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Lean statement

Parisi ground-state value identities and finite Gaussian approximation

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.SKValue.value_consequences

Lean statement file: SKValue.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 281 link it.

See the candidate theoremWho ran the proof-verification programReport 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 value_consequences :    (∀ (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ),      IsMinimizer W γ → IsDiffusion W γ X → ScalarValueConclusion W γ X) ∧    (∀ (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ) (T : ℝ),      0 < T → T < 1 → IsMinimizer W γ → IsDiffusion W γ X →      FiniteGaussianConclusion W γ X T) ∧    (∀ (W : BrownianSpace) (γ : OrderParameter),      IsMinimizer W γ → groundStateValue = parisi W γ) := 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.

  • definition OAI.SKValue.standardGaussian line 10

    noncomputable def standardGaussian : Measure ℝ := gaussianReal 0 1

  • definition OAI.SKValue.gaussianProduct line 14

    noncomputable def gaussianProduct (ι : Type) [Fintype ι] : Measure (ι → ℝ) :=  Measure.pi (fun _ : ι ↦ standardGaussian)

  • definition OAI.SKValue.coordinate line 21

    def coordinate (N : ℕ) (j : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=  z ⟨j % (N+1), Nat.mod_lt _ (Nat.succ_pos _)⟩

  • definition OAI.SKValue.stepSize line 24

    noncomputable def stepSize (T : ℝ) (N : ℕ) : ℝ := T / N

  • definition OAI.SKValue.meshTime line 25

    noncomputable def meshTime (T : ℝ) (N j : ℕ) : ℝ := j * stepSize T N

  • definition OAI.SKValue.euler line 27

    noncomputable def euler (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)    (z : Fin (N+1) → ℝ) : ℕ → ℝ  | 0 => 0  | j+1 => euler T N γ u z j + Real.sqrt (stepSize T N) * coordinate N j z +      stepSize T N * γ (meshTime T N j) * u (meshTime T N j) (euler T N γ u z j)

  • structure OAI.SKValue.OrderParameter line 33

    structure OrderParameter where  coeff : ℝ → ℝ  nonneg : ∀ t ∈ Ico (0 : ℝ) 1, 0 ≤ coeff t  monotone : MonotoneOn coeff (Ico (0 : ℝ) 1)  rightContinuous : ∀ t ∈ Ico (0 : ℝ) 1, ContinuousWithinAt coeff (Ici t) t  integrable : IntegrableOn coeff (Ioc (0 : ℝ) 1)

  • structure OAI.SKValue.BrownianSpace line 40

    structure BrownianSpace where  Ω : Type  measurableSpace : MeasurableSpace Ω  μ : @Measure Ω measurableSpace  probability : @IsProbabilityMeasure Ω measurableSpace μ  B : ℝ≥0 → Ω → ℝ  measurable : ∀ t, @StronglyMeasurable Ω ℝ _ measurableSpace (B t)  brownian : @IsBrownianReal Ω measurableSpace B μ

  • definition OAI.SKValue.BrownianSpace.filtration line 51

    noncomputable def BrownianSpace.filtration (W : BrownianSpace) :    Filtration ℝ≥0 W.measurableSpace := Filtration.natural W.B W.measurable

  • definition OAI.SKValue.shiftedFiltration line 54

    noncomputable def shiftedFiltration (W : BrownianSpace) (t : ℝ) :    Filtration ℝ≥0 W.measurableSpace :=  Filtration.natural (fun s ω ↦ W.B (Real.toNNReal t + s) ω - W.B (Real.toNNReal t) ω)    (fun _ ↦ (W.measurable _).sub (W.measurable _))

  • definition OAI.SKValue.Admissible line 59

    def Admissible (W : BrownianSpace) (t : ℝ) (α : ℝ≥0 → W.Ω → ℝ) : Prop :=  IsProgressive (shiftedFiltration W t) α ∧ ∀ s ω, |α s ω| ≤ 1

