\chapter{Introduction}\label{sec:introduction} \section{The Sherrington--Kirkpatrick model and the Parisi formula}\label{sec:intro-background} Let $(g_{ij})_{1\le i0$ has Hamiltonian \[ H_N(\sigma)=\frac1{\sqrt N}\sum_{1\le i1$, its support is an interval $[0,q_\beta]$ with $01$ close to $1$, this was proved earlier by Zhou~\cite{Zhou2026FRSB}. \section{Finite-size corrections}\label{sec:intro-finite-size} In this paper, we study the rate of convergence in the Parisi formula at low temperature. By Guerra's bound, the \emph{mean bias} $P_\beta-p_{N,\beta}$ is nonnegative for every $N$. Up to a factor $1/\beta$, it is the finite-size correction to the expected free energy per spin (\cref{sec:intro-results}). At high temperature, the mean bias is small. For $0<\beta<1$, Guerra and Toninelli showed by a quadratic coupling of two replicas that it is of order at most $N^{-1}$~\cite[Section~3]{GuerraToninelli2002Quadratic}. With a nonzero external field, Lopatto proved that the limiting pressure equals the replica-symmetric value up to and including the de~Almeida--Thouless line~\cite[Theorem~1.1]{Lopatto2026ATLine}, and Kusuoka and Nakajima proved that the pressure is within $O(N^{-1})$ of this value on compact subsets of the strict de~Almeida--Thouless region~\cite[Theorem~1.1]{KusuokaNakajima2026}. Below the critical temperature, the mean bias is expected to decay more slowly. In~\cite[Theorem~1.1, (1.5)]{AL}, we proved that $0\le P_\beta-p_{N,\beta}\le C\beta N^{-1/2}$ for every $\beta>0$ and $N\ge1$, with a numerical constant $C$. For fixed $\beta>1$, we used the interval support of $\mu_\beta$ to prove the sharper upper bound $C_\beta N^{-2/3}$ and the lower bound $c_\beta N^{-11/12}$, for all sufficiently large $N$~\cite[Theorem~1.4]{AL}. In physics, the correction is expected to decay like $N^{-\omega}$ with $\omega<1$ at low temperature. Two values have been proposed for the full model, $\omega=2/3$ and $\omega=3/4$. A calculation retaining only longitudinal fluctuations gives $\omega=4/5$~\cite{ParisiSarraTalamanca2019}. We review these predictions in \cref{subsec:intro-physics}. We improve the exponent of the upper bound from $2/3$ to $7/10$. \section{Main result}\label{sec:intro-results} Our main result is the following. \begin{theorem}\label{thm:main} For every $\beta>1$ there is a constant $C_\beta<\infty$ such that \[ P_\beta-p_{N,\beta}\le C_\beta N^{-7/10} \qquad(N\ge2). \] \end{theorem} Since $P_\beta-p_{N,\beta}\ge0$ for every $N$~\cite{Guerra2003}, \cref{thm:main} is a two-sided estimate. In physical units, the expected free energy density is $\E[-F_{N,\beta}/(\beta N)]$, and its limit is $-P_\beta/\beta$. By \cref{thm:main}, we have \[ 0\le \E\Bigl[-\frac{F_{N,\beta}}{\beta N}\Bigr] -\Bigl(-\frac{P_\beta}{\beta}\Bigr) =\frac{P_\beta-p_{N,\beta}}{\beta} \le\frac{C_\beta}{\beta}N^{-7/10} \qquad(N\ge2). \] For the expected logarithm of the partition function itself, the corresponding bound is $0\le NP_\beta-\E F_{N,\beta}\le C_\beta N^{3/10}$. The constant $C_\beta$ depends on $\beta$, and we do not control it as $\beta\downarrow1$ or as $\beta\to\infty$. In particular, we do not obtain a corresponding bound for the ground-state energy. Together with the lower bound in~\cite[Theorem~1.4]{AL}, \cref{thm:main} implies that for every fixed $\beta>1$, \[ c_\beta N^{-11/12}\le P_\beta-p_{N,\beta}\le C_\beta N^{-7/10} \] for all sufficiently large $N$. The exponent $7/10$ lies between the exponent $2/3$ of~\cite[Theorem~1.4]{AL} and the exponent $3/4$ predicted by Franz, Parisi, and Ricci-Tersenghi~\cite{FranzParisiRicciTersenghi2026}, which we discuss in \cref{subsec:intro-physics}. In \cref{subsec:intro-openai}, we compare \cref{thm:main} with two conditional consequences of results stated in the OpenAI manuscripts~\cite{OpenAIFluctuationScale,OpenAILimitingLaw}. An uncentered comparison stated there would give an unspecified strict improvement over $2/3$, and the variance asymptotic stated there would imply the lower bound $P_\beta-p_{N,\beta}\ge c_\beta N^{-5/6}$. We now outline the proof; \cref{sec:overview} gives a detailed overview. We follow the interpolation scheme of~\cite{AL}, which is a quantitative form of the Guerra interpolation~\cite{Guerra2003} that Talagrand used for the positive fractional moments of the partition function~\cite{Talagrand2007FractionalMoments}. For a tilt $0<\lambda\le\lambda_0$, let $e_{N,\beta}(\lambda)\ge0$ be the error, in energy units, between the constrained Parisi value and the normalized logarithmic moment of order $\lambda/\beta$ of the completed partition function. The constraint requires mass at least $\lambda/\beta$ at zero (\cref{sec:fnd}). Imposing this mass raises the Parisi