:
\begin{itemize}
\item a closed elementary formula for the random function;
\item a computable error bound at a chosen finite \(n\);
\item a convergence rate in the random-function topology;
\item a finite comparison of stationary and quench curves.
\end{itemize}

The local theorem in Section~\ref{sec:certificate} supplies the
last item for one small finite system. It supplies none of
the missing infinite-system verification.

\begin{example}[A finite certificate cannot establish a limit]
Fix any finite set of system sizes. Two sequences of continuous
functions can agree at every one of those sizes, while one
converges and the other alternates forever between the
constant functions zero and one at later sizes.
Thus even an exact finite certificate cannot logically verify
a claimed limiting law without additional uniform estimates.
\end{example}

\section{Stage 4: change one hypothesis at a time}
\label{sec:hypotheses}

\subsection{Temperature and clock changes}

The family's high-temperature cutoff and low-temperature
barrier claims are retained as external records, not proved
by the present calculation \cite{Lean227,Catalogue}.
Neither regime is substituted for \(\beta=1\).

Changing from rate-one-per-site clocks to a single total
rate-one clock divides the generator by \(n\). The same
physical evolution would then require \(n\) times as many
time units. At the critical scaling this changes
\(n^{2/3}\) physical units into \(n^{5/3}\) single-attempt
units.

Holding at an attempted update is part of the heat-bath
model. It is not silently removed.

\subsection{Initialization, observables, and convergence}

\begin{longtable}{@{}p{.27\linewidth}p{.35\linewidth}p{.30\linewidth}@{}}
\toprule
Proposed change & Mathematical effect & Disposition\\
\midrule
\endfirsthead
\toprule
Proposed change & Mathematical effect & Disposition\\
\midrule
\endhead
\bottomrule
\endfoot
Replace \(U_n\) by \(\pi\) &
The entrance density becomes one from the start. &
This removes the quench-selection problem.\\[5pt]

Keep only stationary convergence &
One loses the waiting-time variable and uniform-start
selection. &
Closer to the stationary companion, not all of 227-01.\\[5pt]

Drop either named disorder &
One no longer compares the two coupling laws. &
The universality clause has been weakened.\\[5pt]

Replace the mean of spin autocorrelations by magnetization
autocorrelation &
Cross-spin terms are introduced. &
A different observable requires its own argument.\\[5pt]

Average disorder before taking the process law &
Random conditional means are replaced by annealed averages. &
The claimed non-point limiting law is no longer the same object.\\[5pt]

Use a mixing-time exponent &
Only a distance-to-equilibrium timescale is controlled. &
No entrance law or random autocorrelation limit follows.\\[5pt]

Use a finite time grid only &
Between-grid oscillation is uncontrolled without a modulus. &
Finite-dimensional information is not automatically functional.\\[5pt]

Use the finite modulus in Lemma~\ref{lem:modulus} at growing \(n\) &
Its constants grow polynomially with \(n\). &
It supplies no uniform asymptotic tightness by itself.\\
\end{longtable}

\subsection{QAOA and quantum SK are different dynamical problems}

A diagonal quantum cost operator built from the classical
energy is
\[
\widehat{\mathcal H}_{n,J}
=
-\frac1{\sqrt n}\sum_{i<j}J_{ij}Z_iZ_j.
\]
It commutes with each \(Z_i\). Consequently unitary evolution
under this operator leaves every \(Z_i\) unchanged.

By contrast,
\[
\left.
\frac{d}{d\ell}C_{n,J}^{\mathrm{stat}}(\ell)
\right|_{\ell=0}
=
-\frac1n\sum_i\pi[1-\tanh^2 h_i]<0.
\]
The strict inequality holds for every finite coupling array.
Thus even this elementary diagonal quantum evolution is not
the heat-bath dynamics.

QAOA adds prescribed unitary mixing layers and optimizes an
energy expectation as a function of its parameters and depth.
It does not install independent heat-bath clocks
\cite{FarhiEtAl}.
The dossier's Family 281 order-of-limits energy statement is
therefore only a catalogue comparison \cite{Catalogue,Dossier}.

Likewise, a Haldane quantum spectral gap, Anderson localization,
and an area-law theorem concern different operators and
questions. None identifies the present Markov generator.

\section{Stage 5: recovery and gravitational interpretations remain separate}
\label{sec:separations}

\subsection{An autocorrelation is not a quantum recovery channel}

A classical probability law \(p(\sigma)\) can be embedded
diagonally as
\[
\rho_p
=
\sum_{\sigma\in\Omega_n}
p(\sigma)\bigotimes_{i=1}^n\proj{\sigma_i}.
\]
This is a convex combination of product states and is therefore
separable across every spin partition. Classical spin
correlations do not by themselves certify quantum entanglement.

