Lean statement

Mézard–Parisi variational equality for diluted even-arity spin glasses

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Theorem selected for verification: OAI.DilutedSpinGlass.mezard_parisi

Lean statement file: DilutedSpin.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 221 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem mezard_parisi {p : ℕ} (M : Model p) (hM : Admissible M) :    Tendsto (pressure M) atTop (𝓝 (variationalValue M)) := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.DilutedSpinGlass.Spin line 12
```
abbrev Spin := Bool
```

definition OAI.DilutedSpinGlass.spin line 14

def spin (s : Spin) : ℝ := if s then 1 else -1

abbreviation OAI.DilutedSpinGlass.Interaction line 16

abbrev Interaction (p : ℕ) := (Fin p → Spin) → ℝ

abbreviation OAI.DilutedSpinGlass.InteractionSample line 19

abbrev InteractionSample (p : ℕ) :=  Interaction p × ℝ × ℝ × (Fin p → Spin → ℝ)

structure OAI.DilutedSpinGlass.Model line 22

structure Model (p : ℕ) where  alpha : ℝ≥0  disorder : ProbabilityMeasure (InteractionSample p)  field : ProbabilityMeasure ℝ

structure OAI.DilutedSpinGlass.Admissible line 29

structure Admissible {p : ℕ} (M : Model p) : Prop where

Show the remaining 22 lines

  arity : 2 ≤ p  even : Even p  density : 0 < M.alpha  interaction_integrable : Integrable (fun z : InteractionSample p => ‖z.1‖)    M.disorder.toMeasure  field_integrable : Integrable (fun h : ℝ => |h|) M.field.toMeasure  factorization : ∀ᵐ z ∂M.disorder.toMeasure,    0 < z.2.1 ∧ ∀ s : Fin p → Spin,      Real.exp (z.1 s) = z.2.1 * (1 + z.2.2.1 * ∏ l, z.2.2.2 l (s l)) ∧      |z.2.2.1 * ∏ l, z.2.2.2 l (s l)| < 1  independent_f : iIndepFun (fun l (z : InteractionSample p) => z.2.2.2 l)    M.disorder.toMeasure  identically_distributed_f : ∀ l j : Fin p,    IdentDistrib (fun z : InteractionSample p => z.2.2.2 l)      (fun z : InteractionSample p => z.2.2.2 j)      M.disorder.toMeasure M.disorder.toMeasure  independent_b : IndepFun (fun z : InteractionSample p => z.2.2.1)    (fun z : InteractionSample p => z.2.2.2) M.disorder.toMeasure  finite_b_moments : ∀ n : ℕ, 1 ≤ n →    Integrable (fun z : InteractionSample p => (-z.2.2.1) ^ n) M.disorder.toMeasure  positivity : ∀ n : ℕ, 1 ≤ n →    0 ≤ ∫ z : InteractionSample p, (-z.2.2.1) ^ n ∂M.disorder.toMeasure

definition OAI.DilutedSpinGlass.logWeight line 54

noncomputable def logWeight {p N k : ℕ} (theta : Fin k → InteractionSample p)    (h : Fin N → ℝ) (indices : Fin k → Fin p → Fin N) (σ : Fin N → Spin) : ℝ :=  (∑ j, (theta j).1 (fun l => σ (indices j l))) + ∑ i, h i * spin (σ i)

definition OAI.DilutedSpinGlass.logPartition line 58

noncomputable def logPartition {p N k : ℕ} (theta : Fin k → InteractionSample p)    (h : Fin N → ℝ) (indices : Fin k → Fin p → Fin N) : ℝ :=  Real.log (∑ σ : Fin N → Spin, Real.exp (logWeight theta h indices σ))