value by at most $C_\beta\lambda^5$ in energy units~\cite[Lemma~7.2]{AL}. Using this bound and the convexity of Gaussian logarithmic moments~\cite[Theorem~1]{ChenConvexity}, as in the proof of~\cite[Theorem~7.8, Step~1]{AL}, we find \[ 0\le\frac{P_\beta-p_{N,\beta}}{\beta} \le C_\beta\lambda^5+2e_{N,\beta}(\lambda). \] For sufficiently large $N$, we take $\lambda=N^{-7/50}$, so it suffices to show that $e_{N,\beta}(N^{-7/50})\le C_\beta N^{-7/10}$. Up to a factor of four, $e_{N,\beta}(\lambda)$ is the integral over $\rho\in(0,1]$ of a defect $\calD_N(\rho)=\beta(A+2J+V)$ along the interpolation, evaluated at a minimizing profile of an $N$-dimensional variational problem. The terms $A$ and $J$ are controlled directly (\cref{sec:fnd,sec:th}). The difficulty is $V$, the integrated variance of a self-overlap. For $\rho$ above a cutoff $\rho_0$, we bound $V$ by a one-site cavity argument (\cref{part:cavity}). Removing one spin leads to an inequality $V\le(1-\rho)(1+\eps_N)V+{}$errors, where $1+\eps_N$ bounds a scalar response operator at the minimizing profile, and this inequality contracts once $\eps_N\le\rho/4$. To prove this bound, we show that the response operator at the Parisi measure is bounded by the identity, which reflects the marginal stability of $\mu_\beta$, and that the minimizing profile is close to a scalar trial profile (\cref{part:profile}). For $\rho\le\rho_0$, we bound $V$ by the quantile-shell estimates of~\cite[Section~3]{AL}, refined in \cref{sec:wi}, and by a localization of the lowest quantiles of the minimizer (\cref{sec:lz-root}), which refines~\cite[Lemma~7.6]{AL}. The cutoff is $\rho_0=\min\{1,K_\beta\eta^{1/7}\}$, where the trial budget $\eta=e_{N,\beta}(\lambda)+C_\beta\lambda^5$ contains the unknown error. The result is an implicit inequality of the form \[ e_{N,\beta}(\lambda)\lesssim N^{-2/3} \bigl(e_{N,\beta}(\lambda)+\lambda^5\bigr)^{1/21}+\text{smaller terms}. \] If $e_{N,\beta}(\lambda)$ and $\lambda^5$ are both of order $N^{-a}$, this inequality is balanced when $a=2/3+a/21$, that is, $a=7/10$. \section{Relation to physical predictions}\label{subsec:intro-physics} \paragraph{Expectations and early calculations.} Below the critical temperature, physicists expect the finite-size correction to the free energy density to be nonanalytic, of order $N^{-\omega}$ with $\omega<1$. This expectation is related to marginal stability. In a truncated replica model near criticality, De~Dominicis and Kondor found that the stability matrix of the Parisi solution has nonnegative eigenvalues and many zero modes~\cite[pp.~606--608]{DeDominicisKondor1983}. These zero modes make the expansion in fluctuations around this solution singular; see~\cite[Section~2]{FranzParisiRicciTersenghi2026}. Parisi, Ritort, and Slanina studied finite-size corrections at criticality~\cite{ParisiRitortSlanina1993Critical}, where the free energy density has a correction $(\log N)/(12N)$ (see~\cite{Billoire2006FiniteSize}), as well as slightly below criticality and along the de~Almeida--Thouless line, including corrections to overlap distributions and to ultrametricity~\cite{ParisiRitortSlanina1993Corrections}. \paragraph{The exponent $2/3$.} At zero temperature, Bouchaud, Krzakala, and Martin found numerically, for Gaussian couplings, that the mean ground-state energy per spin approaches its limit like $N^{-2/3}$~\cite[Section~VI\,B]{BouchaudKrzakalaMartin2003}. In a study of the SK model with binary couplings, Boettcher fits the corrections to the mean of the ground-state energy per spin and to the width of its distribution with exponents close to $2/3$ and $3/4$, respectively~\cite[Section~3]{Boettcher2005}. In a later study of diluted versions of the model, the fit for the undiluted model is again close to $2/3$~\cite[Table~I]{Boettcher2020Diluted}. At positive temperature, Billoire estimated the correction numerically for binary couplings, by integrating overlap observables along the interpolation of Guerra and Toninelli~\cite{GuerraToninelli2002}, and found data compatible with $\omega=2/3$~\cite{Billoire2006FiniteSize}. Aspelmeier, Billoire, Marinari, and Moore derived $\omega=2/3$ from a heuristic argument: a system of $N$ spins resolves about $N^{1/6}$ steps of replica symmetry breaking, and truncation after $K$ steps changes the free energy density by order $K^{-4}$~\cite[Section~2]{AspelmeierEtAl2008FiniteSize}. \paragraph{Longitudinal fluctuations and the exponent $4/5$.} Parisi and Biscari obtained an earlier contribution of order $N^{-3/4}\log N$ from the positive longitudinal eigenvalues near criticality, neglecting interactions between modes~\cite[Section~1 and Eq.