A record-only quantum channel is measure-and-prepare and has
the corresponding entanglement-breaking limitation
\cite{CheungVIIIB,CheungHawk04B}.
That fact does not prevent a classical spin process from
having long temporal correlations. It is a statement about
unknown-state quantum recovery, not about the value of
\eqref{eq:physical-correlation}.

Our probability model uses classical trajectory observations.
It does not insert a destructive quantum measurement into
the heat-bath generator. If an implementation uses Enter for
quantum-encoded records, its stated consumption rules remain
unchanged; no surviving measured quantum wire is inferred
\cite{CheungI}.

\subsection{A clock is not a metric}

The clock normalization specifies attempted updates per unit
time. The SK coupling graph is all-to-all and has no
independently declared spatial embedding here.

Even the invariant law does not determine time normalization:
\(L_J\) and \(cL_J\), \(c>0\), have the same Gibbs law but
different autocorrelation functions at the same numerical
time. A static record therefore cannot supply a temporal
process without further data. This is the appropriate
methodological comparison with Oai265-A
\cite{CheungOai265A}.

Scheduled ancestry, operational influence, and a separately
supplied geometric causal relation also remain different
objects \cite{CheungXXIIA}. No geometric propagation law is
derived from \(T_n=n^{2/3}\).

\subsection{No gravitational dictionary is present}

The present model specifies a finite configuration space,
a classical Gibbs law, and stochastic update rates. It does
not specify a spacetime metric, horizon, radiation algebra,
evaporation dynamics, gravitational mediator, or island
extremization problem.

Accordingly, neither a stationary autocorrelation nor a
uniform-quench two-time correlation is a Page curve, an
island, a horizon, or Hawking radiation. No gravity-generated
entanglement or exclusion of classical gravity follows.

The external thermal and residual-recovery comparisons
explicitly require their own model identifications
\cite{CheungEn07C,CheungHawk02D,CheungHawk02E}.
They supply no such identification for classical SK.

\section{Stage 6: the one explicit finite certificate}
\label{sec:certificate}

\subsection{Freeze the complete finite instance}

We choose
\begin{equation}
n=2,\qquad
\beta=1,\qquad
v=2^{1/3},\qquad
T=2^{2/3},
\qquad
\Kcert=[4,5]\times[1,2],
\qquad
\epscert=\frac1{65536}.
\label{eq:certificate-data}
\end{equation}
The single coupling \(J=J_{12}\) is arbitrary in \(\R\).
Write
\[
\kappa=\frac{J}{\sqrt2},
\qquad
a=\tanh\kappa,
\qquad
b=\frac{1-a}{2},
\qquad
c=\frac{1+a}{2}.
\]
In particular, \(-1<a<1\).

\begin{lemma}[The exact four-state generator]
\label{lem:two-generator}
In the ordered state space
\[
(++),(+-),(-+),(--),
\]
the heat-bath generator and invariant row vector are
\begin{equation}
Q_a=
\begin{pmatrix}
-2b&b&b&0\\
c&-2c&0&c\\
c&0&-2c&c\\
0&b&b&-2b
\end{pmatrix},
\qquad
\pi_a=\frac14(1+a,1-a,1-a,1+a).
\label{eq:two-generator}
\end{equation}
For \(z(\sigma)=\sigma_1\sigma_2\),
\begin{align}
L f_1&=-f_1+af_2,
&
L f_2&=-f_2+af_1,
\notag\\
Lz&=-2z+2a.
\label{eq:two-actions}
\end{align}
\end{lemma}

\begin{proof}
A flip at either site has rate
\[
\frac{1-a\sigma_1\sigma_2}{2}.
\]
It is \(b\) in aligned states and \(c\) in anti-aligned
states, which gives \(Q_a\). The Gibbs weights are
\(e^\kappa\) in aligned states and \(e^{-\kappa}\) otherwise.
Dividing by \(4\cosh\kappa\) gives \(\pi_a\).

