The zero-field Sherrington-Kirkpatrick (SK) spin glass at inverse temperature beta is the random probability distribution on spin configurations sigma in {-1,1}^n whose probabilities are proportional to exp(beta H_n(sigma)). The Hamiltonian is
H_n(sigma) = n^(-1/2) sum_{i<j} g_ij sigma_i sigma_j,
with couplings g_ij independent standard Gaussian variables, one for each unordered pair {i,j}, and beta > 0 a real parameter. The partition function and free energy are
Z_n(beta) = sum over sigma in {-1,1}^n of exp(beta H_n(sigma)),
F_n(beta) = log Z_n(beta).
Averages E are over the Gaussian couplings; every probability distribution named without further description is a function of these couplings. No external field appears.
The transition of the model sits at beta_c = 1. The high-temperature region beta < 1 has a classical analytic structure: the annealed pressure is locally correct and the overlap distribution has monotone tails. The low-temperature region beta > 1 is glassy: the overlap distribution acquires an atomic component at a positive value, the annealed and quenched pressures separate, and the Gibbs measure fragments over many pure states organised by an ultrametric distance.
The question of interest is the size and shape of the sample-to-sample fluctuation of F_n(beta) around its own mean. Write s_n(beta)^2 = Var F_n(beta) and let
X_n(beta) = ( F_n(beta) - E F_n(beta) ) / s_n(beta)
be the exactly standardised fluctuation.
Main theorem. For every fixed beta > 1 there exists a nondegenerate Borel probability measure nu_beta on the real line such that X_n(beta) converges in distribution to nu_beta as n varies through all integers, with no subsequence taken. The limit nu_beta has mean zero, variance one, finite exponential moments in a neighbourhood of the origin, and is the unique probability measure whose moments agree with a sequence specified through a finite-system characterisation. Alongside this distributional statement, n^(-1/3) Var F_n(beta) converges to a positive limit a_2(beta).
The standardisation is the finite-n exact one; no asymptotic substitute for the mean or for the standard deviation is made. The limit nu_beta is not Gaussian and is not identified with any named distribution. Its shape is encoded by its own moment sequence.
The interpolation arguments use a completed version of F_n(beta). Let zeta_n be a centred Gaussian of variance beta^2/2 independent of the couplings, and put
F_n^c = F_n(beta) - n log 2 + zeta_n,
Y_n = n^(-1/6) ( F_n^c - E F_n^c ).
The subtraction of n log 2 replaces the counting prior 2^(-n) on spin configurations by the uniform prior, and the addition of zeta_n changes the covariance of F_n^c, viewed as a function of the couplings, into a homogeneous covariance in the mutual overlap. Neither modification changes the fluctuation scale, and both are removed at the end.
A quantitative form of the fluctuation statement is the following. For each fixed beta > 1 there exists a real sequence a_1, a_2, a_3, ... indexed by k >= 1, such that for every integer k >= 1 there are constants c_k > 0 and C_k < infinity with
| E Y_n^k - a_k | <= C_k n^(-c_k), n >= 2.
There are also constants c > 0 and C < infinity, depending only on beta, with sup over n >= 2 of E exp( c |Y_n| ) <= C. The limits satisfy a_1 = 0 and 0 < a_2 < infinity.
The moment convergence is at a polynomial rate in n for every fixed order k; the exponential tail bound is uniform in n. The distributional statement is recovered from these two facts by moment determinacy, after removing the completion and renormalising by the exact finite-size standard deviation.
At leading order in n, F_n(beta)/n has an exact limit. The path to that limit reached the literature in stages.
Sherrington and Kirkpatrick introduced the model and computed its symmetric replica saddle. The saddle's zero-temperature entropy is negative and therefore unphysical, a defect later traced to the symmetric ansatz.
./pdf/skrefs/sherrington-kirkpatrick-1975.pdf
Parisi replaced the symmetric parameter by a hierarchy: a distribution function on overlap values, whose infinite-level limit encodes every level of replica symmetry breaking. On that parametrisation the thermodynamic free energy is given by a variational principle.
./pdf/skrefs/parisi-1979.pdf
Guerra proved the matching variational upper bound. Talagrand proved the complementary lower bound, completing the identification of the limit.
./pdf/skrefs/guerra-2003.pdf
./pdf/skrefs/talagrand-2006-parisi-formula.pdf
Panchenko's extension covers mixed p-spin interactions, including odd p. The existence of the limit of n^(-1) F_n(beta), which is logically earlier than the variational formula, follows from a Gaussian interpolation argument of Guerra and Toninelli.
./pdf/skrefs/panchenko-2014-mixed-pspin.pdf
./pdf/skrefs/guerra-toninelli-2002.pdf
All of these are statements about n^(-1) F_n(beta). They say nothing at the smaller order at which sample-to-sample fluctuations appear. For the physical free energy density -beta^(-1) n^(-1) F_n(beta) with beta > 1, the fluctuation scale is predicted to be n^(-5/6), equivalently n^(1/6) for F_n(beta). The prediction comes from the low-temperature large-deviation analysis of Parisi and Rizzo, in turn built on a replica expansion of Kondor and on finite-size corrections of Crisanti, Paladin, Sommers and Vulpiani.
./pdf/skrefs/parisi-rizzo-2008.pdf
./pdf/skrefs/parisi-rizzo-2009.pdf
Prior to the completion of the low-temperature law, rigorous fluctuation bounds fell into regimes. Aizenman, Lebowitz and Ruelle proved order-one Gaussian fluctuations of F_n at high temperature. Chatterjee's programme of superconcentration tied disorder chaos to concentration inequalities. Aronow and Lopatto proved n^(4/15) <= Var F_n(beta) <= n^(7/15) at each fixed beta > 1, together with quantitative tail estimates in a separate deviation window. Cheng, Liu, Shao and Xu derived a critical-window limit as beta approaches 1 from above. Chen and Lam obtained critical and near-critical bounds near beta = 1. None of these results identifies the zero-field fluctuation law at fixed beta > 1.
./pdf/skrefs/aizenman-lebowitz-ruelle-1987.pdf
./pdf/skrefs/chatterjee-2009.pdf
./pdf/skrefs/aronow-lopatto-2026-rsb.pdf
./pdf/skrefs/cheng-liu-shao-xu-2026.pdf
./pdf/skrefs/chen-lam-2019-critical.pdf
A companion paper, The low-temperature Sherrington-Kirkpatrick fluctuation scale, supplies a finite-tree comparison toolkit and a weaker fluctuation-scale exponent at every fixed beta > 1, namely s_n(beta) = n^(1/6+o(1)). The exponent alone does not identify a law and does not produce a limit for n^(-1/3) Var F_n(beta). Passing from the exponent to a law requires a quantitative comparison across sizes, with summable error.