~(5.25)]{ParisiBiscari1992}. Their contribution from the longitudinal zero modes diverges when treated separately from the omitted transverse modes~\cite[discussion following Eq.~(5.27)]{ParisiBiscari1992}; see \cite[Section~1]{ParisiSarraTalamanca2019}. Parisi, Sarra, and Talamanca later used a resummation-based scaling argument for the longitudinal fluctuations of a truncated free-energy functional for the SK model near criticality and predicted $\omega=4/5$~\cite[Section~3, Eqs.~(16)--(19)]{ParisiSarraTalamanca2019}. They omit the transverse fluctuations, so this exponent describes their longitudinal approximation. To determine the exponent for the full model, one must also control the omitted modes. \paragraph{The exponent $3/4$.} Franz, Parisi, and Ricci-Tersenghi~\cite{FranzParisiRicciTersenghi2026} argue that $\omega=2/3$ is incorrect and that the correct value is $\omega=3/4$, possibly up to logarithmic factors~\cite[Eq.~(14)]{FranzParisiRicciTersenghi2026}. Their calculation for the Gaussian SK model matches two regimes: one in which the overlaps of two coupled systems lock and the Guerra--Toninelli integrand decays like $(Nt)^{-1/2}$, and a perturbative regime in which this integrand is of order $t^{-3/2}/N$~\cite[Eqs.~(9)--(12) and (15)--(17)]{FranzParisiRicciTersenghi2026}. They estimate the large-$z$ locking exponent semianalytically by sampling random ultrametric trees generated by the Bolthausen--Sznitman coalescent and extrapolating in the number of leaves~\cite[Materials and Methods]{FranzParisiRicciTersenghi2026}. Matching this behavior to the perturbative regime gives the scaling prediction. The fit of the energy uses random regular graphs of degree four~\cite[Eq.~(18) and Fig.~2]{FranzParisiRicciTersenghi2026}, and the simulations of the mode-locking integral use rewired random regular graphs of degree four with binary couplings (see the subsection ``Direct simulations of the mode-locking integral'' of~\cite[Materials and Methods]{FranzParisiRicciTersenghi2026}). These models differ from the Gaussian SK model in their connectivity, and the latter also in the law of the couplings. \paragraph{Consequences of \cref{thm:main}.} \Cref{thm:main} implies that $P_\beta-p_{N,\beta}=O(N^{-7/10})$ for every fixed $\beta>1$; in particular, the correction is $o(N^{-2/3})$. We therefore exclude a nonzero leading correction of order $N^{-2/3}$ for the Gaussian SK model at every fixed temperature below the critical one. This is consistent with the prediction of Franz, Parisi, and Ricci-Tersenghi. \Cref{thm:main} does not determine the leading exponent $\omega$. Since our constants are not uniform as $\beta\to\infty$, it also says nothing about the numerical results at zero temperature described above. The estimates on which our proof rests have a similar structure with two regimes, although they concern a different interpolation. If we ignore logarithms and the dependence on the tilt $\lambda$, the defect $\calD_N(\rho)$ is at most of order $(N\rho)^{-2/3}$ for small $\rho$ and of order $(N\rho^2)^{-1}$ on the interval where our cavity argument applies. If the second bound held for all $\rho\ge N^{-1/4}$, where the two bounds are equal, integration would give $N^{-3/4}$. In our proof, the cavity interval begins only at $\rho_0\ge N^{-1/10}$, and the tilt $\lambda=N^{-7/50}$ enters both bounds; these two features account for the difference between $7/10$ and $3/4$. Marginal stability also enters our proof, through a scalar statement: at the Parisi measure, the multiplier that dominates our scalar response operator equals one on $\supp\mu_\beta$ (\cref{sec:sc}). Crisanti and Rizzo wrote the corresponding identity in physical variables, on the part of the order parameter that varies~\cite[Eqs.~(22)--(23)]{CrisantiRizzo2002}. This scalar statement does not describe the full stability spectrum of the replica calculation. \paragraph{Mean, width, and large deviations.} The exponents $\omega$ for the finite-size correction discussed above concern the mean. The sample-to-sample fluctuations of the free energy and its large deviations are governed by different exponents. In a truncated replica model near criticality, Kondor obtained a fifth-order dependence on $n$ along the Parisi branch for small positive $n$~\cite[pp.~L129--L130, Eq.~(11)]{Kondor1983}. Interpreted as a formal prediction for the SK model, this says that the replica-normalized pressure in the thermodynamic limit, $\lim_{N\to\infty}(Nn)^{-1}\log\E e^{nF_{N,\beta}}$, differs from $P_\beta$ first at order $n^5$ as $n\downarrow0$ with $n>0$ (see~\cite[pp.