The spin actions follow from \eqref{eq:spin-action} because
\(\tanh(\kappa\sigma_j)=a\sigma_j\).
At an update of site one, the conditional mean of the new
product is \(a\), and the same holds at site two.
Therefore \(Lz=2(a-z)\).
\end{proof}

\begin{lemma}[Exact stationary and quench functions]
\label{lem:two-functions}
For physical waiting time \(w\ge0\) and lag \(\ell\ge0\),
\begin{align}
C_{2,J}^{\mathrm{stat}}(\ell)
&=
e^{-\ell}
\left[\cosh(a\ell)+a\sinh(a\ell)\right],
\label{eq:two-stat}\\
C_{2,J}^{\mathrm q}(w,\ell)
&=
e^{-\ell}
\left[
\cosh(a\ell)
+a(1-e^{-2w})\sinh(a\ell)
\right].
\label{eq:two-quench}
\end{align}
Consequently,
\begin{equation}
A_{2,J}(t)-B_{2,J}(s,t)
=
v\,a\,e^{-2Ts-Tt}\sinh(aTt).
\label{eq:exact-difference}
\end{equation}
\end{lemma}

\begin{proof}
The first two equations in \eqref{eq:two-actions} form a
closed two-dimensional linear system. Thus
\[
P_\ell f_1
=
e^{-\ell}
\left[\cosh(a\ell)f_1+\sinh(a\ell)f_2\right],
\]
with the indices interchanged for \(P_\ell f_2\).

Under the uniform start, \(m(w)=\E[z(\sigma(w))]\) solves
\[
m'(w)=-2m(w)+2a,\qquad m(0)=0.
\]
Hence \(m(w)=a(1-e^{-2w})\). Under the invariant law,
\(\pi_a[z]=a\).

Condition on the configuration at waiting time \(w\), multiply
the displayed semigroup formula by the corresponding initial
spin, and average. Both coordinates give the same answer.
This proves \eqref{eq:two-stat} and \eqref{eq:two-quench}.
Insert \(w=Ts\), \(\ell=Tt\), multiply by \(v\), and subtract.
\end{proof}

\begin{theorem}[Oai227-A finite stationary--quench certificate]
\label{thm:finite-certificate}
For the data in \eqref{eq:certificate-data}, every real
coupling \(J\) satisfies
\begin{equation}
0
\le
A_{2,J}(t)-B_{2,J}(s,t)
<
\frac1{98304}
<
\epscert
\qquad
((s,t)\in\Kcert).
\label{eq:main-certificate}
\end{equation}
In particular,
\[
\sup_{(s,t)\in\Kcert}
|A_{2,J}(t)-B_{2,J}(s,t)|
<
\frac1{65536}.
\]

The two functions are explicitly given by
\eqref{eq:two-stat}--\eqref{eq:two-quench} in the frozen units.
Under each of the Gaussian and Rademacher coupling laws,
the displayed uniform comparison holds with probability one.
No exceptional-disorder event, asymptotic theorem, numerical
optimization, or empirical sample is used.

For every \(J\ne0\) and positive \(s,t\), the difference is
strictly positive. Thus the theorem certifies a tolerance,
not exact finite-waiting-time stationarity.
\end{theorem}

\begin{proof}
Since \(a\sinh(a\ell)\ge0\), the difference is nonnegative.
For \(\ell\ge0\),
\[
|a|e^{-\ell}\sinh(|a|\ell)
\le
e^{-\ell}\sinh\ell
=
\frac{1-e^{-2\ell}}2
\le\frac12.
\]
Equation~\eqref{eq:exact-difference} therefore gives, on
\(s\ge4\),
\[
0\le A_{2,J}(t)-B_{2,J}(s,t)
\le\frac v2e^{-8T}.
\]

All remaining comparisons have finite elementary certificates.
Cubing positive quantities gives
\[
T>\frac32,\qquad v<\frac43,
\]
because \(T^3=4>27/8\) and \(v^3=2<64/27\).
The positive exponential series gives
\[
e^3>
1+3+\frac{3^2}{2!}
+\frac{3^3}{3!}
+\frac{3^4}{4!}
=
\frac{131}{8}
>
16.
\]
Consequently,
\[
\frac v2e^{-8T}
<
\frac23e^{-12}
<
\frac23\,16^{-4}
=
\frac1{98304}
<
\frac1{65536}.
\]
The same strict upper bound applies to the supremum because
it is a single bound independent of \(J,s,t\).