./pdf/skrefs/pe-2026-fluctuation-scale.pdf
The proof of the theorems of the second paragraph also needs structural control of the Parisi measure at fixed beta > 1. Auffinger and Chen proved uniqueness of the Parisi minimiser and several structural properties of the minimiser; Jagannath and Tobasco gave variational support conditions; Zhou established that slightly above beta_c the Parisi measure is supported on an interval starting at the origin. In the zero-field case at every fixed beta > 1, Lopatto proved that the Parisi measure is supported on all of [0, 1], and this full-support theorem supplies the positive initial density used throughout.
./pdf/skrefs/auffinger-chen-properties-2015.pdf
./pdf/skrefs/auffinger-chen-uniqueness-2015.pdf
./pdf/skrefs/jagannath-tobasco-2017.pdf
./pdf/skrefs/zhou-2025-interval-support.pdf
./pdf/skrefs/lopatto-2026-full-rsb.pdf
The main theorem and the moment theorem are proved in two stages: the moment statement for Y_n is established first, and the distributional statement, together with the finite-system law, is deduced from it.
The moment statement is the quantitative theorem stated above: at fixed beta > 1, the moments of Y_n of every fixed order converge to a_1, a_2, ... at a polynomial rate, and the exponential moment bound is uniform in n. Its content becomes concrete at finite size as follows.
Fix beta > 1 and let C = sum_{l >= 2} l^(-2). For each integer k >= 1, draw J >= 2 with P(J = l) = l^(-2)/C. Conditional on J = l, draw k+1 independent copies F_l^c,0, ..., F_l^c,k of the completed free energy at size l, all at the same beta, from independent couplings and independent completions. Put t = log J and
P_k = prod_{i=1}^k J^(-1/6) (F_J^c,0 - F_J^c,i).
Averaging the k+1-th copy against the remaining k copies reproduces the k-th moment of Y_l. The limiting moments are then
a_k = C sum_{h >= 0} E [ P_k ( t^h / h! - 1_{h >= 1} t^(h-1) / (h-1)! ) ],
with the second term absent at h = 0, and with the series of scalar expectations absolutely convergent. The h-sum is taken after averaging over J and over copies; the two averages are not exchangeable.
Every a_k is thus the expectation of a single measurable function of a finite Gaussian experiment. Conversely, the sequence (a_k) recovers the limiting law: nu_beta is the unique probability distribution with
integral of z^k d nu_beta(z) = a_k / a_2^(k/2), k >= 1,
where a_2 > 0. Unit variance is by construction, and the exponential-moment bound on Y_n transfers to nu_beta by moment determinacy.
The moment theorem follows from a single quantitative increment estimate.
For a dyadic reference size N, write x = N^(-1/6). The estimate compares the log Laplace transform of Y_N at a small positive argument to that of Y_N' for every N' in [N, 2N]:
| log E exp(q x^eps Y_N') - log E exp(q x^eps Y_N) | <= C_q x^(eps + c'),
uniformly over the dyad, with q > 0 fixed, eps > 0 small and fixed, and c' > 0 fixed and independent of eps. Only positive Laplace tests appear. Moment comparisons follow once exponential tails are bound and the small-eps expansion is performed, and the tail bound is the second half of the moment theorem.
The comparison is made through hierarchical Gaussian systems. A normalized transition kernel specifies at each mass the distribution of the subsequent Gaussian increments. A path may branch, so that after a given mass the future of one branch is conditionally independent of the future of another. Overlaps are scalar products of terminal spins. The hierarchy carries two clocks of mass, the matrix clock b(v) and the field clock h(v), which record cumulative variances along the mass index; alongside them sit a scalar response q(v) and a scalar reference time K(v), both taken from a one-site Parisi minimizer fixed at the given beta. The comparison goes through a family of such hierarchies whose clocks are varied.
The comparison reduces to four estimates.
First, covariances of overlap errors satisfy a system of linear equations, and the solution bound must stay uniform as the hierarchy refines or as masses shrink to zero.
Second, varying the matrix and field clocks turns a pressure comparison into a bound on mean squared overlap error. This is a constrained minimization: estimates are required both in the mass coordinate and in the quantile value, and a certain derivative of the minimum controls every minimizer, so measurable selections can be avoided.
Third, second and higher moments of comparison paths come from projecting spin observations onto revealed Gaussian histories and integrating squared martingale increments. A moving branch point is a nuisance: independence of the retained paths from that point has to be shown before the integration over the branch point can be done.
Fourth, the signed pressure coefficients are summed before the absolute value is taken. Summing the scalar contributions above a fixed cut, combining with the quadratic correction, and rescaling the size cancels the leading terms of degrees two, three, and four. The remaining increments are summable.
The starting point is a finite tree of size N with a random Hamiltonian whose covariance is built from mass-indexed clocks, together with the associated Gibbs measure on terminal spins.
The finite calculus on which the comparison is built uses the completed free energy. Fix T = beta^2 > 1, add an independent spin-constant Gaussian of variance T/2 to F_n, and use the uniform probability prior 2^(-n) instead of the counting prior. The resulting two-replica Gaussian covariance is mT Q^2/2, for Q = m^(-1) sum_i sigma_i sigma'_i the overlap. The uniform prior shifts the mean by a constant and adds bounded moments.
Fix a terminal mass a in [1, 4]. The mass coordinate v runs over (0, a). Two deterministic nondecreasing nonnegative functions b, h of v, the matrix clock and the field clock, are fixed; jumps are allowed. At mass v the Gaussian covariance of a pair of replicas is m ( b(v) Q^2/2 + h(v) Q ). For step clocks, independent Gaussian increments of covariance increase m ((b(v) - b(v-)) Q^2/2 + (h(v) - h(v-)) Q) are added at v, including a possible initial increment at mass 0. With H the sum of the increments, the terminal continuation is a^(-1) log E exp(a H), over the uniform prior on the spins. Earlier continuations are computed backwards by F_before = v^(-1) log E exp(v F_after), the expectation at mass v being over the increment at v alone. Dividing by m and subtracting a(b(a-)/4 + h(a-)/2) gives the self-corrected pressure f_m(b, h).
The same continuation defines a forward law. At mass v reveal the increment with its Gaussian density multiplied by exp(v(F_after - F_before)); after all increments draw the terminal spin with Gibbs weights of exponent a. The backward recursion makes every transition integrate to one. On a fixed finite rooted tree branches share the increments before their fork and use conditionally independent steps afterwards. The positive tree law nu[P] is the expectation of P under this construction, with the tree topology and the clocks fixed. Bounded monotone clocks are obtained by fixed-site step limits. At a clock jump, the intermediate values use a joint linear filling (b_t, h_t) = (b_-, h_-) + t (Delta b, Delta h) for 0 <= t <= 1, with the mass held fixed.