~1 and~3]{ParisiRizzo2008}). Parisi and Rizzo extended this calculation throughout the low-temperature phase in~\cite{ParisiRizzo2008} and~\cite[Appendix~B]{ParisiRizzo2009}. Equivalently, in this formal expansion, the limit of $N^{-1}\log\E e^{n(F_{N,\beta}-NP_\beta)}$ begins at order $n^6$, and its Legendre transform gives a rate with power $6/5$ for upward deviations of $F_{N,\beta}/N$~\cite{ParisiRizzo2008}. Crisanti, Paladin, Sommers, and Vulpiani interpreted the replica number as a tilt over the disorder and predicted that the free energy per spin fluctuates on the scale $N^{-5/6}$~\cite{CrisantiEtAl1992Fluctuations}. Under an asymmetric scaled-law assumption, the same calculation also predicts a correction to the mean density on this scale~\cite[p.~1331, Eq.~(34) and following text]{CrisantiEtAl1992Fluctuations}. This conclusion uses the truncated replica theory near criticality and a matching of the disorder tails to the typical peak. In our convention, in which $F_{N,\beta}$ is the logarithm of the partition function, Parisi and Rizzo's replica predictions give upward density deviations with speed $N$ and power $6/5$, and downward density deviations with speed $N^2$ and power $12/5$~\cite[Eqs.~(12)--(15)]{ParisiRizzo2010}. Smoothly matching these tails to the finite-size peak predicts the typical density scale $N^{-5/6}$. This matching is an additional assumption; the limits $n\downarrow0$ and $N\to\infty$ cannot simply be interchanged~\cite[Section~VI]{ParisiRizzo2009}. Aspelmeier, Billoire, Marinari, and Moore also predicted a variance of order $N^{1/3}$ for $F_{N,\beta}$~\cite{AspelmeierEtAl2008FiniteSize}. Aspelmeier derived exact Gaussian interpolation identities relating the variance of the free energy to two- and four-replica overlaps between systems with correlated disorder~\cite[Eqs.~(14) and~(31)]{Aspelmeier2008Exact}. Combining these identities with replica estimates of the bond-chaos overlap distribution gives a predicted upper bound $1/4$ for the extensive standard-deviation exponent~\cite[Section~4.3]{Aspelmeier2008Exact,Aspelmeier2008Bond}. The identities are exact; this exponent estimate uses replica approximations and concerns the width. At zero temperature, the width of the distribution of the ground-state energy has been studied numerically in~\cite{Boettcher2005,Palassini2008,Boettcher2010Fluctuations}. For Gaussian couplings, Palassini finds data consistent with a mean correction exponent $2/3$ and, after subleading corrections, favors the exponent $5/6$ over $3/4$ for the standard deviation per spin~\cite[Sections~3--4 and~6]{Palassini2008}. Boettcher discusses the difficulty of distinguishing the exponents $3/4$ and $5/6$ after corrections to scaling~\cite[Section~IV]{Boettcher2010Fluctuations}. Monthus and Garel studied rare-disorder scenarios and the matching between typical and large-deviation tails for the ground-state energy of the SK model at zero temperature~\cite[Section~III C--F]{MonthusGarel2010}. Their alternative width--tail relations illustrate the additional assumptions involved in such matching. \section{Related work}\label{sec:intro-related} \paragraph{The Parisi formula.} Besides the proofs of Guerra and Talagrand, the rigorous theory of the Parisi formula includes the following results. Guerra and Toninelli proved by interpolation that the thermodynamic limit exists~\cite{GuerraToninelli2002}, and Aizenman, Sims, and Starr formulated an extended variational principle in terms of random overlap structures~\cite{AizenmanSimsStarr2003}. Panchenko proved that the Ghirlanda--Guerra identities~\cite{GhirlandaGuerra1998} imply ultrametricity of overlaps~\cite[Theorem~1]{Panchenko2013Ultrametricity}, and he proved the Parisi formula for mixed $p$-spin models, including mixtures with odd interactions~\cite{Panchenko2014Parisi}. We refer to the books of Talagrand~\cite{TalagrandVolI2011,TalagrandVolII2011} and Panchenko~\cite{Panchenko2013Book} for accounts of this theory; Panchenko develops it through Ruelle probability cascades. \paragraph{The Parisi measure.} Toninelli proved that the replica-symmetric value is not optimal for $\beta>1$ at zero field~\cite{Toninelli2002}. Auffinger and Chen proved that the minimizer is unique~\cite{AuffingerChen2015Uniqueness}, that zero belongs to its support, and that the largest point of its support is an atom for $1<\beta\le3/2$~\cite[Theorems~1 and~4]{AuffingerChen2015Properties}. They also proved that its distribution function is smooth on $[a,b)$ whenever $(a,b)$ is contained in its support, for $0\le a1$, that $\mu_\beta$ is the sum of a density that is smooth up to the endpoints of $[0,q_\beta]$ and an atom at $q_\beta$~\cite[Theorem~1.1]{Lopatto}. Auffinger, Chen, and Zeng proved that the Parisi minimizer at zero temperature has infinitely many support points~\cite{AuffingerChenZeng2020}. H.-B.