The disorder conclusions follow from the statement for every
real \(J\). If \(J\ne0\), then \(a\ne0\), and
\(a\sinh(aTt)>0\) for \(t>0\), proving strict positivity.
\end{proof}

\subsection{Certificate manifest}

\begin{center}
\begin{tabularx}{\linewidth}{@{}p{.29\linewidth}X@{}}
\toprule
Item & Frozen value or verified statement\\
\midrule
Size and temperature &
\(n=2\), zero field, \(\beta=1\).\\[4pt]
Clock convention &
Independent rate-one clock at each of the two sites;
holds retained.\\[4pt]
Disorder quantifier &
Every \(J\in\R\), hence separately probability one under
\(N(0,1)\) and symmetric \(\{-1,1\}\).\\[4pt]
Initializations &
Stationary law \(\pi_a\) and independent fair-spin law \(U_2\).\\[4pt]
Rescaling &
Amplitude \(v=2^{1/3}\); physical waiting and lag \(Ts,Tt\),
where \(T=2^{2/3}\).\\[4pt]
Compact &
\([4,5]\times[1,2]\), entirely inside positive times.\\[4pt]
Exact difference &
\(v a e^{-2Ts-Tt}\sinh(aTt)\).\\[4pt]
Uniform tolerance &
Strictly less than \(1/98304\), hence less than \(1/65536\).\\[4pt]
Finite arithmetic &
\(32>27\), \(64>54\), \(131>128\),
and \(98304>65536\).\\
\bottomrule
\end{tabularx}
\end{center}

This manifest, the four-state generator, and the proof above
are the finite certificate. No unexecuted numerical assertion
is needed to establish its inequalities.

\subsection{A probability-law formulation of the same certificate}

Extend the stationary function to the compact by
\[
\widetilde A_{2,J}(s,t)=A_{2,J}(t).
\]
For either disorder law, sample the same \(J\) for
\(\widetilde A_{2,J}\) and \(B_{2,J}\).
This is an explicit coupling of their laws on \(C(\Kcert)\),
with
\[
\norm{\widetilde A_{2,J}-B_{2,J}}_{\infty,\Kcert}
<
\epscert
\quad\text{almost surely}.
\]
Thus the within-disorder function laws admit a coupling with
this uniform error.

This restates the finite theorem. It is not a coupling
between Gaussian and Rademacher environments.

\subsection{Why this is not finite-size disorder universality}

\begin{proposition}[The two finite disorder laws are different]
\label{prop:finite-laws}
At \(n=2\), both autocorrelation functions are deterministic
under Rademacher disorder. Under Gaussian disorder, the
stationary function has a non-point law. In fact, for every
fixed positive lag \(t\), the Gaussian and Rademacher laws
of \(A_{2,J}(t)\) have total-variation distance one.
\end{proposition}

\begin{proof}
The functions in
\eqref{eq:two-stat}--\eqref{eq:two-quench}
are even in \(a\), hence even in \(J\). Rademacher disorder
has \(|J|=1\) almost surely, so its function pair is deterministic.

For fixed physical lag \(\ell>0\), put
\[
F_\ell(z)=e^{-\ell}
[\cosh(z\ell)+z\sinh(z\ell)],
\qquad 0\le z<1.
\]
For \(z>0\),
\[
F_\ell'(z)
=
e^{-\ell}
\left[
(\ell+1)\sinh(z\ell)
+z\ell\cosh(z\ell)
\right]>0.
\]
Thus \(F_\ell\) is strictly increasing on \([0,1)\).
The map \(|J|\mapsto|\tanh(J/\sqrt2)|\) is also strictly
increasing. A Gaussian \(|J|\) has a continuous distribution.
Its autocorrelation value is therefore non-atomic.

The Rademacher value is the value corresponding to \(|J|=1\).
The singleton containing that value has Rademacher probability
one and Gaussian probability zero. This proves the
total-variation assertion.
\end{proof}

This does not contradict a common large-\(n\) weak limit.
It shows why the finite certificate must not be mislabeled
as equality of the two disorder laws.

\subsection{Fast comparison from this start is not a uniform mixing gap}

\begin{proposition}[The observable and the worst-start rate differ]
\label{prop:gap}
For the two-spin chain, the eigenvalues of \(-L\) are
\[
0,\qquad 1-a,\qquad 1+a,\qquad 2.
\]
Its spectral gap is \(1-|a|\), which can be arbitrarily small
as \(|J|\to\infty\). Nevertheless
Theorem~\ref{thm:finite-certificate} is uniform in \(J\).
\end{proposition}

\begin{proof}
The four independent eigenfunctions can be taken as
\[
1,\qquad f_1+f_2,\qquad f_1-f_2,\qquad z-a.
\]
Their eigenvalues follow from \eqref{eq:two-actions}.
Since \(a\to\pm1\) as \(J\to\pm\infty\), the gap tends to zero.

The uniform-start product expectation relaxes at rate two,
as used in Lemma~\ref{lem:two-functions}. The finite certificate
depends on that initialized observable, not on a disorder-uniform
lower bound for the full gap.
\end{proof}

This is a concrete demonstration that a stationary--quench
observable comparison and worst-start mixing are different
claims.

