Lean statement

Full support of zero-temperature SK minimizers

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Theorem selected for verification: OAI.ZeroTemperatureSK.full_support

Lean statement file: SKFullSupport.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 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 full_support (W : BrownianSystem Ω) (γ : OrderParameter)    (hmin : IsMinimizer W γ) : FullSupportConclusion 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.

abbreviation OAI.ZeroTemperatureSK.Time line 12
```
abbrev Time := Set.Ico (0 : ℝ) 1
```

definition OAI.ZeroTemperatureSK.extend line 14

def extend (γ : Time → ℝ) (t : ℝ) : ℝ :=  if ht : t ∈ Set.Ico (0 : ℝ) 1 then γ ⟨t, ht⟩ else 0

structure OAI.ZeroTemperatureSK.OrderParameter line 17

structure OrderParameter where  val : Time → ℝ  nonneg : ∀ t, 0 ≤ val t  monotone : Monotone val  right_continuous : ∀ t, ContinuousWithinAt val (Set.Ici t) t  integrable : Integrable (extend val)

structure OAI.ZeroTemperatureSK.BrownianSystem line 24

structure BrownianSystem (Ω : Type*) [MeasurableSpace Ω] where  law : Measure Ω  isProbability : IsProbabilityMeasure law  B : ℝ≥0 → Ω → ℝ  measurable : ∀ t, Measurable (B t)  brownian : IsBrownianReal B law

definition OAI.ZeroTemperatureSK.incrementFiltration line 33

def incrementFiltration (W : BrownianSystem Ω) (t : ℝ≥0) :    Filtration ℝ≥0 ‹MeasurableSpace Ω› :=  Filtration.natural (fun s ω => W.B (t + s) ω - W.B t ω)    (fun s => ((W.measurable (t + s)).sub (W.measurable t)).stronglyMeasurable)

structure OAI.ZeroTemperatureSK.Control line 38

structure Control (W : BrownianSystem Ω) (t : ℝ≥0) where  val : ℝ≥0 → Ω → ℝ  progressive : IsProgressive (incrementFiltration W t) val  bound : ∀ s ω, |val s ω| ≤ 1

definition OAI.ZeroTemperatureSK.controlPayoff line 43

def controlPayoff (W : BrownianSystem Ω) (γ : OrderParameter) (t x : ℝ)    (α : Control W (Real.toNNReal t)) (ω : Ω) : ℝ :=  |x + W.B 1 ω - W.B (Real.toNNReal t) ω +      ∫ s in t..1, extend γ.val s * α.val (Real.toNNReal (s - t)) ω| -    (1 / 2 : ℝ) * ∫ s in t..1, extend γ.val s * (α.val (Real.toNNReal (s - t)) ω)^2

definition OAI.ZeroTemperatureSK.value line 49

def value (W : BrownianSystem Ω) (γ : OrderParameter) (t x : ℝ) : ℝ :=  sSup (Set.range (fun α : Control W (Real.toNNReal t) =>    ∫ ω, controlPayoff W γ t x α ω ∂W.law))

definition OAI.ZeroTemperatureSK.gradient line 53

def gradient (W : BrownianSystem Ω) (γ : OrderParameter) (t x : ℝ) : ℝ :=  deriv (value W γ t) x

definition OAI.ZeroTemperatureSK.curvature line 56

def curvature (W : BrownianSystem Ω) (γ : OrderParameter) (t x : ℝ) : ℝ :=  deriv (gradient W γ t) x

definition OAI.ZeroTemperatureSK.parisi line 59

def parisi (W : BrownianSystem Ω) (γ : OrderParameter) : ℝ :=  value W γ 0 0 - (1 / 2 : ℝ) * ∫ t in (0 : ℝ)..1, t * extend γ.val t

definition OAI.ZeroTemperatureSK.IsMinimizer line 62

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

definition OAI.ZeroTemperatureSK.IsStieltjesMeasure line 65

def IsStieltjesMeasure (γ : OrderParameter) (μ : Measure Time) : Prop :=  ∀ t : Time, μ (Set.Iic t) = ENNReal.ofReal (γ.val t)