definition OAI.DilutedSpinGlass.pressure line 64

noncomputable def pressure {p : ℕ} (M : Model p) (N : ℕ) : ℝ :=  (∫ k : ℕ,    ∫ theta : Fin k → InteractionSample p,      ∫ h : Fin N → ℝ,        ((∑ indices : Fin k → Fin p → Fin N, logPartition theta h indices) /          (Fintype.card (Fin k → Fin p → Fin N) : ℝ))      ∂Measure.pi (fun _ : Fin N => M.field.toMeasure)    ∂Measure.pi (fun _ : Fin k => M.disorder.toMeasure)  ∂poissonMeasure (M.alpha * N)) / N

definition OAI.DilutedSpinGlass.q line 74

noncomputable def q (x : ℝ) (s : Spin) : ℝ :=  Real.exp (x * spin s) / (2 * Real.cosh x)

definition OAI.DilutedSpinGlass.edge line 77

noncomputable def edge {p : ℕ} (theta : Interaction p) (x : Fin p → ℝ) : ℝ :=  ∑ s : Fin p → Spin, Real.exp (theta s) * ∏ l, q (x l) (s l)

definition OAI.DilutedSpinGlass.appendSpin line 81

def appendSpin {p : ℕ} (s : Fin (p - 1) → Spin) (ε : Spin) (l : Fin p) : Spin :=  if h : l.val < p - 1 then s ⟨l.val, h⟩ else ε

definition OAI.DilutedSpinGlass.message line 84

noncomputable def message {p : ℕ} (theta : Interaction p)    (x : Fin (p - 1) → ℝ) (ε : Spin) : ℝ :=  Real.log (∑ s : Fin (p - 1) → Spin,    Real.exp (theta (appendSpin s ε)) * ∏ l, q (x l) (s l))

definition OAI.DilutedSpinGlass.siteLog line 89

noncomputable def siteLog {p k : ℕ} (theta : Fin k → InteractionSample p)    (h : ℝ) (x : (Fin k × Fin (p - 1)) → ℝ) : ℝ :=  Real.log ((∑ ε : Spin, Real.exp    (h * spin ε + ∑ j, message (theta j).1 (fun l => x (j, l)) ε)) / 2)

definition OAI.DilutedSpinGlass.Hierarchy line 100

noncomputable def Hierarchy : ℕ → TopCat  | 0 => TopCat.of ℝ  | r + 1 => TopCat.of (ProbabilityMeasure (Hierarchy r))

definition OAI.DilutedSpinGlass.logMean line 107

noncomputable def logMean {ι : Type} [Fintype ι] :    (r : ℕ) → ((ι → ℝ) → ℝ) → (Fin r → ℝ) → (ι → Hierarchy r) → ℝ  | 0, g, _, x => g x  | r + 1, g, m, eta =>    Real.log (∫ x : ι → Hierarchy r,      Real.exp (m 0 * logMean r g (fun i => m i.succ) x)      ∂Measure.pi (fun i => (eta i).toMeasure)) / m 0

definition OAI.DilutedSpinGlass.trialLog line 117

noncomputable def trialLog {ι : Type} [Fintype ι] (r : ℕ)    (ζ : Hierarchy (r + 1)) (m : Fin r → ℝ) (g : (ι → ℝ) → ℝ) : ℝ :=  ∫ eta : ι → Hierarchy r, logMean r g m eta    ∂Measure.pi (fun _ : ι => ζ.toMeasure)

definition OAI.DilutedSpinGlass.Exponents line 123

def Exponents {r : ℕ} (m : Fin r → ℝ) : Prop :=  StrictMono m ∧ ∀ i, 0 < m i ∧ m i < 1

definition OAI.DilutedSpinGlass.functional line 127

noncomputable def functional {p : ℕ} (M : Model p) (r : ℕ)    (ζ : Hierarchy (r + 1)) (m : Fin r → ℝ) : ℝ :=