\subsection{Finite initialization can also be written explicitly}

For a stationary sample, draw an independent fair sign \(S\)
and a product sign \(Z\) with
\[
\Prob\{Z=1\}=\frac{1+a}{2}.
\]
Set
\[
(\sigma_1,\sigma_2)=(S,SZ).
\]
This gives \(\pi_a\) exactly. A quench sample instead uses
two independent fair signs.

Thereafter, at a site-one ring, sample the new spin with mean
\(a\sigma_2\); at a site-two ring, sample with mean
\(a\sigma_1\). This writes the finite stochastic mechanism
without appealing to Em04-A, Hawk02-E, or En07-C for the
couplings or clocks.

These are ideal mathematical sampling instructions.
Finite-precision implementations require their own rate and
initialization allowances, as in
Proposition~\ref{prop:perturbation}.

\section{Completed release and unchanged boundaries}
\label{sec:release}

\subsection{Seven-stage accounting}

\begin{longtable}{@{}p{.07\linewidth}p{.47\linewidth}p{.38\linewidth}@{}}
\toprule
Stage & Completed task & Exact status\\
\midrule
\endfirsthead
\toprule
Stage & Completed task & Exact status\\
\midrule
\endhead
\bottomrule
\endfoot
0 &
Energy, coupling normalization, per-site clocks,
observables, initialization, scaling, and external claim frozen. &
The common pair law is distinguished from equality of
stationary and quench functions.\\[5pt]

1 &
Five interfaces audited: stationary edge,
Gaussian estimates, warming, entrance selection, and replacement. &
Elementary finite identities proved; the external asymptotic
proof is not imported or verified.\\[5pt]

2 &
Sampling, interpolation, clock histories, and generator
allowances priced on stated interfaces. &
No quantum model or template supplies an SK generator
by analogy.\\[5pt]

3 &
Implicit limiting representation separated from evaluation,
rate, and finite certification. &
The source's existing representation is acknowledged.\\[5pt]

4 &
Temperature, clocks, start, disorder, observable,
mixing, and quantum-algorithm changes separated. &
No silent hypothesis weakening or model substitution.\\[5pt]

5 &
Classical records, recovery, geometry, and gravitational
readings distinguished. &
No Page curve or gravitational identification.\\[5pt]

6 &
One finite certificate for \(n=2\), one positive compact,
and one numerical tolerance. &
Completed by Theorem~\ref{thm:finite-certificate}.\\
\end{longtable}

\subsection{What is proved, and what is not}

The local result is
\[
\boxed{
\begin{gathered}
n=2,\qquad \beta=1,\qquad
\Kcert=[4,5]\times[1,2],\\[2pt]
\sup_{\Kcert}
|A_{2,J}(t)-B_{2,J}(s,t)|
<
\frac1{65536}
\quad\text{for every }J\in\R.
\end{gathered}}
\]

The following remain outside this certificate:
\begin{enumerate}
\item verification of the external Family 227-01 process limit;
\item construction or independent verification of its limiting
edge diffusion and selected entrance;
\item an asymptotic convergence rate or large-\(n\) error certificate;
\item a Lean formalization of the functional autocorrelation theorem;
\item a quantum SK theorem or identification with QAOA;
\item a closed evaluated formula for the infinite random process;
\item a Page curve, island, horizon, Hawking-radiation statement,
gravity-generated entanglement, or classical-gravity exclusion.
\end{enumerate}

Completion of the finite theorem does not discharge any of
these separately quantified problems. No further next theorem
is opened by this manuscript.

\appendix

\section{A direct finite arithmetic check}
\label{app:arithmetic}

The certificate's numerical inequality uses no approximation
to \(\tanh(J/\sqrt2)\).
Its only size-dependent algebraic comparisons are
\[
8T^3-27=32-27=5>0,
\qquad
64-27v^3=64-54=10>0.
\]
The exponential comparison uses
\[
1+3+\frac92+\frac{27}{6}+\frac{81}{24}
=
\frac{131}{8},
\qquad
131-128=3>0.
\]
Finally,
\[
16^4=65536,
\qquad
\frac23\,16^{-4}
=
\frac1{98304}
<
\frac1{65536}.
\]

The proof is therefore a finite analytic certificate with
explicit rational comparisons. No floating-point agreement,
numerical root finder, or simulation is used as a substitute
for an inequality.

\section{The unrelated characteristic-composition programme}
\label{app:other-programme}

The supplied dossier also contains detailed XIV--XV--XVI
material about reconstructed distributions and a missing joint
characteristic composition. Those objects have no identification
with \(\Omega_n\), \(Q_a\), or the finite correlation scores.

