Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability. (OpenAI’s Lean note)
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The paper states
In The Mézard–Parisi formula for diluted spin glasses, 23 September 2026
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.
The other possible matches are Theorem 6.5 (Identity on prescribed trees) in the same paper () and Theorem 7.1 in the same paper ().
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OpenAI released an abridged summary of the model’s reasoning for this family: The Mézard–Parisi formula for diluted spin glasses
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DilutedSpinThe paper states
In The Mézard–Parisi formula for diluted spin glasses, 23 September 2026
The words they share: integrability, limit, model, positivity, thermodynamic.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
DilutedSpin.lean, the release lists it as a supporting statement.
theorem mezard_parisi {p : ℕ} (M : Model p) (hM : Admissible M) : Tendsto (pressure M) atTop (𝓝 (variationalValue M)) := by sorryThis file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
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Exponents Model.lean, line 121MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note: mezard_parisi (M : Model p) (hM : Admissible M) : Tendsto (pressure M) atTop (𝓝 (variationalValue M)) for Poisson-diluted even-arity (p >= 2 even) Ising models with the factorization exp(θ s) = a (1 + b ∏ f_l(s_l)), independence/identical-distribution of the f_l, b-moment positivity 0 ≤ E[(-b)^n] and only first-moment integrability of θ and the field; variationalValue M := ⨅ r, phi M r with phi M r := sInf {functional M r ζ m | ζ ∈ Hierarchy (r+1), 0 < m_1 <... < m_r < 1}. Matches the summary (Viana-Bray etc. are instances of the class, not separately stated). Caveat: everything is Bochner integrals, so a non-integrable integrand silently evaluates to 0.
Definitions the group asks readers to compare with the paper: Custom ‘free energy’: pressure M N := (∫ k ∂Poisson(αN), ∫ θ, ∫ h, avg_indices log Σ_σ exp(logWeight)...) / N and the hierarchical functional functional M r ζ m over Hierarchy : ℕ → TopCat (H_0 = ℝ, H_{r+1} = ProbabilityMeasure (H_r)) with logMean/trialLog; all hand-written, and phi is an sInf over reals (junk value 0 if unbounded below) with Bochner integrals defaulting to 0 off the integrable class. Not trivializing (convergence to a nontrivial inf is asserted) but the junk-value behaviour is a fidelity risk.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
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cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: 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.
OpenAI’s own note on what its Lean covers
The formalization proves the Mézard–Parisi hierarchical cavity formula for diluted even-arity Ising models in the Panchenko–Talagrand class. Under the class’s factorization, independence, integrability, and positivity assumptions, the finite-system pressure converges to the infimum of the trial functional over all finite hierarchy depths and trial laws. The arity is any even integer at least two and the interaction density is positive.
Mézard–Parisi variational equality for diluted even-arity spin glasses DilutedSpin
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the Mézard–Parisi formula, with existence of the thermodynamic limit, for diluted even-arity models in the Panchenko–Talagrand class.
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DilutedSpinTwo reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “CERTIFIED” | Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source
Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.
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Why this passage is shown
Theorem 6.5 (Identity on prescribed trees)
In The Mézard–Parisi formula for diluted spin glasses, 23 September 2026
Why this passage is shown
Theorem 7.1
In The Mézard–Parisi formula for diluted spin glasses, 23 September 2026
Why this passage is shown
Theorem 2.1
In The Mézard–Parisi formula for diluted spin glasses, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, DilutedSpin
Proof-verification report
MathVet, DilutedSpin
Lean statement
DilutedSpin