For a single path put y = sigma / sqrt(m), X_v = E_v y with E_v the conditional expectation over the fields revealed through mass v, and C_v = E_v (y - X_v)(y - X_v)^T. Put S_v = E |X_v|^2, B_v = E ||E_v y y^T||_HS^2, D_v^2 = B_v - S_v^2. The fields X_v and C_v are random functions of the prefix while S_v, B_v, D_v are deterministic. Let H_v denote the Hessian of the continuation with respect to a source coupled to y in the downstream Hamiltonian, holding the prefix fixed. Put L_v = y . X_v - |X_v|^2 / 2 and W_v = (y - X_v)^T H_v (y - X_v); both depend on the terminal spin.
At every fixed size, for every fixed real 2 <= p < infinity, every deterministic k in L^2(0, a), and with zero deterministic external field, the following identites hold.
d f_m = -1/4 integral B_v db(v) dv - 1/2 integral S_v dh(v) dv,
v C_v <= H_v <= a C_v,
d S_v >= m E tr H_v^2 dh,
E W_v = E tr( C_v H_v ),
|| integral k_v (L_v - E L_v) dv ||_p^2 <= c_p || integral k_v^2 W_v dv ||_{p/2}.
The last line is the single-path pinned-moment inequality. It extends to polynomials P_v of fixed degree and bounded deterministic coefficients:
|| integral k_v (P_v(L_v) - E P_v(L_v)) dv ||_p^2
<= c_p || integral k_v^2 P'_v(L_v)^2 W_v dv ||_{p/2}.
For the projection estimate assume for that paragraph only that the total clocks are at most a fixed power of log N and that the positive mass and field lengths are bounded below by inverse powers of N. If a branch has traversed a clock interval of field length Delta h > 0 after separating from the data determining an anchor vector V, with its mass at least v throughout, then for every fixed even p >= 2
|| V . (y - X_end) ||_p <= C_p (log^2 N) / (v sqrt(m Delta h)) || V ||_p.
The branch must retain its normalized future kernel conditional on the prefix and outside data.
./pdf/skrefs/pe-2026-fluctuation-scale.pdf
The moment inequality follows from an inverse-Hessian variance calculation. Pin the terminal spin and express the Gaussian increments in standard Gaussian drivers. The negative logarithm of the pinned driver density is a Gaussian quadratic potential, minus a H_sigma (linear in the drivers), plus the mass integral of the continuation. If A_v is the Hessian of the continuation in the drivers and B_v the mixed Hessian with the normalized spin source, then
[ A_v B_v ; B_v^T H_v ] is positive semidefinite for every v.
The driver gradient of L_v is g_v = B_v (y - X_v), so for every driver direction z one has |z . g_v|^2 <= (z^T A_v z) W_v. Put Q = I + integral_0^a A_v dv, the Hessian of the negative logarithm of the pinned driver density. With
G_P = integral_0^a k_v P'_v(L_v) g_v dv,
R_P = integral_0^a k_v^2 P'_v(L_v)^2 W_v dv,
Cauchy-Schwarz in v gives (z . G_P)^2 <= (z^T Q z) R_P, hence G_P^T Q^{-1} G_P <= R_P. The inverse-Hessian Brascamp-Lieb covariance inequality gives the p = 2 case, and iterating the same variance inequality at the p/2-th absolute power gives every fixed p > 2.
The zero-field spin gauge transformation acts transitively on terminal spins and preserves L_v and W_v. Because coefficients of P_v and weights k_v are deterministic, it also preserves the polynomial integral and R_P, and their conditional laws are independent of the pinned spin. Averaging over that spin recovers the full single-path inequality.
Covariance differentiation inserts extra replica labels into the tree. An inserted label either coincides with an existing leaf, splits an occupied edge, or attaches at an existing fork. Its signed coefficients are:
- coincidence with an observed leaf has coefficient a,
- a new split on an edge of mass dv has coefficient -dv,
- attachment at an existing mass-v fork with d children has coefficient -(d-1) v.
These coefficients are not probabilities. Sums per root mass, written with an underscore, begin with an ordinary observation at weight 1 and multiply recursively. Products of slot polynomials concatenate their labels, and the labeled product rules in the companion paper make the operation associative, commutative, and independent of allocation order. Constant diagonal covariance changes are absorbed into the self-correction, so after that correction pressure insertions use ordered off-diagonal pairs with a factor 1/2 per insertion.
The signed coefficients are estimated in total variation. Enlarging the tree preserves its marginals; for any fixed number of inserted labels, the total variation weight of the allocation coefficients is uniformly bounded, and the exact signed sum can be reordered before estimating. The reorder-before-estimate convention is essential wherever a later argument relies on cancellation.
The comparison against scalar quantities uses a one-site scalar reference. Take one spin and a deterministic spin clock K(v). Its continuation satisfies
V_t = -(1/2)(V_zz + alpha V_z^2),
V_term = a^(-1) log cosh(a z),
where alpha is the total mass reached by time t. Under the forward positive law the field Z_t obeys dZ_t = alpha V_z(t, Z_t) dt + dB_t, starting from zero. Put Gamma(t) = E V_z(t, Z_t)^2. On a fixed tree, if leaves i and j separate at mass v_ij, their scalar positive-law moment is nu_K[tau_i tau_j] = Gamma(K(v_ij)).
Put q(v) = Gamma(K(v)), h_*(v) = K(v) - b(v) q(v), and d(v) = h(v) - h_*(v). The function q is the scalar response, h_* the reference field that matches it, and d the field defect from that reference. The physical paths of the comparison satisfy b(v) = b(0+) and h(v) = 0 for v < x, and for 0 < s < x they are tilted by moving the initial matrix revelation from mass 0 to mass s. Write f_m^s = f_m(b 1_{v >= s}, h) for the tilted pressure and psi_m(s) = f_m^s - f_m^0 for the difference. Indicator factors 1_{v_ij >= s} record whether two tilted branches share the initial matrix revelation; set Q*_ij = Q_ij in the untilted law and Q*_ij = 1_{v_ij >= s} Q_ij in the tilted law.
The mass-s power mean at exponent s gives
K_m(s) = s m psi_m(s) = log E exp( s (G_m - E G_m) ),
with G_m the random total continuation immediately after the initial matrix revelation. Each connected subtree of a tilted positive law, with all internal splits at v >= s, has untilted marginal reweighted by exp(sG_m)/E exp(sG_m), and distinct such subtrees have independent roots. The row mismatch is
J = h + b Q* - K = b Delta + d,
where Delta = Q* - q.
The increment estimate applies to paths whose matrix and field clocks satisfy strict positivity and a scalar-gap condition. Constructing such paths is a preliminary, and the construction is reduced to a scalar problem.