  • definition OAI.SKValue.controlPayoff line 62

    noncomputable def controlPayoff (W : BrownianSpace) (γ : OrderParameter)    (t x : ℝ) (α : ℝ≥0 → W.Ω → ℝ) : ℝ :=  ∫ ω, (|x + W.B 1 ω - W.B (Real.toNNReal t) ω +    ∫ s in (0 : ℝ)..(1-t), γ.coeff (t+s) * α (Real.toNNReal s) ω| -    (1/2 : ℝ) * ∫ s in (0 : ℝ)..(1-t), γ.coeff (t+s) * α (Real.toNNReal s) ω ^ 2) ∂W.μ

  • definition OAI.SKValue.phi line 68

    noncomputable def phi (W : BrownianSpace) (γ : OrderParameter) (t x : ℝ) : ℝ :=  sSup {r | ∃ α, Admissible W t α ∧ r = controlPayoff W γ t x α}

  • definition OAI.SKValue.gradient line 71

    noncomputable def gradient (W : BrownianSpace) (γ : OrderParameter) (t x : ℝ) : ℝ :=  deriv (phi W γ t) x

  • definition OAI.SKValue.curvature line 74

    noncomputable def curvature (W : BrownianSpace) (γ : OrderParameter) (t x : ℝ) : ℝ :=  deriv (gradient W γ t) x

  • definition OAI.SKValue.parisi line 77

    noncomputable def parisi (W : BrownianSpace) (γ : OrderParameter) : ℝ :=  phi W γ 0 0 - (1/2 : ℝ) * ∫ t in (0 : ℝ)..1, t * γ.coeff t

  • definition OAI.SKValue.IsMinimizer line 80

    def IsMinimizer (W : BrownianSpace) (γ : OrderParameter) : Prop :=  ∀ η : OrderParameter, parisi W γ ≤ parisi W η

  • definition OAI.SKValue.IsDiffusion line 83

    def IsDiffusion (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ) : Prop :=  Adapted W.filtration (fun t ω ↦ X (min (t : ℝ) 1) ω) ∧  (∀ᵐ ω ∂W.μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) 1) ∧    ∀ t ∈ Icc (0 : ℝ) 1,      X t ω = W.B (Real.toNNReal t) ω +        ∫ s in (0 : ℝ)..t, γ.coeff s * gradient W γ s (X s ω))

  • definition OAI.SKValue.spin line 90

    noncomputable def spin (b : Bool) : ℝ := if b then 1 else -1

  • definition OAI.SKValue.hamiltonian line 92

    noncomputable def hamiltonian (n : ℕ) (J : (Fin n × Fin n) → ℝ)    (σ : Fin n → Bool) : ℝ :=  (Real.sqrt (n : ℝ))⁻¹ *    ∑ i : Fin n, ∑ j : Fin n, if i < j then J (i,j) * spin (σ i) * spin (σ j) else 0

  • definition OAI.SKValue.groundStateSequence line 97

    noncomputable def groundStateSequence (n : ℕ) : ℝ :=  (n : ℝ)⁻¹ * ∫ J, sSup (range (hamiltonian n J)) ∂gaussianProduct (Fin n × Fin n)

  • definition OAI.SKValue.groundStateValue line 100

    noncomputable def groundStateValue : ℝ := limUnder atTop groundStateSequence

  • definition OAI.SKValue.gradientStopped line 102

    noncomputable def gradientStopped (W : BrownianSpace) (γ : OrderParameter)    (X : ℝ → W.Ω → ℝ) (U : W.Ω → ℝ) (t : ℝ≥0) (ω : W.Ω) : ℝ :=  if (t : ℝ) < 1 then gradient W γ t (X t ω) else U ω

