Lean statement

Variational formula for spherical-perceptron pressure

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.SphericalPerceptronFreeEnergy.main

Lean statement file: PerceptronFreeEnergy.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 222 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 main (P : Measure BrownianPath) [IsProbabilityMeasure P]    (hB : IsBrownianReal brownianEval P)    (α β : ℝ) (hα : 0 < α) (hβ : 0 < β) (φ : ℝ →ᵇ ℝ) :    ∃ p : ℝ, variationalValue P α β φ = (p : EReal) ∧      Tendsto (fun N : ℕ => expectedPressure α β φ (N + 1)) atTop (𝓝 p) ∧      ∀ ε : ℝ, 0 < ε →        Tendsto (fun N : ℕ => patternLaw α (N + 1)          {g | ε < |pressure α β φ (N + 1) g - p|}) atTop (𝓝 0) := 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.SphericalPerceptronFreeEnergy.Time line 11
```
abbrev Time := unitInterval
```
abbreviation OAI.SphericalPerceptronFreeEnergy.BrownianPath line 12
```
abbrev BrownianPath := C(ℝ≥0, ℝ)
```
abbreviation OAI.SphericalPerceptronFreeEnergy.Spin line 17
```
abbrev Spin (N : ℕ) := EuclideanSpace ℝ (Fin N)
```

definition OAI.SphericalPerceptronFreeEnergy.patternCount line 19

def patternCount (α : ℝ) (N : ℕ) : ℕ := ⌊α * (N : ℝ)⌋₊

abbreviation OAI.SphericalPerceptronFreeEnergy.Patterns line 20

abbrev Patterns (α : ℝ) (N : ℕ) := Fin (patternCount α N) → Fin N → ℝ

definition OAI.SphericalPerceptronFreeEnergy.patternLaw line 22

def patternLaw (α : ℝ) (N : ℕ) : Measure (Patterns α N) :=  Measure.pi fun _ => Measure.pi fun _ => gaussianReal 0 1

definition OAI.SphericalPerceptronFreeEnergy.unitSphereLaw line 25

def unitSphereLaw (N : ℕ) : Measure (Metric.sphere (0 : Spin N) 1) :=  let s := (volume : Measure (Spin N)).toSphere  (s Set.univ)⁻¹ • s

definition OAI.SphericalPerceptronFreeEnergy.sphereLaw line 29

def sphereLaw (N : ℕ) : Measure (Spin N) :=  (unitSphereLaw N).map fun x => Real.sqrt (N : ℝ) • x.val

definition OAI.SphericalPerceptronFreeEnergy.patternField line 32

def patternField {α : ℝ} {N : ℕ} (g : Patterns α N)    (a : Fin (patternCount α N)) (x : Spin N) : ℝ :=  (∑ i : Fin N, g a i * x i) / Real.sqrt (N : ℝ)

definition OAI.SphericalPerceptronFreeEnergy.hamiltonian line 36

def hamiltonian (α : ℝ) (φ : ℝ →ᵇ ℝ) (N : ℕ) (g : Patterns α N)    (x : Spin N) : ℝ :=  ∑ a : Fin (patternCount α N), φ (patternField g a x)

definition OAI.SphericalPerceptronFreeEnergy.pressure line 40

def pressure (α β : ℝ) (φ : ℝ →ᵇ ℝ) (N : ℕ) (g : Patterns α N) : ℝ :=  Real.log (∫ x, Real.exp (β * hamiltonian α φ N g x) ∂sphereLaw N) / (N : ℝ)

definition OAI.SphericalPerceptronFreeEnergy.expectedPressure line 43

def expectedPressure (α β : ℝ) (φ : ℝ →ᵇ ℝ) (N : ℕ) : ℝ :=  ∫ g, pressure α β φ N g ∂patternLaw α N

structure OAI.SphericalPerceptronFreeEnergy.Trial line 46

structure Trial where  toFun : Time → ℝ  monotone : Monotone toFun  measurable : Measurable toFun  nonneg : ∀ t, 0 ≤ toFun t  le_one : ∀ t, toFun t ≤ 1

definition OAI.SphericalPerceptronFreeEnergy.timeLaw line 55

def timeLaw : Measure Time := Measure.comap Subtype.val volume

definition OAI.SphericalPerceptronFreeEnergy.tailIntegral line 57

def tailIntegral (m : Trial) (t : Time) : ℝ :=  ∫ s in Set.Ici t, m s ∂timeLaw

definition OAI.SphericalPerceptronFreeEnergy.entropy line 60

def entropy (m : Trial) : ℝ≥0∞ :=  (∫⁻ t, (ENNReal.ofReal (tailIntegral m t))⁻¹ -    (ENNReal.ofReal (1 - (t : ℝ)))⁻¹ ∂timeLaw) / 2