~Chen proved that $q_\beta\to1$ as $\beta\to\infty$, and that the Stieltjes measure associated with the Parisi order parameter at zero temperature has a smooth density and support $[0,1)$~\cite[Theorems~1.2 and~1.3]{Chen2026ZeroTemperatureSupport}. The full support at zero temperature is also stated in~\cite[Theorem~1.1]{OpenAIFullSupport}. We use the interval support at fixed $\beta>1$ throughout, and the regularity of $\mu_\beta$ in \cref{part:profile}. \paragraph{Fluctuations of the free energy.} The fluctuations of $F_{N,\beta}$ depend on $\beta$ and on the external field. For $0<\beta<1$, Aizenman, Lebowitz, and Ruelle proved a Gaussian limit for its fluctuations, which are of order one~\cite{AizenmanLebowitzRuelle1987}, and Comets and Neveu studied this limit by stochastic calculus~\cite{CometsNeveu1995}. With a nonzero external field, Chen, Dey, and Panchenko proved that the variance is of order $N$ and that a central limit theorem holds, under assumptions on the interactions that include the SK model~\cite[Theorems~1 and~2]{ChenDeyPanchenko2017}. At $\beta=1$, Chen and Lam proved a variance bound of order $\log^2N$~\cite[Theorem~1.1]{ChenLam2019}. Schertzer proved that the variance is at most $(\log N)/4+C$ and at least of order $\log\log\log N$~\cite[Theorems~1 and~2]{Schertzer2026Critical}, and Du and Huang proved that $\Var(F_{N,1})=(\log N)/6+O(1)$ and that the fluctuations have a Gaussian limit~\cite[Theorem~1.3]{DuHuang2026}. Dey and Kang obtained variance asymptotics and a Gaussian limit as $\beta$ approaches $1$ from below along sequences $\beta=\beta_N$ satisfying $N^{1/3}(1-\beta_N^2)\to c\in(0,\infty]$~\cite[Theorem~1.1]{DeyKang2026}. For fixed $\beta>1$, Chatterjee proved the superconcentration bound $\Var(F_{N,\beta})\le C_\beta N/\log N$ and related it to disorder chaos~\cite[Theorem~1.5]{Chatterjee2009Chaos}. In~\cite[Theorem~1.4]{AL}, we proved that $\Var(F_{N,\beta})$ lies between constant multiples of $N^{4/15}$ and $N^{7/15}$ for large $N$. The OpenAI manuscripts state the order $N^{1/3+o(1)}$, a positive limit for $N^{-1/3}\Var(F_{N,\beta})$, and a limiting law for the normalized fluctuations; see \cref{subsec:intro-openai}. A variance bound alone does not bound the mean bias from above, but a lower bound on the variance gives a lower bound on the mean bias through~\cite[Lemma~7.4]{AL}. For the spherical SK model, Baik and Lee proved that the free energy per spin has Gaussian fluctuations of order $N^{-1}$ at high temperature and Tracy--Widom fluctuations of order $N^{-2/3}$ at low temperature, governed by the largest eigenvalue of the coupling matrix~\cite[Theorems~2.10 and~2.11]{BaikLee2016}. \paragraph{Gaussian-process comparisons.} Chu and Raginsky use Gaussian-process geometry and information-theoretic tensorization to bound expected soft maxima~\cite{ChuRaginsky2026}. Their application to the SK model gives a finite-volume comparison with the Parisi variational value up to universal multiplicative constants~\cite[Section~6, Theorem~10]{ChuRaginsky2026}. It uses a uniform spin prior and normalizes the Hamiltonian differently. Its conclusion is a multiplicative comparison; the present theorem controls the additive error to the limiting value. \paragraph{Fractional moments.} Talagrand computed the limits of normalized logarithms of positive moments of the partition function~\cite{Talagrand2007FractionalMoments}. For moments of order $01$, the increase $P_\beta(s)-P_\beta$ is of order $s^5$ as $s\downarrow0$. This is a rigorous counterpart of the replica calculations of Kondor and of Parisi and Rizzo described in \cref{subsec:intro-physics}, and Pei identified the leading coefficient of this increase~\cite[Theorem~1(a)]{Pei2026}. The tilt $\lambda=N^{-7/50}$ in our proof balances this cost against the fractional-moment error. \paragraph{Origins of the methods.} The interpolation of~\cite[Section~2]{AL} is a finite-dimensional, quantitative form of Guerra's interpolation, with which he proved the upper bound in the Parisi formula~\cite{Guerra2003}. For posterior estimates under its tilted laws, we combine the Brascamp--Lieb inequality~\cite[Theorem~4.1]{BrascampLieb1976}, in its inverse-Hessian form~\cite[Proposition~2.2]{AL}, with a joint-source argument from the proof of~\cite[Lemma~2.3]{OpenAIFluctuationScale} (\cref{sec:gauss}). Near the singular end of the interpolation, our estimates refine the quantile-window bound of~\cite[Lemma~3.3]{AL} and the localization of lowest quantiles in~\cite[Lemma~7.6]{AL}. We close the final implicit inequality as in the proof of~\cite[Proposition~7.7]{AL}. The cavity method originates in physics. An early precursor is the single-site description of Thouless, Anderson, and Palmer~\cite[Section~3, Eqs.