  • definition OAI.SKValue.ScalarValueConclusion line 106

    def ScalarValueConclusion (W : BrownianSpace) (γ : OrderParameter)    (X : ℝ → W.Ω → ℝ) : Prop :=
    Show the remaining 11 lines
      Tendsto groundStateSequence atTop (𝓝 groundStateValue) ∧  ∃ U : W.Ω → ℝ,    Martingale (gradientStopped W γ X U) W.filtration W.μ ∧    (∀ᵐ ω ∂W.μ, |U ω| ≤ 1 ∧ (U ω)^2 = 1 ∧      Tendsto (fun t ↦ gradient W γ t (X t ω)) (𝓝[<] (1 : ℝ)) (𝓝 (U ω))) ∧    Tendsto (fun t ↦ ∫ ω, (gradient W γ t (X t ω) - U ω)^2 ∂W.μ)      (𝓝[<] (1 : ℝ)) (𝓝 (0 : ℝ)) ∧    groundStateValue = (∫ ω, |X 1 ω| ∂W.μ) - ∫ t in (0 : ℝ)..1, t * γ.coeff t ∧    groundStateValue = ∫ ω, U ω * W.B 1 ω ∂W.μ ∧    groundStateValue = ∫ t in (0 : ℝ)..1, ∫ ω, curvature W γ t (X t ω) ∂W.μ ∧    IntervalIntegrable (fun t ↦ ∫ ω, curvature W γ t (X t ω) ∂W.μ) volume 0 1

9 more definitions are reached the same way; the full file lists them.

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: equivalently, field, finite, model and zero.

Statements in QAOA attains the SK ground-state energy in the thermodynamic-first limit come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of QAOA attains the SK ground-state energy in the thermodynamic-first limit

Theorem

For the zero-field SK model (eq:sk-model),

\lim_{p\to\infty}Q_p=\Pstar.

Equivalently, for every \varepsilon>0 there exist a finite integer p and deterministic real vectors \gamma,\beta\in\R^p, independent of n and J, such that v_p(\gamma,\beta)\ge\Pstar-\varepsilon.

Section 1, Introduction and statement of the result Read it in the paper TeX source, line 111
Show the TeX
For the zero-field SK model~\eqref{eq:sk-model},
\begin{equation}
 \lim_{p\to\infty}Q_p=\Pstar.
\end{equation}
Equivalently, for every $\varepsilon>0$ there exist a finite integer $p$ and
deterministic real vectors $\gamma,\beta\in\R^p$, independent of $n$ and $J$,
such that $v_p(\gamma,\beta)\ge\Pstar-\varepsilon$.

Other possible paper matches, closest wording first

  • Proposition 4.2 of QAOA attains the SK ground-state energy in the thermodynamic-first limit. Words of the Lean statement, its names split into words, that also appear in the theorem: space, top, spins, strongly, sequence, scalar and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: asserted, proved, convergence and tends.
  • Proposition 7.2 of Full support of the zero-temperature Sherrington-Kirkpatrick order parameter. Words of the Lean statement, its names split into words, that also appear in the theorem: standard, mesh, minimizer, sign, gaussian, integer and five more.
  • Corollary 2.13 of QAOA attains the SK ground-state energy in the thermodynamic-first limit. Words of the Lean statement, its names split into words, that also appear in the theorem: time, hamiltonian, spin, sequence, scalar, continuous and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: circuit and tend.
  • Proposition 2.2 of QAOA attains the SK ground-state energy in the thermodynamic-first limit. Words of the Lean statement, its names split into words, that also appear in the theorem: shift, coordinates, top, strongly, sqrt and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: resulting and energy.
  • Corollary 7.1 of Full support of the zero-temperature Sherrington-Kirkpatrick order parameter. Words of the Lean statement, its names split into words, that also appear in the theorem: martingale, minimizer, diffusion, value, phi and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: terminal and integral.
  • Theorem 2.12 of QAOA attains the SK ground-state energy in the thermodynamic-first limit. Words of the Lean statement, its names split into words, that also appear in the theorem: measurable, sequence, scalar, value, parameters, coefficients and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: resulting, convergence and energy.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

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

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 281’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

  • MathVet

    Finds a supporting or a different result stated Their term: “supporting-only”.