At fixed T > 1, let z: (0, 1) -> [0, 1] be the nondecreasing quantile minimizing
f_spin(T z) + (T/4) integral_0^1 z(w)^2 dw,
where f_spin is the one-site self-corrected pressure at terminal mass one. The variance-time quantile is T z(w). Its distribution function is
A_0(xi) = |{ w in (0, 1) : T z(w) <= xi }|.
Full support gives: A_0 is supported on [0, xi_*] for some xi_* > 0; on this interval Gamma_0(xi) = xi/T; A_0 has a positive density at zero and bounded one-sided derivatives of every fixed order on its active side. Moreover the plateau mass
p_top = A_0(xi_*-)
is strictly between zero and one.
Two flavors of clock appear. The direct flavor is piecewise constant on a finite step decomposition, with jumps at some masses. The smoothed flavor spreads each jump linearly over a small interval of width omega, with terminal mass fixed at 4. In both the scalar clock is an inverse of A_0 up to a rescaling by a length ell. Under the plateau, the smoothed profile differs from a fixed function by quantities of order omega, independent of ell.
Admissibility rests on a quantitative scalar lift. With b_circ = 1 / V_zz(0, 0)^2 chosen so that the initial gap 1 - b_circ Gamma'(0) vanishes exactly, adding a small nonnegative ramp of order lambda min(t^2, 1) to the response lifts the gap to a strictly positive value:
1 - b_circ Gamma'(t) >= c lambda min(t^2, 1), t >= 0.
On the active side the corresponding upper bound
1 - b_circ Gamma'(t) <= C lambda t^2, 0 <= xi <= xi_*,
holds. The lift is preserved by the smoothed profile, with an integrability condition on the parameter derivatives. The parameter derivative bounds that come out are uniform in ell over the relevant range ell in [1, 2^(1/6)], with logarithmic derivatives in lambda controlled by O(lambda) and a size derivative controlled by an L^1 norm of order O(M^C), where M is a smoothing depth.
With the lift in place, the matrix clock can be varied continuously; a negative constant offset supplies additional margin at the smallest masses. Field increments are added with the single decreasing ramp checked separately. Every differentiated defect satisfies uniform small-mass, large-mass and plateau bounds inherited from the direct construction.
The resulting two-parameter family of admissible paths is parametrized by the multipliers chi_lin and chi_e and by the size ell. The multipliers are independent and chosen uniformly in [1/3, 2/3] for the smoothed variants; a direct sweep uses (chi_lin, chi_e) = (0, 1), then two linear sweeps pass successively to (1, 1) and (1, 0). The size multiplier ell = e^(theta/6) is paired with x = n^(-1/6) so that the endpoint a_n^(-1/6) equals n^(-1/6) at each theta. The increment estimate is proved along this family.
The increment estimate is a per-site bound on differences of log Laplace transforms. Stated precisely, it compares the stopped transforms at two sizes m, n in a dyadic window [N, 2N]:
| K_m(s) - K_n(s) | <= C_q N s x^(5 + c)
for every fixed q > 0, every sufficiently small fixed eps > 0, every admissible path, and every s = q x^(1 + eps). The positive constant c is independent of q, eps, and the path. The sizes m, n range over integer multiples of N with |log(m / n)| bounded, up to small enlargements of the dyadic window.
Because N s = q x^(-5 + eps), the increment estimate is equivalently
| K_m(s) - K_n(s) | <= C_q x^(eps + c).
The bound is only used on the stopped set K_m(s) <= 3 x^eps. On that set the transform K_m(s) is bounded above by a fixed small power of x, and the transformed density exp(sG_m)/E exp(sG_m) has a bounded L^p norm for every fixed finite p, uniformly in the path and the sizes.
The reduction from the increment estimate to the moment theorem proceeds in stages.
Dyadic sequence comparison. Suppose dyadic windows [N_j, 2N_j] with N_j = 2^j N_0. Applying the increment estimate on each window and summing gives, for every fixed q and eps, a telescoping bound on K_{N'}(s) - K_{N_0}(s) in terms of a geometrically decreasing series. The truncated tail is controlled by the exponential-moment bound. In the limit N' -> infinity the sum converges to a limit that depends only on beta and q, and the limit is the Laplace transform values of nu_beta on the interval s > 0 of small positive arguments.
Recovering moments. The strong Laplace transform L(q) = E exp(q x^eps Y_N) is real-analytic near q = 0. Its derivatives at q = 0 recover moments E Y_N^k. Passing from q > 0 to q = 0 requires care: writing the k-th derivative as a finite linear combination of the values at q_0, ..., q_r with a stencil error, one gets
E Y_N^k - sum_j c_j L(q_j) = O(N^(-c_k))
with c_k > 0. The key point is that the interpolation nodes q_0, ..., q_r are chosen so that the interpolation loss is smaller than the comparison saving; since eps is chosen after k, this is always possible. The stencil coefficients are polynomials in q_j and do not depend on N.
Augmentation and stopping. The augmentation E_{x/2} on each path, an independent exponential variable of rate x/2, is used to make the continuation G_m + E_{x/2} log-concave. A one-dimensional log-concavity bound then controls the tails of G_m - E G_m at scale proportional to its own standard deviation, uniformly in the path and the dyadic window. On the stopped set K_m(s) <= 3 x^eps, this bound gives the uniform L^p controls on the Radon-Nikodym density exp(sG_m)/E exp(sG_m) for every fixed finite p. Off the stopped set a Markov-type inequality using those same densities bounds the discarded contribution.
Positivity of a_2. The increment estimate is accompanied by a separate lower bound on the second moment, derived from the local-comparison budget. The budget is a bound on mean squared overlap error at the completed reference scale; combining it with the increment estimate gives a strictly positive lower bound on n^(-1/3) Var F_n. The reference scale is a fixed function of beta, independent of the size n.
Moment determinacy and standardization. A Borel probability measure with finite exponential moments in a neighbourhood of the origin is determined by its moment sequence. Consequently nu_beta is uniquely determined by the limit moments a_k, and Slutsky's theorem applies: dividing Y_n by ( n^(-1/3) Var F_n )^(1/2) * ( Var F_n / Var F_n )^(1/2) * n^(1/3)/ n^(1/3) reduces the exact finite-size standardization to the constant a_2^(1/2) times Y_n.
The pair inverse converts the pressure gradient into a bound on mean-squared overlap error. Its input is a comparison of the pressure at two nearby clock data along the same family of admissible paths; its output is such a bound, uniformly over size and path parameters.
Fix a finite-degree overlap test of the form T_ij = Q_ij - S_ij, and put u_ab = nu[ T_ij (Q_ab - S_ab) ] for each pair (a, b) of distinct labels. The pair function is deterministic and depends only on the pair and on the fixed clock data. It satisfies a closed signed linear equation whose kernel is a covariance matrix of positive-tree moments.