~(10)--(12)]{ThoulessAndersonPalmer1977}, and M\'ezard, Parisi, and Virasoro formulated the cavity construction for broken replica symmetry in~\cite{MezardParisiVirasoro1986}; see also the exposition in~\cite{MezardParisiVirasoro1987}. We use Talagrand's rigorous version of the one-site argument~\cite[Section~1.6]{TalagrandVolI2011}. The expansion under hierarchical Gaussian laws relies on normalized replica trees and covariance insertions of the type developed in~\cite[Section~2]{OpenAIFluctuationScale}. In the scalar stability argument, the first variation of $\mathcal P_\beta$ under transports of levels is due to Talagrand~\cite[Lemma~3.7]{Talagrand2006ParisiMeasures}; see~\cite[Section~3, (19)]{AuffingerChen2015Properties}. We compute the second variation and compare it with a response multiplier, using convexity of $\mathcal P_\beta$ along segments of distribution functions~\cite{AuffingerChen2015Uniqueness}. This multiplier equals one on $\supp\mu_\beta$, by an identity of Talagrand~\cite{Talagrand2006ParisiMeasures}, in the form stated in~\cite[Theorem~5]{AuffingerChen2015Properties} and~\cite[Lemma~2.6]{Lopatto}, and by the interval support~\cite[Theorem~1.1]{Lopatto}. Above the support, we bound it by one using the curvature of the forward density and its heat evolution from~\cite[Proposition~4.1 and Lemma~2.4(ii)]{Lopatto}. \paragraph{Our related work.} This paper continues~\cite{AL}, where we proved the quantitative Parisi formulas recalled in \cref{sec:intro-finite-size}, the variance bounds above, and upper-tail estimates for $F_{N,\beta}$ with exponent $6/5$~\cite[Theorem~1.5]{AL}. The interval-support theorem used in both papers was proved by the second author~\cite{Lopatto}. In~\cite{AronowLopatto2026EA}, we studied a different approximation of the SK pressure. On $d$-dimensional tori of side length $L\ge3$ and on hypercubes, with independent Gaussian couplings of variance $1/(2d)$ and $1/d$, respectively, the expected pressure per site converges to the SK pressure as $d\to\infty$, for every fixed inverse temperature $\beta\ge0$ and every uniform field $h\in\R$; on tori, the convergence is uniform in the side length. The present paper concerns the finite-size bias of the SK model itself. \section{Comparison with the OpenAI preprints}\label{subsec:intro-openai} We obtained the exponent $7/10$ on September~23, 2026, following a suggestion from Silvio Franz on September~15, 2026. The OpenAI mathematics repository~\cite{OpenAIMathRepository} contains \emph{The low-temperature Sherrington--Kirkpatrick fluctuation scale}~\cite{OpenAIFluctuationScale} and its companion \emph{The low-temperature Sherrington--Kirkpatrick free-energy limiting law}~\cite{OpenAILimitingLaw}, both dated September~24, 2026. We refer to their initial repository version. \paragraph{Stated results.} For every fixed $\beta>1$, \cite[Theorem~1.1]{OpenAIFluctuationScale} states that $\Var(F_{N,\beta})=N^{1/3+o(1)}$. The companion paper states that $N^{-1/3}\Var(F_{N,\beta})$ converges to a positive limit and that $F_{N,\beta}$, standardized by its exact mean and variance, converges in law to a nondegenerate limit~\cite[Theorems~1.1 and~1.2]{OpenAILimitingLaw}. These results concern fluctuations about the finite-size mean $\E F_{N,\beta}$, and they do not by themselves estimate $P_\beta-p_{N,\beta}$. We do not use these results, or the comparison discussed next, in the proof of \cref{thm:main}. \paragraph{A conditional upper bound.} An improvement of the exponent $2/3$ for the mean bias is implicit in an uncentered comparison stated in~\cite{OpenAILimitingLaw}. We use the notation of that paper. Let $T=\beta^2$ and $x=N^{-1/6}$, and fix an admissible choice of its auxiliary exponents and reference parameters. For its direct, unmasked, and unseeded reference paths, \cite[Proposition~6.1, (G1)]{OpenAILimitingLaw} states that there exists $c_G>0$ such that \begin{equation}\label{eq:intro-openai-G1} 0\le F_*(\lambda)-f_N(b_\lambda,h_{*,\lambda})\le x^{4+c_G} \end{equation} for all sufficiently large $N$. Here $F_*$ is a scalar trial objective, $f_N$ is a self-corrected hierarchical pressure evaluated at the clocks $(b_\lambda,h_{*,\lambda})$, and $\lambda$ is a perturbation strength that is unrelated to our tilt. We take terminal mass one and the allowed strength $\lambda=x^{12}$. From the definitions in~\cite[Section~3.1]{OpenAILimitingLaw}, one checks that the scalar objective at $\lambda=0$ equals $P_\beta-\log2-T/4$. Using the perturbation estimates for the direct scalar lift~\cite[Section~3.2, proof of Lemma~3.2]{OpenAILimitingLaw} and the construction of the clocks~\cite[Section~3.4, (P3)--(P4)]{OpenAILimitingLaw}, we find \[ \|b_\lambda-T\|_\infty+\|h_{*,\lambda}\|_\infty\le C_\beta\lambda, \qquad \Bigl|F_*(\lambda)-\Bigl(P_\beta-\log2-\frac T4\Bigr)\Bigr| \le C_\beta\lambda. \] With the uniform spin prior and centered Gaussian completion of~\cite[Section~2.1]{OpenAILimitingLaw}, the unperturbed clocks satisfy $f_N(T,0)=p_{N,\beta}-\log2-T/4$. Integrating the pressure variation formula~\cite[Lemma~2.1, (A1)]{OpenAILimitingLaw} along the segment between the two pairs of clocks, and using that its integrands lie in $[0,1]$ at terminal mass one, we obtain \[ \bigl|f_N(b_\lambda,h_{*,\lambda})-f_N(T,0)\bigr| \le\frac14\|b_\lambda-T\|_1+\frac12\|h_{*,\lambda}\|_1 \le C_\beta\lambda. \] We write $P_\beta-p_{N,\beta}$ as the sum of $P_\beta-\log2-T/4-F_*(\lambda)$, $F_*(\lambda)-f_N(b_\lambda,h_{*,\lambda})$, and $f_N(b_\lambda,h_{*,\lambda})-f_N(T,0)$. We bound the first and last differences by $C_\beta\lambda$ and the middle one by \eqref{eq:intro-openai-G1}. Since $x^{4+c_G}=N^{-2/3-c_G/6}$ and $x^{12}=N^{-2}$, we obtain \begin{equation}\label{eq:intro-openai-bias} 0\le P_\beta-p_{N,\beta} \le x^{4+c_G}+C_\beta x^{12} \le C_\beta N^{-2/3-\delta_\beta},\qquad \delta_\beta=c_G/6>0, \end{equation} after decreasing $c_G$ so that $c_G<8$ if necessary. The lower bound is Guerra's, and \eqref{eq:intro-openai-bias} holds for the finitely many remaining $N\ge2$ after we enlarge $C_\beta$. This bound follows from \eqref{eq:intro-openai-G1} and the perturbation estimates above; it is not a theorem on the mean bias stated in either paper. The proof of~\cite[Proposition~6.1]{OpenAILimitingLaw} leaves $c_G$ unspecified, and we do not extract from it the saving $c_G\ge1/5$ that the exponent $7/10$ would require. The budgets of order $x^{5+c}$ in~\cite[Proposition~4.1]{OpenAILimitingLaw} concern increments of a centered pressure~\cite[(A4)]{OpenAILimitingLaw}, from which the finite-size mean has been subtracted, so they do not bound the mean bias. \paragraph{Conditional lower bounds.} In the other direction, the variance--mean-deficit inequality of~\cite[Lemma~7.4 and Remark~7.14, (7.68)]{AL} states that for every $N\ge1$, \[ P_\beta-p_{N,\beta}\ge c_\beta \Bigl(\frac{\Var(F_{N,\beta})+\beta^2/2}{N}\Bigr)^{5/4}. \] Inserting $\Var(F_{N,\beta})\ge N^{1/3-4\varepsilon/5}$, we see that the variance exponent stated in~\cite[Theorem~1.1]{OpenAIFluctuationScale} would imply, for every fixed $\varepsilon>0$, that $P_\beta-p_{N,\beta}\ge c_{\beta,\varepsilon}N^{-5/6-\varepsilon}$ for all sufficiently large $N$. The positive limit of $N^{-1/3}\Var(F_{N,\beta})$ stated in~\cite[Theorem~1.2]{OpenAILimitingLaw} would give the stronger bound \begin{equation}\label{eq:intro-openai-lower} P_\beta-p_{N,\beta}\ge c_\beta N^{-5/6} \qquad\text{for all sufficiently large }N. \end{equation} For \eqref{eq:intro-openai-lower}, an eventual lower bound $\Var(F_{N,\beta})\ge k_\beta N^{1/3}$ with $k_\beta>0$ would suffice. Combined with \cref{thm:main}, the variance exponent of~\cite{OpenAIFluctuationScale} would imply that if $P_\beta-p_{N,\beta}$ is of exact order $N^{-a}$, then $7/10\le a\le5/6$. \paragraph{Verification status.} We have checked the statements quoted above and the conditional deductions from them. We have not independently reconstructed complete proofs of the global comparison and fluctuation estimates of~\cite{OpenAIFluctuationScale,OpenAILimitingLaw}, and we do not assume their conclusions. The repository also contains Lean sources on the fluctuations of the SK model. Their main theorem on the fluctuation exponents, as we inspected it, takes as an explicit hypothesis a two-sided bound on the centered logarithmic Laplace transform of the completed free energy, \texttt{PositiveLaplaceWindow}; in the fluctuation-scale paper, this bound is the conclusion of~\cite[Proposition~7.5]{OpenAIFluctuationScale}. The conclusion of that theorem concerns the fluctuation exponents, not a positive variance prefactor or a limiting law.\footnote{We inspected the 151 files of the directory \nolinkurl{lean/OAI/Probability/SKFluctuations} at commit \nolinkurl{adc7f1241b42e322a6451854ab7e4b4c146bf78a}. The theorem \nolinkurl{main_of_positiveLaplaceWindow}, at line~294 of \nolinkurl{EventuallyNatPos.lean}, takes a hypothesis \texttt{hw} of type \nolinkurl{PositiveLaplaceWindow}, which is defined at lines~453--459 of \nolinkurl{Spin.lean}; the conclusion of the theorem, \nolinkurl{MainFluctuationStatement}, is defined at lines~531--540 of the same file. We found no proof supplying the hypothesis in this directory. The directory was unchanged at commit \nolinkurl{fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb}, which we checked on October~9, 2026. This was an inspection of the sources, not a compilation.