The supplied source relationship remains:
\begin{itemize}
\item XIV-I constructs its named internal amplitude at I2 but
does not construct the complete joint map at I3;
\item XIV-J gives reconstruction descent and a supported-sector
off-characteristic reference, while the full map remains
unresolved at J2;
\item its subsequent section, transverse, contact, output,
and admission requirements are not discharged by a finite
SK calculation;
\item the supplied XV-B revision develops abstract lifting and
extension interfaces without constructing the inherited
full graph-specific map.
\end{itemize}
These are documentary boundaries, not theorem premises
\cite{CheungXIVI,CheungXIVJ,CheungXVB}.

In particular, this manuscript neither constructs those missing
maps nor proves that no possible map exists.

\section*{Declarations}
\addcontentsline{toc}{section}{Declarations}

\paragraph{Mathematical release.}
The main local theorem is the explicit two-spin uniform
stationary--quench certificate. Its supporting finite-state,
sampling, perturbation, and separation results are proved
on their stated interfaces.

\paragraph{Asymptotic status.}
The universal non-point functional limit remains an external
claim. This manuscript does not assert that it is false,
and does not assert that it has been independently verified.

\paragraph{Formalization and computation.}
No successful Lean execution, removal of a Comparator
placeholder, simulation run, finite-precision implementation,
independent referee verification, or successful
\texttt{pdflatex} execution is asserted.
The displayed analytic proof does not depend on such execution.

\paragraph{Source use.}
The supplied Cheung manuscripts are cited by available
identifier, version, section, and label. Incomplete excerpts
do not license claims about the full corpus or uninspected
proof dependencies. No source registration is asserted.

\paragraph{Physical scope.}
The model is classical statistical mechanics.
Its stochastic generator is not identified with a quantum
Hamiltonian, Lindblad generator, QAOA circuit, spacetime metric,
or gravitational dynamics.

\paragraph{Standing programme boundaries.}
In the separate programme described in the dossier,
\[
\boxed{
\begin{gathered}
\text{O.1 remains }\NOTCLOSED,\\
\text{the downstream XV-B admission exit is not opened},\\
\text{the remaining XVI-N obligations are not discharged},\\
\text{no characteristic quantum coefficient is admitted here}.
\end{gathered}}
\]
This does not deny the existence of technical continuation
manuscripts or replace their version-specific conclusions
by a universal nonexistence assertion.

\begin{thebibliography}{99}

\bibitem{SK1975}
D.~Sherrington and S.~Kirkpatrick.
Solvable model of a spin-glass.
\emph{Physical Review Letters}
\textbf{35} (1975), 1792--1796.
doi:10.1103/PhysRevLett.35.1792.

\bibitem{LevinPeres}
D.~A.~Levin and Y.~Peres, with contributions by E.~L.~Wilmer.
\emph{Markov Chains and Mixing Times}.
Second edition, American Mathematical Society, 2017.
Finite heat-bath chains, reversibility, mixing conventions,
and continuous time.
\url{https://pages.uoregon.edu/dlevin/MARKOV/mcmt2e.pdf}

\bibitem{Hoeffding}
W.~Hoeffding.
Probability inequalities for sums of bounded random variables.
\emph{Journal of the American Statistical Association}
\textbf{58} (1963), 13--30.
doi:10.1080/01621459.1963.10500830.

\bibitem{Chatterjee}
S.~Chatterjee.
A generalization of the Lindeberg principle.
\emph{Annals of Probability}
\textbf{34} (2006), 2061--2076.
doi:10.1214/009117906000000575.
\url{https://arxiv.org/abs/math/0508519}

\bibitem{FarhiEtAl}
E.~Farhi, J.~Goldstone, S.~Gutmann, and L.~Zhou.
The quantum approximate optimization algorithm and the
Sherrington--Kirkpatrick model at infinite size.
\emph{Quantum}
\textbf{6} (2022), 759.
doi:10.22331/q-2022-07-07-759.
Cited for the quantum-circuit and energy-expectation interface,
not for identification with heat-bath dynamics.
\url{https://quantum-journal.org/papers/q-2022-07-07-759/}

\bibitem{OAI227Quench}
OpenAI.
\emph{Universality of critical quench autocorrelations in the
Sherrington--Kirkpatrick model}.
External Family 227-01 manuscript, dated 5 October 2026.
Relevant source: Section 1, especially
\texttt{thm:main}.
Documentary claim record, not a premise of the finite theorem.
\url{https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/build/sections/01-introduction.tex}