Show the remaining 9 lines

  Real.log 2 +    (∫ k : ℕ,      ∫ theta : Fin k → InteractionSample p,        ∫ h : ℝ, trialLog r ζ m (siteLog theta h) ∂M.field.toMeasure      ∂Measure.pi (fun _ : Fin k => M.disorder.toMeasure)    ∂poissonMeasure (M.alpha * p)) -    (M.alpha : ℝ) * (p - 1 : ℕ) *      (∫ theta : InteractionSample p,        trialLog r ζ m (fun x => Real.log (edge theta.1 x)) ∂M.disorder.toMeasure)

definition OAI.DilutedSpinGlass.phi line 139

noncomputable def phi {p : ℕ} (M : Model p) (r : ℕ) : ℝ :=  sInf {v : ℝ | ∃ (ζ : Hierarchy (r + 1)) (m : Fin r → ℝ),    Exponents m ∧ v = functional M r ζ m}

definition OAI.DilutedSpinGlass.variationalValue line 143

noncomputable def variationalValue {p : ℕ} (M : Model p) : ℝ :=  ⨅ r : ℕ, phi M r

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: integrability, limit, model, positivity and thermodynamic.

Theorem 2.1 of The Mézard–Parisi formula for diluted spin glasses

Theorem

Under (model:integrability)–(model:positivity), the thermodynamic limit exists and

\lim_{N\to\infty}F_N=\inf_{r\ge0}\Phi_r.

Section 2, The model and the variational principle Read it in the paper TeX source, line 130

Show the TeX

Under \eqref{model:integrability}--\eqref{model:positivity}, the thermodynamic
limit exists and
\begin{equation}
 \lim_{N\to\infty}F_N=\inf_{r\ge0}\Phi_r.
\end{equation}

Other possible paper matches, closest wording first

Corollary 10.1 of The Mézard–Parisi formula for diluted spin glasses. Words of the Lean statement, its names split into words, that also appear in the theorem: real, field, theta, variational, independent, spin and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: diluted and formula.
Proposition 4.1 of The Mézard–Parisi formula for diluted spin glasses. Words of the Lean statement, its names split into words, that also appear in the theorem: admissible, hypotheses, level, finite, trial, zeta and two more.
Corollary 10.3 of The Mézard–Parisi formula for diluted spin glasses. Words of the Lean statement, its names split into words, that also appear in the theorem: max, sigma, expected, log, weight, variational and four more.
Corollary 10.2 of The Mézard–Parisi formula for diluted spin glasses. Words of the Lean statement, its names split into words, that also appear in the theorem: choice, gives, weight, theta, independent, even and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: formula.
Corollary 8.4 of The Mézard–Parisi formula for diluted spin glasses. Words of the Lean statement, its names split into words, that also appear in the theorem: functionals, field, interaction, phi, law and model.
Theorem 7.1 of The Mézard–Parisi formula for diluted spin glasses. Words of the Lean statement, its names split into words, that also appear in the theorem: averaged, topology, eta and depths.

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Reported Comparator runs

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Carlos Toledo’s cert-machine

How they ran it, in their words

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 (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

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Do family 221’s Lean statements express the advertised result?

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MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
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.
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement 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.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

154 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 66 to its own declarations, 90 to Mathlib’s documentation. 24 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves mezard_parisi exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches 21 of the file’s 23 other declarations; with the theorem itself they span 96 lines. 4 of them are named in the statement itself.

Names in the namespace OAI.DilutedSpinGlass are listed without it.

abbrev Spinline 12library
def spinline 14library
abbrev Interactionline 16library
abbrev InteractionSampleline 19library
structure Modelline 22in the statementlibrary
structure Admissibleline 29in the statementlibrary
def logWeightline 54library
def logPartitionline 58library
def pressureline 64in the statementlibrary
def qline 74library
def edgeline 77library
def appendSpinline 81library
def messageline 84library
def siteLogline 89library
def Hierarchyline 100library
def logMeanline 107library
def trialLogline 117library
def Exponentsline 123library
def functionalline 127library
def philine 139library
def variationalValueline 143in the statementlibrary
theorem mezard_parisiline 148selected theoremlibrary