} Neither this conditional formal statement nor our incomplete reconstruction shows that the stated results are false. Our inspection did not identify an invalid step in the portions examined. \paragraph{Proof techniques.} The limiting-law paper contains a one-site cavity expansion whose structure is similar to that of \cref{part:cavity}. In~\cite[Lemma~2.2]{OpenAILimitingLaw}, one site is chosen by exchangeability and removed, and its Gaussian kernels are interpolated to those of an independent scalar model on one site; the linear cavity kernel is half of a connected four-spin correlation of the removed spin~\cite[(A8)]{OpenAILimitingLaw}. In the proof of~\cite[Proposition~6.1]{OpenAILimitingLaw}, the resulting linear equation is inverted under a gap condition on the scalar response~\cite[Proposition~5.1, (H1)]{OpenAILimitingLaw}, and~\cite[Lemma~5.2, (H2)]{OpenAILimitingLaw} writes the response on the additive sector as a susceptibility multiplier minus a positive semidefinite Hessian kernel. In our argument, the cavity inequality contracts because the scalar response operator is at most $1+\eps_N$ with $\eps_N\le\rho/4$, while the interpolation contributes the factor $1-\rho$. We derive the bound on the response operator from the second variation of the Parisi functional under transports of levels (\cref{lem:Jdom}); this operator acts on a space of transports, not on the additive sector of~\cite{OpenAILimitingLaw}. The fluctuation-scale paper also contains a one-site closure for the mean overlap~\cite[Proposition~4.1]{OpenAIFluctuationScale}. We use two tools from that paper. \Cref{part:cavity} relies on normalized replica trees and covariance insertions of the type developed in~\cite[Section~2]{OpenAIFluctuationScale} and used in~\cite[Sections~2.1--2.3]{OpenAILimitingLaw}. In \cref{lem:fnd-posterior}, we adapt the joint-source Brascamp--Lieb argument of the proof of~\cite[Lemma~2.3]{OpenAIFluctuationScale}, which is also used in the proof of~\cite[Lemma~2.1]{OpenAILimitingLaw}, to squared conditional means on selected coordinates and to anisotropic laws. \section{Organization and conventions}\label{sec:intro-organization} \Cref{sec:overview} contains an overview of the proof and a map of its dependencies. The proof itself occupies five parts. \begin{itemize} \item \Cref{part:interpolation} concerns the interpolation. \Cref{sec:fnd} recalls the interpolation of~\cite{AL}, its tilted law, and the decomposition of the defect, and reduces the mean bias to the fractional-moment error. \Cref{sec:gauss} proves two Gaussian tools for tilted laws defined by arbitrary profiles, which we use in \cref{part:profile,part:cavity}. \item \Cref{part:overlap} contains overlap estimates at a minimizing profile. \Cref{sec:wi} estimates moments of the self-overlap on quantile intervals, and \cref{sec:th} bounds the mean--covariance term. \item \Cref{part:profile} concerns the minimizing profile and the scalar response. \Cref{sec:lz} shows that the minimizing profile is close in $L^1$ to the scalar trial profile, and \cref{sec:lz-root} localizes its lowest quantiles. \Cref{sec:sc} studies the scalar response operator and proves its stability at the minimizing profile. \item \Cref{part:cavity} contains the cavity argument. \Cref{sec:c7} sets up the cavity interpolation and its tree laws, \cref{sec:c7-endpoint} computes the derivative at the independent endpoint, and \cref{sec:c7-closure} closes the cavity inequality and bounds the defect. \item In \cref{part:proof}, \cref{sec:b7} integrates these estimates and proves \cref{thm:main}. \end{itemize} We rely on~\cite{AL} for the interpolation identities and the scalar floor estimates, and on~\cite{Lopatto} and~\cite{AuffingerChen2015Properties} for the structure and regularity of $\mu_\beta$. We restate these results where they are used. Locators in~\cite{AL} refer to its arXiv version~1, locators in~\cite{Lopatto} to its arXiv version~4, and locators in the OpenAI manuscripts to their initial repository version. Throughout, $\beta>1$ is fixed, and $C_\beta$ denotes a constant depending only on $\beta$ whose value may change from line to line; a numerical constant does not depend on any parameter. All profiles defining a tilted law are deterministic: minimizers may depend on $N$, $\rho$, and $\lambda$ but not on the couplings. For symmetric matrices or self-adjoint operators, $B\preceq C$ means that $C-B$ is positive semidefinite. We write $\log_+x=\log(\max\{1,x\})$ for $x\ge0$. Distribution functions, such as $\alpha$ and the profiles $a$ of \cref{sec:fnd}, are right-continuous. Atoms of measures are retained in Stieltjes integrals, and an interval $(q,1]$ excludes an atom at $q$.