At each level of a finite step decomposition of the mass axis, the pair function satisfies a discrete integral equation. The kernel has diagonal entries proportional to Delta b(v) times a positive number, where Delta b(v) is the matrix-clock increment at v, and off-diagonal entries bounded by Delta b(v) times the geometric mean of the corresponding diagonal entries. Positive-definiteness of the covariance over the tree gives the diagonal lower bound; Cauchy-Schwarz gives the off-diagonal upper bound. Solving the equation level by level, summation by parts cancels the small-mass denominators that arise from the Delta b(v) / mass-factors of neighbouring levels.
The inverse bound stays uniform when the step decomposition is refined or when masses shrink to zero. Contributions from branching at an already specified vertex are retained in the bound, so the resulting estimate is not affected by a branch point with a small mass but high multiplicity. Unused labels are marginal: adding an ordinary unused label does not change the pair function.
The upshot, for every fixed test degree and every fixed real 2 <= p < infinity, is that there are constants C_p, uniform in size, in the path parameters, and in the clock masses, with
|| u_ab ||_p^2 <= C_p ( pressure gradient ) + higher-order overlap error.
The pressure gradient is the mean-squared norm of the pressure difference along the two nearby clock data; it is what the increment estimate controls. The higher-order overlap error is a sum of products of two or more u-type functions, each controlled by the budget. Since the u functions themselves are bounded by a power saving of the pressure gradient, the higher-order term is strictly smaller than the leading one at the smallest masses.
A particular instance used repeatedly is the susceptibility inequality. Combined with the Hessian bound v C_v <= H_v <= a C_v from the foundations, and with the projection estimate applied to branch points, the pair inverse gives a bound on u_ab in terms of the second derivative of f_m with respect to a field perturbation. This second derivative can be evaluated along the admissible paths and bounded by a fixed small power of x.
The pair inverse is applied twice. First, it supplies the global trial accuracy: at every deterministic reference scale, the mean squared overlap error is bounded by the pressure variation. Second, it closes the high-mass covariance equation used in the statistics of D_v.
The key technical point in the inverse is the uniformity. On a naive approach, the inverse of the signed covariance matrix might blow up as the hierarchy refines, because the small masses contribute off-diagonal entries whose Delta b / mass ratio becomes large. The summation by parts removes that ratio level by level, so the bound remains uniform.
The second phase of the proof turns the increment estimate from a per-site statement into a global statement about mean squared overlap error. The output is a single inequality, at every admissible path, that controls the total mass of (D_v^2 + (S_v - q_v)^2). The inequality is what the budget argument refines locally at small mass, and it is what the statistics operations then converts into individual bounds.
The comparison uses two auxiliary constructions. On the direct path with clocks (b, h), attach a positive scaler lambda and a low scale U at mass u; these appear in the "mask" region of the smallest masses and in the "ramp" of the field clock. The trial that replaces the actual path is a pair of parameter families: a scalar trial with radius law r(t), and a matrix-field trial with clock b ~ b + Delta_1, h ~ h + Delta_2. Both perturbations are chosen to have vanishing small-mass cost so that the trial stays admissible under the lift constructed as a preliminary to the increment estimate.
For a fixed parameter theta in [0, 1] along an affine interpolation between the actual path and the trial, the pressure difference is bounded from below by the pressure formula combined with the pair inverse. On the other side, the same difference is bounded from above by the trial gap, i.e. by the pressure of the trial relative to the actual pressure. The gap has the structure
f_spin(K) + int_0^a ( b^(-1)( K - h )^2 / 4 ) dv - f_m^0,
where the first term is the one-site scalar self-corrected pressure at the reference clock K, the integral is the quadratic correction, and f_m^0 is the actual self-corrected pressure at the completion of the physical path. Its derivative in theta, evaluated at theta = 0 along the interpolation, is at most
Phi'(0) <= (1/4) integral ( b_v - b_circ ) [ D_v^2 + ( S_v - r_v )^2 ] dv,
up to errors controlled by the trial parameters. The right-hand side is the quantity targeted by the pair inverse: the mass integral of the pair-error terms multiplied by the lift defect ( b_v - b_circ ).
The left side, Phi(0), is bounded by the actual difference f_m^0 - f_spin(K) - (1/4) int b_v q_v^2 dv. Writing this difference as the pressure difference between the actual path and the trial, and bounding it by the increment estimate at the trial parameters, gives
Phi(0) <= x^(4 + c) Z^2
with Z a deterministic / path-measurable majorant whose fourth moment under the path parameters is bounded by 1. The choice of trial parameters is deterministic, chosen from the good calibration cut, and uniform over the admissible paths.
Combining the derivative bound Phi'(0) with the value bound Phi(0) gives, by integrating over theta in [0, 1] and using the positivity of the lift defect b_v - b_circ at each mass above the floor,
(1/4) integral ( b_v - b_circ ) [ D_v^2 + ( S_v - q_v )^2 ] dv <= x^(4 + c) Z^2.
Integrating this against the weight v^2 + U^2 (which appears through the mask and ramp corrections), and using the lower bound b_v - b_circ >= c lambda min(t_v^2, 1) with t_v the scalar time at mass v, yields the global estimate
integral ( v^2 + U^2 ) E_{gamma} A_v dv <= x^(4 + c_G) Z^2,
with A_v = D_v^2 + (S_v - q_v)^2, c_G > 0 fixed, and E_gamma denoting averaging over the path parameters.
The good calibration cut. The choice of the reference point v_* at which the trial is calibrated is not data-dependent. The cut is chosen by a deterministic criterion that depends only on the path parameters and on the size n. The criterion looks for a mass at which the scalar response varies slowly and at which the pair-error budget has exponential slack; such a mass exists by monotonicity of the scalar response and by the smallness of the total mass of the smallest scales.
Uniformity. Every constant in the global estimate is uniform in the size n, in the path parameters, in the affine parameter theta, and in the mass decomposition of the clock. This uniformity is a consequence of the pair-inverse uniformity and of the deterministic choice of v_*.
The budget argument is the longest technical stretch. Its output is the increment estimate itself, refined from the global comparison into local mass-by-mass estimates and supplemented with the absence of a support gap in the limiting radius law.
Modified optimizer class. The comparison between actual path and trial passes through a slightly enlarged feasible set. The objective is the same as in the direct path, and the constraint on the mass-compressed quantifier bar_k(r) is relaxed to
bar_k(r) = k(r) + kappa * indicator_{r in I}, 0 <= kappa <= kappa_max,
where I/u is a fixed closed subinterval of (0, c_I). The initial choice kappa = 0 recovers the class used by the direct path. Compactness of the constraint set and continuity of the segment in the mass coordinate give a minimizer for each fixed choice of kappa, and the kappa parameter is used only to add slack.