Other Lean rules that may affect the statement 2

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance instMeasurableSpaceCarrier_challenge line 96
instance instBorelSpaceCarrier_challenge line 97

import Mathlibnamespace OAI open MeasureTheory ProbabilityTheory Filteropen scoped BigOperators ENNReal NNReal Topologynamespace DilutedSpinGlassabbrev Spin := Booldef spin (s : Spin) : ℝ := if s then 1 else -1abbrev Interaction (p : ℕ) := (Fin p → Spin) → ℝ/-- One interaction together with its factorization witness: theta,a,b,f. -/abbrev InteractionSample (p : ℕ) :=  Interaction p × ℝ × ℝ × (Fin p → Spin → ℝ)structure Model (p : ℕ) where  alpha : ℝ≥0  disorder : ProbabilityMeasure (InteractionSample p)  field : ProbabilityMeasure ℝ/-- Exactly the standing hypotheses (model:integrability through positivity).`norm` on the finite function space is max_s |theta(s)|. -/structure Admissible {p : ℕ} (M : Model p) : Prop where  arity : 2 ≤ p  even : Even p  density : 0 < M.alpha  interaction_integrable : Integrable (fun z : InteractionSample p => ‖z.1‖)    M.disorder.toMeasure  field_integrable : Integrable (fun h : ℝ => |h|) M.field.toMeasure  factorization : ∀ᵐ z ∂M.disorder.toMeasure,    0 < z.2.1 ∧ ∀ s : Fin p → Spin,      Real.exp (z.1 s) = z.2.1 * (1 + z.2.2.1 * ∏ l, z.2.2.2 l (s l)) ∧      |z.2.2.1 * ∏ l, z.2.2.2 l (s l)| < 1  independent_f : iIndepFun (fun l (z : InteractionSample p) => z.2.2.2 l)    M.disorder.toMeasure  identically_distributed_f : ∀ l j : Fin p,    IdentDistrib (fun z : InteractionSample p => z.2.2.2 l)      (fun z : InteractionSample p => z.2.2.2 j)      M.disorder.toMeasure M.disorder.toMeasure  independent_b : IndepFun (fun z : InteractionSample p => z.2.2.1)    (fun z : InteractionSample p => z.2.2.2) M.disorder.toMeasure  finite_b_moments : ∀ n : ℕ, 1 ≤ n →    Integrable (fun z : InteractionSample p => (-z.2.2.1) ^ n) M.disorder.toMeasure  positivity : ∀ n : ℕ, 1 ≤ n →    0 ≤ ∫ z : InteractionSample p, (-z.2.2.1) ^ n ∂M.disorder.toMeasure/-- Minus the Hamiltonian, with all index choices including repetitions. -/noncomputable def logWeight {p N k : ℕ} (theta : Fin k → InteractionSample p)    (h : Fin N → ℝ) (indices : Fin k → Fin p → Fin N) (σ : Fin N → Spin) : ℝ :=  (∑ j, (theta j).1 (fun l => σ (indices j l))) + ∑ i, h i * spin (σ i)noncomputable def logPartition {p N k : ℕ} (theta : Fin k → InteractionSample p)    (h : Fin N → ℝ) (indices : Fin k → Fin p → Fin N) : ℝ :=  Real.log (∑ σ : Fin N → Spin, Real.exp (logWeight theta h indices σ))/-- F_N, using an explicit uniform average over the independent index labels.The value at N=0 is irrelevant to the limit; valid system sizes are N>=1. -/noncomputable def pressure {p : ℕ} (M : Model p) (N : ℕ) : ℝ :=  (∫ k : ℕ,    ∫ theta : Fin k → InteractionSample p,      ∫ h : Fin N → ℝ,        ((∑ indices : Fin k → Fin p → Fin N, logPartition theta h indices) /          (Fintype.card (Fin k → Fin p → Fin N) : ℝ))      ∂Measure.pi (fun _ : Fin N => M.field.toMeasure)    ∂Measure.pi (fun _ : Fin k => M.disorder.toMeasure)  ∂poissonMeasure (M.alpha * N)) / Nnoncomputable def q (x : ℝ) (s : Spin) : ℝ :=  Real.exp (x * spin s) / (2 * Real.cosh x)noncomputable def edge {p : ℕ} (theta : Interaction p) (x : Fin p → ℝ) : ℝ :=  ∑ s : Fin p → Spin, Real.exp (theta s) * ∏ l, q (x l) (s l)/-- Append the distinguished cavity spin in the last position. -/def appendSpin {p : ℕ} (s : Fin (p - 1) → Spin) (ε : Spin) (l : Fin p) : Spin :=  if h : l.val < p - 1 then s ⟨l.val, h⟩ else εnoncomputable def message {p : ℕ} (theta : Interaction p)    (x : Fin (p - 1) → ℝ) (ε : Spin) : ℝ :=  Real.log (∑ s : Fin (p - 1) → Spin,    Real.exp (theta (appendSpin s ε)) * ∏ l, q (x l) (s l))noncomputable def siteLog {p k : ℕ} (theta : Fin k → InteractionSample p)    (h : ℝ) (x : (Fin k × Fin (p - 1)) → ℝ) : ℝ :=  Real.log ((∑ ε : Spin, Real.exp    (h * spin ε + ∑ j, message (theta j).1 (fun l => x (j, l)) ε)) / 2)/-- Use literally the Borel sigma-field of the weak topology at each level,as stipulated in model.tex. -/local instance instMeasurableSpaceCarrier_challenge (X : TopCat) : MeasurableSpace X := borel Xlocal instance instBorelSpaceCarrier_challenge (X : TopCat) : BorelSpace X := ⟨rfl⟩/-- Nested probability laws: H_0=R and H_{j+1}=P(H_j), with weak topologies. -/noncomputable def Hierarchy : ℕ → TopCat  | 0 => TopCat.of ℝ  | r + 1 => TopCat.of (ProbabilityMeasure (Hierarchy r))/-- Logarithmic form of the conditional power means T_0 using independentmessage labels. The current eta's are held fixed; only their descendantsare integrated. Physical disorder is a fixed argument of g throughout. -/noncomputable def logMean {ι : Type} [Fintype ι] :    (r : ℕ) → ((ι → ℝ) → ℝ) → (Fin r → ℝ) → (ι → Hierarchy r) → ℝ  | 0, g, _, x => g x  | r + 1, g, m, eta =>    Real.log (∫ x : ι → Hierarchy r,      Real.exp (m 0 * logMean r g (fun i => m i.succ) x)      ∂Measure.pi (fun i => (eta i).toMeasure)) / m 0/-- Expectation over the root laws eta_0 of each independent message label.This also gives the r=0 convention (iid real messages, no power means). -/noncomputable def trialLog {ι : Type} [Fintype ι] (r : ℕ)    (ζ : Hierarchy (r + 1)) (m : Fin r → ℝ) (g : (ι → ℝ) → ℝ) : ℝ :=  ∫ eta : ι → Hierarchy r, logMean r g m eta    ∂Measure.pi (fun _ : ι => ζ.toMeasure)/-- Admissible exponents 0<m_1<...<m_r<1; vacuous at r=0. -/def Exponents {r : ℕ} (m : Fin r → ℝ) : Prop :=  StrictMono m ∧ ∀ i, 0 < m i ∧ m i < 1/-- The exact finite-depth cavity functional B_r. -/noncomputable def functional {p : ℕ} (M : Model p) (r : ℕ)    (ζ : Hierarchy (r + 1)) (m : Fin r → ℝ) : ℝ :=  Real.log 2 +    (∫ k : ℕ,      ∫ theta : Fin k → InteractionSample p,        ∫ h : ℝ, trialLog r ζ m (siteLog theta h) ∂M.field.toMeasure      ∂Measure.pi (fun _ : Fin k => M.disorder.toMeasure)    ∂poissonMeasure (M.alpha * p)) -    (M.alpha : ℝ) * (p - 1 : ℕ) *      (∫ theta : InteractionSample p,        trialLog r ζ m (fun x => Real.log (edge theta.1 x)) ∂M.disorder.toMeasure)noncomputable def phi {p : ℕ} (M : Model p) (r : ℕ) : ℝ :=  sInf {v : ℝ | ∃ (ζ : Hierarchy (r + 1)) (m : Fin r → ℝ),    Exponents m ∧ v = functional M r ζ m}noncomputable def variationalValue {p : ℕ} (M : Model p) : ℝ :=  ⨅ r : ℕ, phi M r/-- main:theorem. No boundedness of interactions, fields, or messages is added.Existence of the thermodynamic limit is part of this convergence assertion. -/theorem mezard_parisi {p : ℕ} (M : Model p) (hM : Admissible M) :    Tendsto (pressure M) atTop (𝓝 (variationalValue M)) := by  sorryend DilutedSpinGlassend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