\bibitem{OAI227Stationary}
OpenAI.
\emph{Functional universality of critical SK autocorrelations}.
External stationary companion, dated 5 October 2026.
Non-point stationary limit is recorded, not independently
verified here.
\url{https://github.com/openai/math/blob/main/preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/build/main.tex}

\bibitem{OAI227Background}
OpenAI.
\emph{The stationary edge input}.
Section 2 of the external critical-quench manuscript.
Finite forms, replica coordinates, conditioned edge inputs,
and relaxed-energy interface.
\url{https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/build/sections/02-background.tex}

\bibitem{OAI227Edge}
OpenAI.
\emph{Gaussian estimates at the bottom edge scales}.
Section 3 of the external critical-quench manuscript.
Relevant source label: \texttt{prop:edge-estimates}.
Audited without importing its estimates.
\url{https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/build/sections/03-edge-estimates.tex}

\bibitem{OAI227Warming}
OpenAI.
\emph{Warming at the critical scale}.
Section 4 of the external critical-quench manuscript.
Relevant source label: \texttt{thm:warming}.
Audited without importing its positive-time estimates.
\url{https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/build/sections/04-warming.tex}

\bibitem{OAI227Entrance}
OpenAI.
\emph{Selection of the entrance law}.
Section 6 of the external critical-quench manuscript.
Relevant source label:
\texttt{thm:entrance-selection}.
Audited without importing its uniqueness or existence theorem.
\url{https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/build/sections/06-entrance-selection.tex}

\bibitem{OAI227History}
OpenAI.
\emph{Clock-history cutoffs}.
Section 7 of the external critical-quench manuscript.
Documentary replacement interface.
\url{https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/build/sections/07-history-cutoffs.tex}

\bibitem{OAI227Scores}
OpenAI.
\emph{Clock scores}.
Section 8 of the external critical-quench manuscript.
Documentary score-projection interface.
\url{https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/build/sections/08-clock-scores.tex}

\bibitem{OAI227Replacement}
OpenAI.
\emph{Replacement}.
Section 9 of the external critical-quench manuscript.
Documentary disorder-comparison and functional-tightness
interface, not a locally verified universality theorem.
\url{https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/build/sections/09-replacement.tex}

\bibitem{Lean227}
OpenAI.
\emph{Critical SK autocorrelation processes and dynamics across
the temperature transition}.
Repository scope note \texttt{lean/docs/227.md}.
Consulted 14 October 2026.
A scope description is distinguished from an executed proof.
\url{https://github.com/openai/math/blob/main/lean/docs/227.md}

\bibitem{LeanSKRatio}
OpenAI.
\emph{SKRatio.lean}.
Repository Comparator target.
The inspected \texttt{ratio\_cutoff} theorem body contains
\texttt{sorry}. Consulted 14 October 2026.
\url{https://github.com/openai/math/blob/main/lean/ComparatorChallenges/SKRatio.lean}

\bibitem{LeanCriticalMixing}
OpenAI.
\emph{CriticalSKMixing.lean}.
Repository Comparator target.
The inspected \texttt{critical\_mixing\_bounds} theorem body
contains \texttt{sorry}. Consulted 14 October 2026.
\url{https://github.com/openai/math/blob/main/lean/ComparatorChallenges/CriticalSKMixing.lean}

\bibitem{Catalogue}
OpenAI.
\emph{External mathematics catalogue}, repository
\texttt{overview.tex}.
Family identifiers and their nominated comparisons are
documentary records, not theorem premises.
\url{https://github.com/openai/math/blob/main/overview.tex}

\bibitem{CheungEm04A}
N.~Cheung.
\emph{Em04-A}, version 2.
Supplied manuscript \texttt{Em04-A-v2.tex},
section ``Physicality and exact reduction.''
Relevant label: \texttt{thm:reduction}.
A specified zero-temperature XY-chain reduction;
not a classical SK heat-bath generator.
Enter Calculus Companion.

\bibitem{CheungHawk02D}
N.~Cheung.
\emph{Hawk02-D}, version 1.
Supplied manuscript \texttt{Hawk02-D.tex},
section ``Actual complete-residual recovery.''
Relevant labels:
\texttt{thm:main} and \texttt{cor:window}.
A specified quadratic fermionic model and its complete
residual, not the present classical spin chain.
Enter Calculus Companion.

\bibitem{CheungHawk02E}
N.~Cheung.
\emph{Hawk02-E}, version 1.
Supplied manuscript \texttt{Hawk02-E.tex},
section ``Gravitational correspondence and a common validity
interval,'' subsection ``Audit of the proposed auxiliary
mathematics.''
Explicitly separates classical SK gap estimates from
quantum SYK dilation estimates.
Enter Calculus Companion.