definition OAI.ZeroTemperatureSK.IsDiffusion line 68

def IsDiffusion (W : BrownianSystem Ω) (γ : OrderParameter)    (X : ℝ≥0 → Ω → ℝ) : Prop :=  IsProgressive (Filtration.natural W.B (fun t => (W.measurable t).stronglyMeasurable)) X ∧  (∀ᵐ ω ∂W.law, ContinuousOn (fun t : ℝ => X (Real.toNNReal t) ω) (Set.Icc 0 1) ∧    X 0 ω = 0 ∧ ∀ t ∈ Set.Icc (0 : ℝ) 1,      X (Real.toNNReal t) ω = W.B (Real.toNNReal t) ω +        ∫ s in (0 : ℝ)..t, extend γ.val s *          gradient W γ s (X (Real.toNNReal s) ω))

definition OAI.ZeroTemperatureSK.squareMoment line 77

def squareMoment (W : BrownianSystem Ω) (γ : OrderParameter)    (X : ℝ≥0 → Ω → ℝ) (t : ℝ) : ℝ :=  ∫ ω, (gradient W γ t (X (Real.toNNReal t) ω))^2 ∂W.law

definition OAI.ZeroTemperatureSK.FullSupportConclusion line 81

def FullSupportConclusion (W : BrownianSystem Ω) (γ : OrderParameter) : Prop :=  (∃ μ : Measure Time, IsStieltjesMeasure γ μ) ∧  (∃ X : ℝ≥0 → Ω → ℝ, IsDiffusion W γ X) ∧  (∀ μ : Measure Time, IsStieltjesMeasure γ μ → μ.support = Set.univ) ∧  (∀ a b : Time, a < b → γ.val a < γ.val b) ∧  ∀ X : ℝ≥0 → Ω → ℝ, IsDiffusion W γ X → ∀ t : Time,    squareMoment W γ X t = (t : ℝ) ∧    (∫ ω, (curvature W γ t (X (Real.toNNReal t) ω))^2 ∂W.law) = 1

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: support, full and diffusion. Found in the paper’s title, which its main theorem states: order and parameter. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: temperature, model and zero.

Statements in Full support of the zero-temperature Sherrington-Kirkpatrick order parameter come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of 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}

Other possible paper matches, closest wording first

Proposition 5.1 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: curvature and diffusion. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: strictly and increasing.
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: minimizer, diffusion, value and bounded.
Corollary 6.4 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: measure and diffusion. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: zero.
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: minimizer, 1/2 and value. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: zero.
Proposition 4.2 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: integrable, extend and probability. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: zero.
Theorem 1.1 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: real. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: field, equivalently, model, sk and zero.

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

“The formalization proves that every admissible integrable minimizer of the zero-temperature Parisi functional for the pure zero-field Sherrington–Kirkpatrick model has full relative Stieltjes support on $[0,1)$. Equivalently, the order parameter increases strictly between every two overlap values below one, so its support has no gap. The covariance normalization is $\xi(t)=t^2/2$.”
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

98 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: 56 to its own declarations, 42 to Mathlib’s documentation. 17 declared names also link to the same declaration in OpenAI’s library. 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 full_support 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 all 16 other declarations of the file; with the theorem itself they span 63 lines. 4 of them are named in the statement itself.

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

abbrev Timeline 12library
def extendline 14library
structure OrderParameterline 17in the statementlibrary
structure BrownianSystemline 24in the statementlibrary
def incrementFiltrationline 33library
structure Controlline 38library
def controlPayoffline 43library
def valueline 49library
def gradientline 53library
def curvatureline 56library
def parisiline 59library
def IsMinimizerline 62in the statementlibrary
def IsStieltjesMeasureline 65library
def IsDiffusionline 68library
def squareMomentline 77library
def FullSupportConclusionline 81in the statementlibrary
theorem full_supportline 90selected theoremlibrary