    The challenge statements state a lemma or auxiliary result, not the headline. Their record

    Their note: No QAOA statement exists. The challenges are SKValue.value_consequences (conditional on IsMinimizer W γ and a diffusion X: the finite-size SK ground-state energy per spin groundStateSequence n converges to groundStateValue = parisi W γ, with martingale/curvature-integral formulas and convergence of finite Gaussian coefficient sums) and SKFullSupport.full_support (full support of zero-temperature Parisi minimizers). Docs: 'it does not itself assert convergence of QAOA circuit energies', and 'does not separately assert existence of a minimizer'. The summary's headline (QAOA with finite depth and size/disorder-independent angles reaches the SK ground-state energy per spin, size before depth, plus MaxCut on random regular graphs) is absent.

    Definitions the group asks readers to compare with the paper: Custom Parisi/stochastic-control objects: phi W γ t x := sSup {controlPayoff ..} over Admissible progressive controls (real sSup), gradient := deriv (phi W γ t), curvature := deriv (gradient ..), parisi, IsMinimizer, groundStateValue := limUnder atTop groundStateSequence (convergence asserted in ScalarValueConclusion); derivatives are Mathlib deriv (junk 0 where not differentiable).

  • Carlos Toledo’s cert-machine

    Finds a supporting or a different result stated Their term: “DIFFERENT”.

    The challenge statements state a different theorem from the headline. Their record

    Their note: The whole headline. The challenges state (i) value identities for the zero-temperature SK model conditional on a Parisi minimizer and its diffusion — convergence of the finite-n ground-state energy, equality with the Parisi value, terminal-martingale and integrated-curvature formulas, convergence of finite Gaussian coefficient sums — and (ii) full support of every zero-temperature Parisi minimizer. These are ingredients of the QAOA argument; none mentions QAOA.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“It also proves convergence of finite Gaussian coefficient sums to the curvature integral, which approaches the ground-state value as the terminal time tends to one. The selected statement covers these value and approximation results; it does not itself assert convergence of QAOA circuit energies.”

An AI agent drafted an interpretation of that note, classifying this statement as “states a supporting result”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

175 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: 122 to its own declarations, 81 to Mathlib’s documentation. 35 declared names also link to the same declaration in OpenAI’s library. 5 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names 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.