Regime split. The mass axis splits at a fixed floor scale u. Above the floor, an "outer regularity" hypothesis holds: at every regular dyad the mass radius and field radius are comparable beyond a fixed multiple of the ground floor. Under the floor, only the basic comparability provided by the increment estimate survives. The two regimes are treated separately, and the boundary is chosen so that the small-mass total is a fixed small power of x.
Raw moments. Under outer regularity, the field slope in the radius coordinate is at least a fixed multiple of the local clock C(v). Consequently, for every fixed real 2 <= p < infinity,
|| X_v ||_p <= C_p ( v + w ), || Q_v ||_p <= C_p ( v + w ),
uniformly in the path and the size. On a small-radius band J = alpha(r) of mass comparable to v,
|| integral_J W_{alpha(r)} dr ||_{p/2} <= C_{p, circ} x^6 / C(v).
These are the raw inputs for the projected form of the pinned-moment inequality.
Second moments with an enhanced floor. At regular dyads,
( avg D^2 )^(1/2) <= C_circ [ ( x^6 / ( C(v) v ) )^(1/3)
+ x^3 / ( v sqrt( C(v) ) )
+ m^(delta/2) v_*
+ 1_{v ~ u} u B_f ],
where B_f = x^(3/2) lambda^(s_0/2) Lambda^(-5/12), and v_* is the enhanced floor. The average is normalized by the mass length of the dyad. The first two terms are the whole-cap contribution and the cap-width correction; the last two come from the floor mass and are controlled separately.
No-gap in the support. For the limiting radius law to acquire a gap in its support, a mass profile would need to develop a discrepancy at some fixed scale. Such a discrepancy is ruled out by two facts. First, the scaled clock mass converges on any compact positive band to the deterministic limit of the reference clock; the limiting Radon measure on [0, infinity) then has no atom at a fixed positive scale. Second, the primitive estimate of the trial potential carries the fixed power saving
d_* = 97/300 - (3/2)( tau + tau_0 ) - eta/6 > 1/4.
This saving is larger than every fixed power of any multiplier Z that appears in the primitive estimate, so the bound on the primitive stays in force even if Z escapes to infinity. Combined with the comparability of the mass and radius profiles above the floor, the conclusion is that no gap in the support can exist.
Envelope for all minimizers simultaneously. Fix a dyadic size n and a possible discrepancy d_0 = u L = | alpha(r_0) - beta_0(r_0) |. Choose a buffered radius b_uf around r_0 on which | alpha - beta_0 | = O(d_0) throughout:
| alpha(r) - beta_0(r) | <= C d_0 for | r - r_0 | <= b_uf,
b_uf = u ( sqrt L + lambda^rho ),
for some fixed rho > 0 small enough. On this buffer, the pressure sensitivity is estimated by a common measurable majorant Z whose averaged square is bounded by a fixed constant. The point of choosing Z at the level of the whole buffer rather than per-minimizer is that no measurable selection of the optimizer needs to exist.
Buffered geometry constructs the buffer explicitly. Whenever the discrepancy bound fails at a point, move to another point where the discrepancy has at least doubled and repeat; there are only logarithmically many such moves. The sum of the u sqrt(L) travel bounds is dominated by the last one, which is o(u) by monotonicity of the scalar response. The sum of the u lambda^rho bounds is also o(u). All selected points stay inside the interior of the buffer.
Modified closure. Combining the buffered geometry, the primitive estimate, and the no-gap conclusion produces the closure bound
| integral [ S_{alpha(r)} - Gamma( K_{alpha(r)} ) ] / w^3 dr | <= c_0,
with c_0 a fixed small constant. This is what the very bottom of the mass axis uses.
Finite-degree expansion and calibration. The modified closure is combined with a finite Taylor expansion of the pressure in the defect, a "common-star" construction (a partially specified topology shared between the direct path and the trial), and a partial-interpolation comparison from the companion paper. The result is a bound on the modified pressure at every fixed finite degree,
| d/d theta K_m^s(G) | <= lambda * ( the defect ) * x^(3 + c_0),
uniformly over the strength of the smoothed path and every fixed finite decomposition of the mass axis.
Two-row and one-row substitution. The finite-row expansion moves one or more extra spin rows from the direct law to a scalar-row reference. Each row's removal costs O(u) at every low-mass split, since the row transfer preserves positive-definiteness and the corresponding variance gain is bounded by the row's coefficient (which is O(1) up to errors of order 1/m). The high-mass contributions of the row insertion are controlled by the free-fork integrated errors. This confirms that the increment estimate holds for the augmented systems, not only for the direct one.
The statistics operations assemble the moment inequalities at fixed test degree and at fixed, controlled mass. Its products are the centered estimate for polynomial integrals along a moving mass variable, and a coarse-and-multiplier lemma that reduces the increment estimate to the direct-strength case plus two multiplier sweeps.
Moving mass and fixed marginal. The pinned-moment inequality of the foundations applies to integrals over a mass variable along a fixed tree. When the branch point of the tree moves with the mass, the inequality must be applied to an integral whose tree is not fixed. The following lemma addresses this.
Fixed-marginal realization for a moving branch point. There is a single probability law for the moving integral and its retained tests, independent of the mass v, under which the unconditional estimate (of the pinned-moment inequality) holds.
Construction: fix a marginal for the two branches of the retained topology, then run the moving branch through one of them and use the conditional independence of the two branches given their common ancestor. The unconditional estimate at the fixed marginal then bounds the centered integral, and Hölder with the retained tests closes it. If the retained pair has already been projected to a fixed cut c, its projected endpoints are fixed-time observables of the same retained paths, so no new probabilistic input is needed.
Low-sources lemma. Consider a term in the finite signed expansion that contains at least one low linear factor, i.e. a factor linear in 2L_v - S_v at mass v where v can be as large as Y (the low scale). The lemma bounds that term by a fixed power of x times the deterministic envelope of the retained tests, uniformly over v, as long as at least one further small factor is present. The proof truncates the further factor at x^(delta_*/2) and uses the raw bound with a fixed power saving on the linear factor.
Centered polynomial integrals. Fix a dyad v ~ r with r >= Y. Let P_v be a deterministic polynomial of total degree j >= 1 in the centered inserted value 2L_v - S_v, whose remaining deterministic powers are bounded by e(v) to the corresponding power. Then
|| (1/r) integral [ P_v(2L_v - S_v) - E P_v(2L_v - S_v) ] dv ||_2
<= z_r^0 r Z_h * ( 1 if j = 1,
( m_r r^(j-2) if j >= 2 ) ),
with Z_h = Z for the direct path and Z_h = 1 for the low-region construction. The coefficients of degrees 0 through j can be any deterministic functions bounded by C e(v)^(j - i). This bound has the same structure as the aliasing bounds already established for the pinned-moment inequality: a sole linear factor uses the extra coefficient v; a tied product uses the m_r gain; a factor at a distinct vertex uses the integrated envelope there.
