All-temperature pressure of orthogonally invariant Ising spin glasses
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Gives an exact variational formula for the limiting pressure of orthogonally invariant Ising spin glasses at every fixed temperature, both almost surely and in expectation. The coupling matrix is a Haar-random rotation of a deterministic spectrum converging to a compactly supported law, with extreme eigenvalues converging to its support edges. The zero-field ground-state energy follows as temperature tends to zero. (OpenAI’s Lean note)
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Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Let \Lambda_N be deterministic real diagonal matrices whose empirical spectral measures converge weakly to a compactly supported probability law \mu. Assume that their smallest and largest eigenvalues converge to the respective edges of \supp\mu. Let U_N be Haar distributed on O(N), and define P_N using the uniform spin probability in Equation (1.1). With S and G_1 as in Equations (1.7)–(1.8),
We determine the limiting pressure of an Ising spin glass with Haar orthogonal eigenvectors, a compact limiting spectral law, and no asymptotic outliers, at every fixed temperature. The pressure converges in expectation and almost surely to a variational formula. We also give the limiting pressure with deterministic external fields whose empirical laws converge in first-moment transport distance, and derive a formula for the zero-field ground-state energy by taking temperature to zero.
One Lean statement in this family
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Let \Lambda_N be deterministic real diagonal matrices whose empirical spectral measures converge weakly to a compactly supported probability law \mu. Assume that their smallest and largest eigenvalues converge to the respective edges of \supp\mu. Let U_N be Haar distributed on O(N), and define P_N using the uniform spin probability in Equation (1.1). With S and G_1 as in Equations (1.7)–(1.8),
InvariantIsing.lean, the release lists it as a supporting statement.
theoremlimiting_pressure_of_extreme_limits_unconditional : ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] (U : (N : ℕ) → Ω → Orthogonal N) (_hU : ∀ N, Measurable (U N)) (_hHaar : ∀ N, (P.map (U N)).IsMulRightInvariant) (eig : (N : ℕ) → Fin N → ℝ) (ν : ProbabilityMeasure ℝ) (a b : ℝ) (_hcompact : IsCompact (ν : Measure ℝ).support) (_hbound : (ν : Measure ℝ).support ⊆ Icc a b) (_ha : a∈(ν : Measure ℝ).support) (_hb : b∈(ν : Measure ℝ).support) (_hmin : Tendsto (fun k => spectralMinimum (eig (k+1))) atTop (𝓝 a)) (_hmax : Tendsto (fun k => spectralMaximum (eig (k+1))) atTop (𝓝 b)) (_hweak : Tendsto (fun k => empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1))) atTop (𝓝 ν)), Tendsto (fun N => ∫ ω, rotatedPressure (eig N) (matrixRotation (U N ω)⁻¹) (fun _ => 0) ∂P) atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) ∧ ∀ᵐ ω ∂P, Tendsto (fun N => rotatedPressure (eig N) (matrixRotation (U N ω)⁻¹) (fun _ => 0)) atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) := by⋮
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Independent comparisons and proof-verification reports
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MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note:limiting_pressure_of_extreme_limits_unconditional: for Haar-distributed orthogonal U N ((P.map (U N)).IsMulRightInvariant), deterministic spectra eig N with empiricalSpectralLaw → ν (compact support, extreme eigenvalues → a, b), ∫ rotatedPressure.. ∂P → (variationalFunctional (measureR ν b)).toReal and the same almost surely; positive_temperature_pressure_unconditional (all β > 0 via scaledSpectralLaw), ground_state_limit_unconditional (Tendsto (thermalVariationalValue ν b) atTop (𝓝 M) with M the ground-state limit), plus magnetic-field, random-spectrum and Gaussian-pattern variants. All summary claims are stated.
Definitions the group asks readers to compare with the paper:Custom ‘free energy’: logPartition H := log ((card X)⁻¹ * Σ_x exp(H x)) (normalized by 2^{-N}, i.e. pressure minus log 2), rotatedPressure, the Parisi-type variationalFunctional R := ⨅ p : OverlapPath, entropyFunctional p + spectralFunctional R p (EReal) and measureR μ e x (R-transform via a h.choose inverse of the resolvent on (e, ∞)); hand-written variational formulas, with the final value taken via .toReal of an EReal (junk if infinite).
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
The formalization gives variational limits for orthogonally invariant Ising spin glasses whose eigenvectors have Haar law and whose empirical eigenvalue distributions converge to a compactly supported law with the stated control of extreme eigenvalues. The pressure converges both in expectation and almost surely to an explicit functional of the limiting spectral law. A version with external fields assumes convergence of their empirical laws in Wasserstein distance and gives the corresponding magnetic-field functional.
The formalization also covers every positive temperature, a ground-state limit, random spectral data with the stated conditional Haar law, and Gaussian-pattern specializations. These hypotheses identify the invariant models to which the limits apply.
AI-drafted comparisons, with no recorded human review
Pressure and ground-state limits for invariant Ising modelsInvariantIsing
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem (limiting pressure of orthogonally invariant Ising spin glasses, in expectation and almost surely) with the field, ground-state, random-spectrum and Gaussian-pattern versions.
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Pressure and ground-state limits for invariant Ising modelsInvariantIsing
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The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state the headline as OpenAI wrote it.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator with Lean’s kernel: accepted (the check took 580 s; peak memory 7.3 GiB).
Date
10 October 2026
Version
at the release’s recorded version
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Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.