import Mathlibnamespace OAIopen MeasureTheory ProbabilityTheory Filter Setopen scoped Topology ENNReal NNReal BigOperatorsnamespace SKValuenoncomputable def standardGaussian : Measure ℝ := gaussianReal 0 1instance : IsProbabilityMeasure standardGaussian := by unfold standardGaussian; infer_instancenoncomputable def gaussianProduct (ι : Type) [Fintype ι] : Measure (ι → ℝ) :=  Measure.pi (fun _ : ι ↦ standardGaussian)instance gaussianProduct_probability (ι : Type) [Fintype ι] :    IsProbabilityMeasure (gaussianProduct ι) := by  unfold gaussianProduct  infer_instancedef coordinate (N : ℕ) (j : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=  z ⟨j % (N+1), Nat.mod_lt _ (Nat.succ_pos _)⟩noncomputable def stepSize (T : ℝ) (N : ℕ) : ℝ := T / Nnoncomputable def meshTime (T : ℝ) (N j : ℕ) : ℝ := j * stepSize T Nnoncomputable def euler (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)    (z : Fin (N+1) → ℝ) : ℕ → ℝ  | 0 => 0  | j+1 => euler T N γ u z j + Real.sqrt (stepSize T N) * coordinate N j z +      stepSize T N * γ (meshTime T N j) * u (meshTime T N j) (euler T N γ u z j)structure OrderParameter where  coeff : ℝ → ℝ  nonneg : ∀ t ∈ Ico (0 : ℝ) 1, 0 ≤ coeff t  monotone : MonotoneOn coeff (Ico (0 : ℝ) 1)  rightContinuous : ∀ t ∈ Ico (0 : ℝ) 1, ContinuousWithinAt coeff (Ici t) t  integrable : IntegrableOn coeff (Ioc (0 : ℝ) 1)structure BrownianSpace where  Ω : Type  measurableSpace : MeasurableSpace Ω  μ : @Measure Ω measurableSpace  probability : @IsProbabilityMeasure Ω measurableSpace μ  B : ℝ≥0 → Ω → ℝ  measurable : ∀ t, @StronglyMeasurable Ω ℝ _ measurableSpace (B t)  brownian : @IsBrownianReal Ω measurableSpace B μattribute [instance] BrownianSpace.measurableSpace BrownianSpace.probabilitynoncomputable def BrownianSpace.filtration (W : BrownianSpace) :    Filtration ℝ≥0 W.measurableSpace := Filtration.natural W.B W.measurablenoncomputable def shiftedFiltration (W : BrownianSpace) (t : ℝ) :    Filtration ℝ≥0 W.measurableSpace :=  Filtration.natural (fun s ω ↦ W.B (Real.toNNReal t + s) ω - W.B (Real.toNNReal t) ω)    (fun _ ↦ (W.measurable _).sub (W.measurable _))def Admissible (W : BrownianSpace) (t : ℝ) (α : ℝ≥0 → W.Ω → ℝ) : Prop :=  IsProgressive (shiftedFiltration W t) α ∧ ∀ s ω, |α s ω| ≤ 1noncomputable def controlPayoff (W : BrownianSpace) (γ : OrderParameter)    (t x : ℝ) (α : ℝ≥0 → W.Ω → ℝ) : ℝ :=  ∫ ω, (|x + W.B 1 ω - W.B (Real.toNNReal t) ω +    ∫ s in (0 : ℝ)..(1-t), γ.coeff (t+s) * α (Real.toNNReal s) ω| -    (1/2 : ℝ) * ∫ s in (0 : ℝ)..(1-t), γ.coeff (t+s) * α (Real.toNNReal s) ω ^ 2) ∂W.μnoncomputable def phi (W : BrownianSpace) (γ : OrderParameter) (t x : ℝ) : ℝ :=  sSup {r | ∃ α, Admissible W t α ∧ r = controlPayoff W γ t x α}noncomputable def gradient (W : BrownianSpace) (γ : OrderParameter) (t x : ℝ) : ℝ :=  deriv (phi W γ t) xnoncomputable def curvature (W : BrownianSpace) (γ : OrderParameter) (t x : ℝ) : ℝ :=  deriv (gradient W γ t) xnoncomputable def parisi (W : BrownianSpace) (γ : OrderParameter) : ℝ :=  phi W γ 0 0 - (1/2 : ℝ) * ∫ t in (0 : ℝ)..1, t * γ.coeff tdef IsMinimizer (W : BrownianSpace) (γ : OrderParameter) : Prop :=  ∀ η : OrderParameter, parisi W γ ≤ parisi W ηdef IsDiffusion (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ) : Prop :=  Adapted W.filtration (fun t ω ↦ X (min (t : ℝ) 1) ω) ∧  (∀ᵐ ω ∂W.μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) 1) ∧    ∀ t ∈ Icc (0 : ℝ) 1,      X t ω = W.B (Real.toNNReal t) ω +        ∫ s in (0 : ℝ)..t, γ.coeff s * gradient W γ s (X s ω))noncomputable def spin (b : Bool) : ℝ := if b then 1 else -1noncomputable def hamiltonian (n : ℕ) (J : (Fin n × Fin n) → ℝ)    (σ : Fin n → Bool) : ℝ :=  (Real.sqrt (n : ℝ))⁻¹ *    ∑ i : Fin n, ∑ j : Fin n, if i < j then J (i,j) * spin (σ i) * spin (σ j) else 0noncomputable def groundStateSequence (n : ℕ) : ℝ :=  (n : ℝ)⁻¹ * ∫ J, sSup (range (hamiltonian n J)) ∂gaussianProduct (Fin n × Fin n)noncomputable def groundStateValue : ℝ := limUnder atTop groundStateSequencenoncomputable def gradientStopped (W : BrownianSpace) (γ : OrderParameter)    (X : ℝ → W.