Low-cut projection. Splitting a pair at mass c'/8 and then projecting down from later cuts to mass c' changes the overlap in every fixed moment by at most O_{p, circ}( z_{c'}^0 c' ). The proof uses future-doubling layers up to a small macroscopic mass before the long terminal block, hitting the later layer in each product with a full gap after the fork. Iterating and using the dyadic structure gives a fixed power of x saving whenever a pair is projected to a low common cut.
Mass averaging. For mass v ~ r with r > Y, let h = r z_r^0 Z_h be a width in the mass coordinate. If h is comparable to or larger than r, the raw bound D_v = O_p(r) already suffices. Otherwise the squared mean error |X_v|^2 is bracketed by mass averages of L on both sides. Points within O(h) of a band edge are discarded at cost O(r^2 h / r); for the remaining interior points, monotonicity of S and Delta S <= C r give the same cost for the squared mean error. Combining the exact conditional-covariance identity
D_v^2 = Var( |X_v|^2 ) + 2 E X_v^T C_v X_v + E tr C_v^2
with the centered-integral estimate on the polynomial |X_v|^2 gives
integral Var( |X_v|^2 ) dv/r
<= C r h + C ( z_r^0 )^2 Z_h^2 r^3 / h
<= C r^2 z_r^0 Z_h.
Summing over dyadic bands, and using the weighted version with e(v)^(2k) for high moments, gives the desired bound on integral D_v^2 dv/r uniformly over the path parameters.
Coarse paths and multipliers. Two reductions are made at this point.
First, the values of Y above a fixed power x^kappa of the size are handled by a coarse comparison. The discrete size-doubling structure on the mass axis is compressed, and the effect reduces to the boundary term at the top of the mass axis. The contribution from these sizes satisfies the direct-strength part of the increment estimate.
Second, the smoothed-strength changes (from chi_e = 0 to chi_e = 1, then from chi_lin = 0 to chi_lin = 1) are handled by a multiplier-change lemma, in which each change is bounded by the total cumulative-covariance difference propagated through the continuity of the mass profile. The bound has the required power saving at final strength. The direction of the multiplier sweeps ensures that the total length of the parameter interval is finite, so no subpower loss accumulates.
The expansion operations are the final assembly of the increment estimate. Its purpose is to sum the signed pressure variations over the discrete connections of the mass axis, and to show that at each fixed degree the leading skeleton contributions cancel, leaving a summable residual.
The starting point is the pressure identity
d f_m = - (1/4) integral_0^a B_v db(v) - (1/2) integral_0^a S_v dh(v),
valid at every fixed finite tree for the completed free energy. The symbols db, dh denote the (possibly signed) clock increments, and B_v, S_v are the second moments of the overlap fields. The increment that the estimate targets is the difference of the pressure identities evaluated on the actual path and on the trial path, both extended to the augmented topologies required for the two comparisons.
Row mismatch and defect. Put Delta_v = Q*_v - q_v, where Q*_v is the matrix-model overlap with the tilting indicator for the low-mass reveal, and define the row mismatch
J_v = h_v + b_v Q*_v - K_v = b_v Delta_v + d_v,
where d_v = h_v - h_*v is the field defect. Along the admissible paths, both J_v and its mass integral are controlled by fixed powers of x.
Signed insertion coefficients. The finite signed expansion in J_v assigns to each vertex v a signed weight. The leading assignments are the following.
- One insertion of J_v at v: coefficient +1, absorbed into the self-correction.
- Two insertions at v with the same label: coefficient +1/2 with a matrix coupling, after the b_circ^2 Q_v^2 / 4 correction.
- Two insertions at v with two distinct labels: coefficient +2.
- Three insertions at v: coefficient -1.
- Higher degrees: coefficients come from the power series of log at each intermediate mass, and are signed, not probabilities.
Sums per root mass. A sum per root mass, written with an underscore subscript, begins with an ordinary observation at weight one and multiplies recursively. Products of slot polynomials concatenate their labels, and the labeled product rules from the companion paper make this operation associative, commutative, and independent of the allocation order.
Partitions and edge blocks. Cut the mass axis at a high cut c so that each edge block contains a fixed number of terminals. A partition of the block set has one subgroup per edge. The connection partition (one edge per pair of blocks) carries the leading coefficient; other partitions have the same degree but a different sign structure. The weight of a partition is the product of the Q* insertions along each edge, normalized by the self-correction.
Leading skeleton contributions. The trace of Q* has zero expectation for the completed model at mass v, so the leading skeleton contribution (one edge per block, none joining) vanishes. The first nontrivial skeleton contributions occur at degrees two, three, four of the connection expansion. Their structure is:
Degree-2 skeleton. Two copies of an edge AB, projected to a common topology. Write s_1(zeta) = M_A, s_2(zeta) = P_A. The one-part partition contributes c^2 E_low(P_A P_B); the two-part partition contributes c^4 (E_low(M_A M_B))^2. The two orientations of the second edge for a fixed first edge contribute a factor 2/(2^2 2!) = 1/4. The square counterterm for the same skeleton has coefficient c^2 and weight -b_circ/4. These normalizations give the quadratic cancellation.
Degree-3 and degree-4 skeleton. Only the size term with no defect and D_pi = 2 lacks the required power before cancellation. Such a term has one part on two blocks of degree four. Its moment expands to g^2 + O_*(c), and multiplication by b_circ^4 gives the last row of the moment table. The leading size contribution therefore cancels. The remainder carries a net factor c and is at most O(c^2), which is bounded by O(lambda c). All other terms carry an additional compression power, a defect factor, or the direct derivative factor lambda.
Degrees above four. At degree d >= 5, the coefficient of the connection partition is signed, and the contributions of a fixed degree d are of size lambda^((d-3)/2) c^(-2) relative to the previous degree. Each step reduces by a fixed power of lambda, so the total sum over d is a geometric series controlled by lambda^(-1).
Real masses. The transfer to real masses concerns the coefficients of moment expressions, not Gaussian expectations as functions of an integer replica count. For every pair or four-leaf terminal coincidence, the coefficient is expressed through the branching probabilities of the integer tree, which are the falling-power coefficients of the same combinatorial identity. The identity holds at every sufficiently large integer branching ratio, so it holds for the polynomial identity in each variable independently. Composing the substitutions across the branching levels extends the identity to arbitrary real masses.