import Mathlibnamespace OAIopen MeasureTheory ProbabilityTheory Set Filteropen scoped ENNReal NNReal Topologynoncomputable sectionnamespace ZeroTemperatureSKabbrev Time := Set.Ico (0 : ℝ) 1def extend (γ : Time → ℝ) (t : ℝ) : ℝ :=  if ht : t ∈ Set.Ico (0 : ℝ) 1 then γ ⟨t, ht⟩ else 0structure OrderParameter where  val : Time → ℝ  nonneg : ∀ t, 0 ≤ val t  monotone : Monotone val  right_continuous : ∀ t, ContinuousWithinAt val (Set.Ici t) t  integrable : Integrable (extend val)structure BrownianSystem (Ω : Type*) [MeasurableSpace Ω] where  law : Measure Ω  isProbability : IsProbabilityMeasure law  B : ℝ≥0 → Ω → ℝ  measurable : ∀ t, Measurable (B t)  brownian : IsBrownianReal B lawvariable {Ω : Type*} [MeasurableSpace Ω]def incrementFiltration (W : BrownianSystem Ω) (t : ℝ≥0) :    Filtration ℝ≥0 ‹MeasurableSpace Ω› :=  Filtration.natural (fun s ω => W.B (t + s) ω - W.B t ω)    (fun s => ((W.measurable (t + s)).sub (W.measurable t)).stronglyMeasurable)structure Control (W : BrownianSystem Ω) (t : ℝ≥0) where  val : ℝ≥0 → Ω → ℝ  progressive : IsProgressive (incrementFiltration W t) val  bound : ∀ s ω, |val s ω| ≤ 1def controlPayoff (W : BrownianSystem Ω) (γ : OrderParameter) (t x : ℝ)    (α : Control W (Real.toNNReal t)) (ω : Ω) : ℝ :=  |x + W.B 1 ω - W.B (Real.toNNReal t) ω +      ∫ s in t..1, extend γ.val s * α.val (Real.toNNReal (s - t)) ω| -    (1 / 2 : ℝ) * ∫ s in t..1, extend γ.val s * (α.val (Real.toNNReal (s - t)) ω)^2def value (W : BrownianSystem Ω) (γ : OrderParameter) (t x : ℝ) : ℝ :=  sSup (Set.range (fun α : Control W (Real.toNNReal t) =>    ∫ ω, controlPayoff W γ t x α ω ∂W.law))def gradient (W : BrownianSystem Ω) (γ : OrderParameter) (t x : ℝ) : ℝ :=  deriv (value W γ t) xdef curvature (W : BrownianSystem Ω) (γ : OrderParameter) (t x : ℝ) : ℝ :=  deriv (gradient W γ t) xdef parisi (W : BrownianSystem Ω) (γ : OrderParameter) : ℝ :=  value W γ 0 0 - (1 / 2 : ℝ) * ∫ t in (0 : ℝ)..1, t * extend γ.val tdef IsMinimizer (W : BrownianSystem Ω) (γ : OrderParameter) : Prop :=  ∀ η : OrderParameter, parisi W γ ≤ parisi W ηdef IsStieltjesMeasure (γ : OrderParameter) (μ : Measure Time) : Prop :=  ∀ t : Time, μ (Set.Iic t) = ENNReal.ofReal (γ.val t)def IsDiffusion (W : BrownianSystem Ω) (γ : OrderParameter)    (X : ℝ≥0 → Ω → ℝ) : Prop :=  IsProgressive (Filtration.natural W.B (fun t => (W.measurable t).stronglyMeasurable)) X ∧  (∀ᵐ ω ∂W.law, ContinuousOn (fun t : ℝ => X (Real.toNNReal t) ω) (Set.Icc 0 1) ∧    X 0 ω = 0 ∧ ∀ t ∈ Set.Icc (0 : ℝ) 1,      X (Real.toNNReal t) ω = W.B (Real.toNNReal t) ω +        ∫ s in (0 : ℝ)..t, extend γ.val s *          gradient W γ s (X (Real.toNNReal s) ω))def squareMoment (W : BrownianSystem Ω) (γ : OrderParameter)    (X : ℝ≥0 → Ω → ℝ) (t : ℝ) : ℝ :=  ∫ ω, (gradient W γ t (X (Real.toNNReal t) ω))^2 ∂W.lawdef FullSupportConclusion (W : BrownianSystem Ω) (γ : OrderParameter) : Prop :=  (∃ μ : Measure Time, IsStieltjesMeasure γ μ) ∧  (∃ X : ℝ≥0 → Ω → ℝ, IsDiffusion W γ X) ∧  (∀ μ : Measure Time, IsStieltjesMeasure γ μ → μ.support = Set.univ) ∧  (∀ a b : Time, a < b → γ.val a < γ.val b) ∧  ∀ X : ℝ≥0 → Ω → ℝ, IsDiffusion W γ X → ∀ t : Time,    squareMoment W γ X t = (t : ℝ) ∧    (∫ ω, (curvature W γ t (X (Real.toNNReal t) ω))^2 ∂W.law) = 1theorem full_support (W : BrownianSystem Ω) (γ : OrderParameter)    (hmin : IsMinimizer W γ) : FullSupportConclusion W γ := by  sorryend ZeroTemperatureSKendend 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

File: lean/ComparatorChallenges/SKFullSupport.lean
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.SKSupport.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: 16 local vocabulary declarations
Lines of Lean, without comments or blank lines: 63
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.

Plectis Atlas