Ω → ℝ) (U : W.Ω → ℝ) (t : ℝ≥0) (ω : W.Ω) : ℝ :=  if (t : ℝ) < 1 then gradient W γ t (X t ω) else U ωdef ScalarValueConclusion (W : BrownianSpace) (γ : OrderParameter)    (X : ℝ → W.Ω → ℝ) : Prop :=  Tendsto groundStateSequence atTop (𝓝 groundStateValue) ∧  ∃ U : W.Ω → ℝ,    Martingale (gradientStopped W γ X U) W.filtration W.μ ∧    (∀ᵐ ω ∂W.μ, |U ω| ≤ 1 ∧ (U ω)^2 = 1 ∧      Tendsto (fun t ↦ gradient W γ t (X t ω)) (𝓝[<] (1 : ℝ)) (𝓝 (U ω))) ∧    Tendsto (fun t ↦ ∫ ω, (gradient W γ t (X t ω) - U ω)^2 ∂W.μ)      (𝓝[<] (1 : ℝ)) (𝓝 (0 : ℝ)) ∧    groundStateValue = (∫ ω, |X 1 ω| ∂W.μ) - ∫ t in (0 : ℝ)..1, t * γ.coeff t ∧    groundStateValue = ∫ ω, U ω * W.B 1 ω ∂W.μ ∧    groundStateValue = ∫ t in (0 : ℝ)..1, ∫ ω, curvature W γ t (X t ω) ∂W.μ ∧    IntervalIntegrable (fun t ↦ ∫ ω, curvature W γ t (X t ω) ∂W.μ) volume 0 1noncomputable def eulerPath (W : BrownianSpace) (γ : OrderParameter) (T : ℝ) (N : ℕ)    (z : Fin (N+1) → ℝ) (j : ℕ) : ℝ :=  euler T N γ.coeff (gradient W γ) z jnoncomputable def rawCoefficient (W : BrownianSpace) (γ : OrderParameter)    (T : ℝ) (N j : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=  curvature W γ (meshTime T N j) (eulerPath W γ T N z j)noncomputable def normalizer (W : BrownianSpace) (γ : OrderParameter)    (T : ℝ) (N j : ℕ) : ℝ :=  (Real.sqrt (∫ z, (rawCoefficient W γ T N j z)^2 ∂gaussianProduct (Fin (N+1))))⁻¹noncomputable def gaussianIncrement (W : BrownianSpace) (γ : OrderParameter)    (T : ℝ) (N j : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=  normalizer W γ T N j * rawCoefficient W γ T N j z * coordinate N j znoncomputable def roundedSign (W : BrownianSpace) (γ : OrderParameter)    (T : ℝ) (N : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=  Real.sign (gradient W γ T (eulerPath W γ T N z N) +    2 * cdf standardGaussian (coordinate N N z) - 1)noncomputable def coefficientA (W : BrownianSpace) (γ : OrderParameter)    (T : ℝ) (N j : ℕ) : ℝ :=  if j = 0 then 0 else    ∫ z, roundedSign W γ T N z * gaussianIncrement W γ T N (j-1) z      ∂gaussianProduct (Fin (N+1))noncomputable def coefficientB (W : BrownianSpace) (γ : OrderParameter)    (T : ℝ) (N i : ℕ) : ℝ :=  ∫ z, roundedSign W γ T N z * coordinate N (i-1) z ∂gaussianProduct (Fin (N+1))noncomputable def shiftedCoefficientSum (W : BrownianSpace) (γ : OrderParameter)    (T : ℝ) (N : ℕ) : ℝ :=  ∑ j ∈ Finset.range (N+1), coefficientA W γ T N j * coefficientB W γ T N (j+1)def FiniteGaussianConclusion (W : BrownianSpace) (γ : OrderParameter)    (X : ℝ → W.Ω → ℝ) (T : ℝ) : Prop :=  (∀ᶠ N in atTop, ∀ j < N, 0 < normalizer W γ T N j) ∧  Tendsto (shiftedCoefficientSum W γ T) atTop    (𝓝 (∫ t in (0 : ℝ)..T, ∫ ω, curvature W γ t (X t ω) ∂W.μ)) ∧  Tendsto (fun S ↦ ∫ t in (0 : ℝ)..S, ∫ ω, curvature W γ t (X t ω) ∂W.μ)    (𝓝[<] (1 : ℝ)) (𝓝 groundStateValue)theorem value_consequences :    (∀ (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ),      IsMinimizer W γ → IsDiffusion W γ X → ScalarValueConclusion W γ X) ∧    (∀ (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ) (T : ℝ),      0 < T → T < 1 → IsMinimizer W γ → IsDiffusion W γ X →      FiniteGaussianConclusion W γ X T) ∧    (∀ (W : BrownianSpace) (γ : OrderParameter),      IsMinimizer W γ → groundStateValue = parisi W γ) := by  sorryend SKValueend 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.