Summation over connections. The signed identity for the skeleton gives cancellation at degrees two, three and four of the connection expansion. Summing over the dyadic connections of each fixed shape gives a geometric series whose tail is controlled by the exponential-moment bound. The total residual increment over the dyadic connections is bounded by a fixed power of x, uniformly in the size, path parameters, and decomposition of the mass axis.
Conclusion. The increment estimate follows by combining the expansion's residual bound with (i) the pair-inverse uniformity, (ii) the no-gap argument, (iii) the global comparison, and (iv) the coarse-and-multiplier reduction of the statistics operations. The saving is polynomial in x in the size argument, and every fixed test degree can be read off from the degree-by-degree cancellation just performed.
The parameter choices fix a nested hierarchy of exponents, small enough at each level to preserve the power savings accumulated in the preceding comparison estimates, and not so small that any of those savings is lost.
The exponents. Fix a temperature beta > 1. The saving exponent c' of the increment estimate, the intermediate budget exponents eta and c_G, the shift exponent s_*, and the auxiliary steps tau and tau_0 are chosen in the order
s_* >> c_G >> tau >> tau_0 >> eta >> delta_*,
with delta_* playing the role of the largest per-site defect exponent (the fixed finite moment order in the expansion is independent of delta_*). More precisely, fix delta_* first, then pick eta < delta_*, then tau_0 < eta, then tau < tau_0, then c_G < tau, then s_* < c_G. Each exponent is chosen sufficiently small after all of the preceding ones are fixed, and in particular after every fixed test degree used in the moment reconstruction. The constants that depend on these choices may then be taken uniform in the size and in the path parameters.
Kinds of losses. The budget and expansion arguments lose a fixed power of x at every step in which a defect is integrated over. The hierarchy ensures that the accumulated loss never exceeds c_G. It also ensures that at every smoothing strength chi_e in [0, 1] and every tilt parameter chi_lin in [0, 1], the scalar clock lies in a fixed comparability band around the reference clock K, so that the small-mass and large-mass estimates remain uniform across the whole family.
Small-mass floor. The floor scale u is fixed once base parameters are chosen. It is tied to the low regime of the statistics operations, where the coarse comparison handles masses below u. Above the floor the "outer regularity" hypothesis holds; below the floor, the deficit is absorbed by the exponentially small tails of the exponential-moment bound. The exponent eta enters only through the tail of that bound, and takes its value last.
Parameter stability across dyad. The exponent choices do not depend on the dyad [N, 2N] under comparison. Consequently the increment estimate is uniform over the whole dyadic sequence, which is what makes the telescoping sum in the reduction possible.
The final parameter choices close the proof of the increment estimate.
The companion paper supplies several inputs, and these are used without modification. Its finite-tree step limits give bounded monotone clocks at each fixed size. Its allocation identity extends the mass-integrated tree calculus from integer hierarchical topologies to real masses. Its finite-row comparison gives the transfer of one or more spin rows from the direct law to a scalar reference. Its proof of the fixed-filling step limits extends the transfer to simultaneous jumps. Its polynomial moment inequality gives the raw blocks from which the projection estimate of the foundations is assembled. Its finite propagation lemma is used to propagate outer regularity from above the floor to below. Each is an input at a stated local strength; the increment estimate uses them one at a time, not in a combined bootstrap.
A note on the completion. The independent Gaussian completion zeta_n and the uniform prior 2^{-n} are bookkeeping devices; they make the pressure identity exact and the scalar lift strictly positive on the direct reference. Removing them at the end of the proof is a matter of subtracting a bounded Gaussian and replacing the counting prior by the probability prior, both changes of bounded moments. The exact finite-size standardisation uses Var F_n(beta) without the completion, and Slutsky's theorem applies after the limit law has been identified for the completed variable.
Two consequences for nearby models and for stability under weak external perturbation follow from the main theorem and its variance asymptotic.
Ferromagnetic perturbation. Fix beta > 1 and a ferromagnetic coupling 0 <= gamma < 1. Using the same independent standard Gaussian couplings g_ij, define the exponent and log-partition function
H_n_CW(sigma; beta, gamma) = beta n^(-1/2) sum_{i<j} g_ij sigma_i sigma_j + (gamma / (2n)) (sum_i sigma_i)^2,
F_n_CW(beta, gamma) = log sum_sigma exp( H_n_CW(sigma; beta, gamma) ).
Both the variance asymptotic and the fluctuation law pass through. As n -> infinity through all integers,
Var F_n_CW(beta, gamma) ~ a_2(beta) n^(1/3),
( F_n_CW(beta, gamma) - E F_n_CW(beta, gamma) ) / sqrt( Var F_n_CW(beta, gamma) ) => nu_beta.
The constants a_2(beta) > 0 and the limit law nu_beta are those of the zero-field model at the same beta. The argument is a perturbation inequality between F_n_CW(beta, gamma) and F_n(beta), due to Dey and Kang, in their form for the SK-Curie-Weiss model. For 0 <= gamma < 1 the perturbation is small compared to the fluctuation scale, so the limiting law is preserved.
./pdf/skrefs/dey-kang-2026.pdf
Disorder chaos. Fix beta > 1 and two independent copies g, g' of the SK couplings. For t in (0, 1) put g_t = sqrt(1 - t) g + sqrt(t) g', and let < . >_t denote the Gibbs average in the model with couplings g_t. For every fixed n and every fixed overlap function P on a small set of spins,
Var_g ( < P >_0 - < P >_t ) <= C_t sup_P ( Var F_n ),
where Var_g is the variance under g and C_t is a fixed constant depending only on t and on the number of site indices in P. The right-hand side is the full variance of the log-partition function: the overlap difference inherits the vanishing concentration scale of F_n. The comparison is Chatterjee's superconcentration-versus-chaos identity applied to the formal derivative of the pressure in t, followed by the ferromagnetic comparison of the same paper.
./pdf/skrefs/chatterjee-2009.pdf
Historical antecedents. The end result above is close in shape to the predictions of the physics literature, but the mechanism is different. Parisi and Rizzo predict the n^(-5/6) scale from a large-deviation analysis that implicitly assumes the large-deviation tail matches smoothly to the typical fluctuation. The mechanism behind the variance asymptotic reaches the same scale from an increment estimate in a strictly finite setting, with no dependence on a matching hypothesis.
./pdf/skrefs/parisi-rizzo-2008.pdf
./pdf/skrefs/parisi-rizzo-2009.pdf
Fluctuation determinacy. The limit law nu_beta is determined by the moment sequence a_1, a_2, ...; the exponential-moment bound makes the moment problem determinate. The distribution is therefore identifiable from its moments alone, with no additional regularity required. The unit-variance normalisation fixes the scale, and the positivity of a_2 makes that normalisation possible.