\bibitem{CheungEn07C}
N.~Cheung.
\emph{En07-C}, version 1.
Supplied manuscript \texttt{En07-C.tex}.
Relevant supplied labels:
\texttt{thm:exact} and \texttt{thm:release}.
Complete-complement recovery and external thermal inputs,
not an identification of SK inverse temperature.
Enter Calculus Companion.

\bibitem{CheungI}
N.~Cheung.
\emph{Landmark I}, version 2.
Supplied manuscript \texttt{I.tex}.
Relevant label: \texttt{thm:main}.
Finite Born probabilities and destructive projective
residuals within a stated strategy class.
Enter Calculus Companion.

\bibitem{CheungVIIIB}
N.~Cheung.
\emph{Landmark VIII-B}, version 3.
Supplied manuscript \texttt{VIII-B-v3.tex},
section ``Classical table decoding is not quantum recovery.''
Relevant label: \texttt{prop:classical-recovery}.
Record-only quantum recovery obstruction, not a classical
autocorrelation theorem.
Enter Calculus Companion.

\bibitem{CheungHawk04B}
N.~Cheung.
\emph{Hawk04-B}, version 2.
Supplied manuscript \texttt{Hawk04-B.tex}.
Complete-complement recovery, typed destructive readout,
and record-only recovery distinctions.
Enter Calculus Companion.

\bibitem{CheungEn01B}
N.~Cheung.
\emph{En01-B}, version 2.
Supplied manuscript \texttt{En01-B-v2.tex}.
The supplied dossier's bounded-statistic and calibration
coverage comparison is used only as a methodological template.
No collapse-model generator is imported.
Enter Calculus Companion.

\bibitem{CheungEn01C}
N.~Cheung.
\emph{En01-C}, version 1.
Supplied manuscript \texttt{En01-C.tex}.
Relevant comparison:
\texttt{thm:certificate}, bounded complete-record
finite-sample coverage.
Its named physical statistic is not an SK autocorrelation.
Enter Calculus Companion.

\bibitem{CheungEm03G}
N.~Cheung.
\emph{Em03-G}, version 1.
Supplied manuscript \texttt{Em03-G-v1.tex}.
Relevant labels:
\texttt{thm:lower},
\texttt{thm:compacttransmission}, and
\texttt{prop:Fobstruction}.
Fixed finite-resource lower bounds and growing-resource
transmission are kept separate.
Enter Calculus Companion.

\bibitem{CheungOai265A}
N.~Cheung.
\emph{Oai265-A}, supplied version 1.
Relevant labels:
\texttt{def:determination} and \texttt{thm:separation}.
Static-record, complete-channel, and temporal-history
separations.
Enter Calculus Companion.

\bibitem{CheungXXIIA}
N.~Cheung.
\emph{XXII-A}, version 1.
Supplied manuscript \texttt{XXII-A.tex}.
Relevant label: \texttt{thm:separation}.
Scheduled ancestry, operational influence, and a separately
supplied causal relation.
Enter Calculus Companion.

\bibitem{CheungXIVI}
N.~Cheung.
\emph{Landmark XIV-I}, version 1.
Supplied manuscript \texttt{XIV-I.tex}.
Relevant labels:
\texttt{thm:I2}, \texttt{thm:I3}, and \texttt{thm:release}.
Internal amplitude construction and the distinct unresolved
joint characteristic map.
Documentary boundary only.

\bibitem{CheungXIVJ}
N.~Cheung.
\emph{Landmark XIV-J}, version 1.
Supplied manuscript \texttt{XIV-J.tex}.
Relevant labels:
\texttt{prop:descent}, \texttt{thm:reference},
\texttt{thm:J2}, and \texttt{cor:noadmission}.
Supported-sector references do not constitute the full
joint characteristic construction.
Documentary boundary only.

\bibitem{CheungXVB}
N.~Cheung.
\emph{Landmark XV-B}, supplied version 2.
Relevant labels:
\texttt{thm:lifting} and \texttt{thm:release}.
Abstract reconstruction-lifting and extension interfaces;
the supplied graph-specific construction remains unresolved.
Documentary boundary only.

\bibitem{Dossier}
\emph{User-supplied Oai227-A plan and comparative source dossier}.
Includes version-specific Cheung excerpts, family catalogue
descriptions, and nominations of Oai272-A and Oai283-A.
Supplied retrieval label:
\texttt{178f233f (retrieval-3)}.
Not an exhaustive corpus scan or independent proof verification.

\end{thebibliography}

\end{document}