QAOA attains the SK ground-state energy in the thermodynamic-first limit

Theorem

For the zero-field SK model (eq:sk-model),

\lim_{p\to\infty}Q_p=\Pstar.

Equivalently, for every \varepsilon>0 there exist a finite integer p and deterministic real vectors \gamma,\beta\in\R^p, independent of n and J, such that v_p(\gamma,\beta)\ge\Pstar-\varepsilon.

Section 1, Introduction and statement of the result Read it in the paper TeX source, line 111
Show the TeX
For the zero-field SK model~\eqref{eq:sk-model},
\begin{equation}
 \lim_{p\to\infty}Q_p=\Pstar.
\end{equation}
Equivalently, for every $\varepsilon>0$ there exist a finite integer $p$ and
deterministic real vectors $\gamma,\beta\in\R^p$, independent of $n$ and $J$,
such that $v_p(\gamma,\beta)\ge\Pstar-\varepsilon$.
Corollary (Regular MaxCut in successive limits)

For every \eta>0 there are an integer p\ge1, deterministic normalized angles \gamma,\beta\in\R^p, and an integer D_0\ge2 such that, for every integer D\ge D_0 and every finite simple unweighted (D+1)-regular graph G of girth greater than 2p+1,

q_{p,G}(2\gamma/\sqrt D,\beta) \ge \frac12+\frac{\Pstar-\eta}{\sqrt D}.

The word p,\gamma,\beta is independent of D, the graph size, and G; the displayed usual MaxCut cost angles, unlike the normalized angles \gamma, are scaled with the degree.

Let G_{n,d} be uniform among simple labelled d-regular graphs on n vertices, along admissible n with nd even. For the same word, put

\Delta_D(G)=\operatorname{OPT}(G) -q_{p,G}(2\gamma/\sqrt D,\beta).

The word may be chosen so that

\begin{gathered}\lim_{D\to\infty} \limsup_{\substack{n\to\infty\\ n(D+1)\ \mathrm{even}}} \Pr_G\!\left[\sqrt D\,\Delta_D(G_{n,D+1})>\eta\right]=0.\end{gathered}

Here the inner limit keeps D,p,\gamma,\beta fixed, and \Pr_G is only over the graph: q_{p,G} already averages quantum measurement.