The Mézard–Parisi formula for diluted spin glasses

Theorem

Under (model:integrability)–(model:positivity), the thermodynamic limit exists and

\lim_{N\to\infty}F_N=\inf_{r\ge0}\Phi_r.

Section 2, The model and the variational principle Read it in the paper TeX source, line 130

Show the TeX

Under \eqref{model:integrability}--\eqref{model:positivity}, the thermodynamic
limit exists and
\begin{equation}
 \lim_{N\to\infty}F_N=\inf_{r\ge0}\Phi_r.
\end{equation}

Theorem (Identity on prescribed trees)

For the perturbation family defined above, along every sequence from Lemma pert:selection,

\sum_{E\in\mathfrak E(\mathcal T,a_*,\mathcal C)} c_E\,\mathbb E[fR_E] -\mathbb E f \sum_{E\in\mathfrak E(\{a_*\},a_*,\mathcal C)} c_E\,\mathbb E[R_E] \longrightarrow0.

This holds for every fixed old tree, either kind of anchor, every bounded polynomial f in its multioverlaps, every finite number of ordered added colors, and every prescribed constraint \mathcal C. The empty addition is allowed. The error is taken to zero at fixed L, fixed tree, and fixed constraint, before any limit in L.

Section 6, Perturbations and identities on prescribed trees Read it in the paper TeX source, line 451

Show the TeX

For the perturbation family defined above, along every sequence from
Lemma~\ref{pert:selection},
\begin{equation}
 \sum_{E\in\mathfrak E(\mathcal T,a_*,\mathcal C)}
       c_E\,\mathbb E[fR_E]
 -\mathbb E f
  \sum_{E\in\mathfrak E(\{a_*\},a_*,\mathcal C)}
       c_E\,\mathbb E[R_E]
 \longrightarrow0.
 
\end{equation}
This holds for every fixed old tree, either kind of anchor, every
bounded polynomial $f$ in its multioverlaps, every finite number of
ordered added colors, and every prescribed constraint $\mathcal C$.
The empty addition is allowed.  The error is taken to zero at fixed
$L$, fixed tree, and fixed constraint, before any limit in $L$.

Theorem

For every fixed labeled topology S,

\lim_{L\to\infty}\limsup_{N\to\infty} \mathcal A_{L,\eta}\operatorname{Var}(R_S)=0 \qquad(\eta>0),

where only the b(S) branching depths are averaged. For a singleton, whose average has no coordinates, \operatorname{Var}(R_S)\to0 as N\to\infty at every fixed L.

Section 7, Decorrelation on prescribed trees Read it in the paper TeX source, line 63

Show the TeX

For every fixed labeled topology $S$,
\begin{equation}
 \lim_{L\to\infty}\limsup_{N\to\infty}
       \mathcal A_{L,\eta}\operatorname{Var}(R_S)=0
       \qquad(\eta>0),
 
\end{equation}
where only the $b(S)$ branching depths are averaged. For a
singleton, whose average has no coordinates,
$\operatorname{Var}(R_S)\to0$ as $N\to\infty$ at every fixed $L$.

Configuration and file facts

File: lean/ComparatorChallenges/DilutedSpin.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 221 link it.
Solution module: OAI.Probability.DilutedSpin.Main
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 23 local vocabulary declarations
Lines of Lean, without comments or blank lines: 98
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

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