Section 1, Introduction and statement of the result Read it in the paper TeX source, line 218
Show the TeX
For every $\eta>0$ there are an integer $p\ge1$, deterministic normalized
angles $\gamma,\beta\in\R^p$, and an integer $D_0\ge2$ such that, for every
integer $D\ge D_0$ and every finite simple unweighted $(D+1)$-regular graph
$G$ of girth greater than $2p+1$,
\begin{equation}
 q_{p,G}(2\gamma/\sqrt D,\beta)
 \ge \frac12+\frac{\Pstar-\eta}{\sqrt D}.
\end{equation}
The word $p,\gamma,\beta$ is independent of $D$, the graph size, and $G$;
the displayed usual MaxCut cost angles, unlike the normalized angles
$\gamma$, are scaled with the degree.

Let $G_{n,d}$ be uniform among simple labelled $d$-regular graphs on $n$
vertices, along admissible $n$ with $nd$ even.  For the same word, put
\[
 \Delta_D(G)=\operatorname{OPT}(G)
             -q_{p,G}(2\gamma/\sqrt D,\beta).
\]
The word may be chosen so that
\begin{equation}
 \lim_{D\to\infty}
 \limsup_{\substack{n\to\infty\\ n(D+1)\ \mathrm{even}}}
 \Pr_G\!\left[\sqrt D\,\Delta_D(G_{n,D+1})>\eta\right]=0.
\end{equation}
Here the inner limit keeps $D,p,\gamma,\beta$ fixed, and $\Pr_G$ is only
over the graph: $q_{p,G}$ already averages quantum measurement.

Full support of the zero-temperature Sherrington-Kirkpatrick order parameter

Theorem (Full support at zero temperature)

If \gamma\in\mathcal U minimizes \mathcal P, then

\supp\mu_\gamma=[0,1).

In particular \gamma(b)>\gamma(a) whenever 0\leq a<b<1. For the solution of (eq:model-diffusion),

\E[u_\gamma(t,X_t)^2]=t,\qquad \E[\Phi_{\gamma,xx}(t,X_t)^2]=1, \qquad 0\leq t<1.
Section 1, Introduction Read it in the paper TeX source, line 119
Show the TeX
If $\gamma\in\mathcal U$ minimizes $\mathcal P$, then
\[
 \supp\mu_\gamma=[0,1).
\]
In particular $\gamma(b)>\gamma(a)$ whenever $0\leq a<b<1$.
For the solution of~\eqref{eq:model-diffusion},
\begin{equation}
 \E[u_\gamma(t,X_t)^2]=t,\qquad
 \E[\Phi_{\gamma,xx}(t,X_t)^2]=1,
 \qquad 0\leq t<1.
\end{equation}
Configuration and file facts
OpenAI’s Lean catalogue
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 281 link it.
Solution module
OAI.Probability.SKValue.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
35 local vocabulary declarations
Lines of Lean, without comments or blank lines
128
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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Plectis Atlas

Parisi ground-state value identities and finite Gaussian approximation

Family 281: QAOA attains the SK optimum in the thermodynamic-first limit

Candidate pairing

Theorem

For the zero-field SK model (eq:sk-model),

\lim_{p\to\infty}Q_p=\Pstar.

Equivalently, for every \varepsilon>0 there exist a finite integer p and deterministic real vectors \gamma,\beta\in\R^p, independent of n and J, such that v_p(\gamma,\beta)\ge\Pstar-\varepsilon.

QAOA attains the SK ground-state energy in the thermodynamic-first limit

TeX label thm:main, build/sections/01-introduction.tex, line 111

The Lean statement says

theorem value_consequences :    (∀ (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ),      IsMinimizer W γ → IsDiffusion W γ X → ScalarValueConclusion W γ X) ∧    (∀ (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ) (T : ℝ),      0 < T → T < 1 → IsMinimizer W γ → IsDiffusion W γ X →      FiniteGaussianConclusion W γ X T) ∧    (∀ (W : BrownianSpace) (γ : OrderParameter),      IsMinimizer W γ → groundStateValue = parisi W γ) := by  sorry

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35 local vocabulary declarationsAllowed axioms: propext, Quot.sound, Classical.choiceopenai/math at fd4aeebwcook04.github.io/plectis-atlas/statements/SKValue.html

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