definition OAI.SphericalPerceptronFreeEnergy.brownianEval line 64

def brownianEval (t : ℝ≥0) (ω : BrownianPath) : ℝ := ω t

definition OAI.SphericalPerceptronFreeEnergy.nullSigma line 66

@[instance_reducible] def nullSigma (P : Measure BrownianPath) : MeasurableSpace BrownianPath :=  MeasurableSpace.generateFrom {s | P s = 0}

definition OAI.SphericalPerceptronFreeEnergy.usualBrownianSigma line 69

@[instance_reducible] def usualBrownianSigma (P : Measure BrownianPath) (t : Time) :    MeasurableSpace BrownianPath :=  ⨅ s : {s : ℝ≥0 // (t : ℝ) < s},    (⨆ r : {r : ℝ≥0 // r ≤ s.val},      MeasurableSpace.comap (brownianEval r.val) (borel ℝ)) ⊔ nullSigma P

definition OAI.SphericalPerceptronFreeEnergy.Progressive line 75

def Progressive (P : Measure BrownianPath) (v : Time → BrownianPath → ℝ) : Prop :=  ∀ t : Time,    @Measurable (Set.Iic t × BrownianPath) ℝ      (MeasurableSpace.prod inferInstance (usualBrownianSigma P t)) (borel ℝ)      (fun p => v p.1.val p.2)

definition OAI.SphericalPerceptronFreeEnergy.controlCost line 81

def controlCost (P : Measure BrownianPath) (m : Trial)    (v : Time → BrownianPath → ℝ) : ℝ≥0∞ :=  ∫⁻ ω, ∫⁻ t, ENNReal.ofReal (m t * v t ω ^ 2) ∂timeLaw ∂P

definition OAI.SphericalPerceptronFreeEnergy.controlPayoff line 85

def controlPayoff (P : Measure BrownianPath) (f : ℝ →ᵇ ℝ) (m : Trial)    (v : Time → BrownianPath → ℝ) : ℝ :=  (∫ ω, f (brownianEval 1 ω + ∫ t, m t * v t ω ∂timeLaw) ∂P) -    (controlCost P m v).toReal / 2

definition OAI.SphericalPerceptronFreeEnergy.controlValue line 90

def controlValue (P : Measure BrownianPath) (f : ℝ →ᵇ ℝ) (m : Trial) : ℝ :=  sSup {a | ∃ v : Time → BrownianPath → ℝ,    Progressive P v ∧ controlCost P m v < ∞ ∧ a = controlPayoff P f m v}

definition OAI.SphericalPerceptronFreeEnergy.variationalValue line 94

def variationalValue (P : Measure BrownianPath) (α β : ℝ) (φ : ℝ →ᵇ ℝ) : EReal :=  ⨅ m : Trial, ((α * controlValue P (β • φ) m : ℝ) : EReal) + (entropy m).toEReal

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: patterns, expected, borel, measure, pressure, gaussian, value and probability. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: random, continuous, converge and bounded.

Theorem 8.5 of The free energy of the Ising random perceptron

Theorem (Random-sign patterns)

Replace the Gaussian pattern entries by independent uniform signs in \{-1,1\}. If f is bounded Borel and its discontinuity set has Gaussian measure zero, then its expected pressure and its pressure in probability converge to the same value \mathcal P_f(\alpha) as in Theorem thm:main. In particular, this holds for every bounded continuous activation.

Section 8, General activations and random-sign patterns Read it in the paper TeX source, line 245

Show the TeX

Replace the Gaussian pattern entries by independent uniform signs in
$\{-1,1\}$. If $f$ is bounded Borel and its discontinuity set has Gaussian
measure zero, then its expected pressure and its pressure in probability
converge to the same value $\mathcal P_f(\alpha)$ as in
Theorem~\ref{thm:main}. In particular, this holds for every bounded
continuous activation.

Other possible paper matches, closest wording first

Corollary 3.3 of The free energy of the Ising random perceptron. Words of the Lean statement, its names split into words, that also appear in the theorem: trials, path, variational, entropy and value. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
Theorem 2.3 of The free energy of the Ising random perceptron. Words of the Lean statement, its names split into words, that also appear in the theorem: real, borel, variational, gaussian, value and probability. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: perceptron, finite, continuous, converges and bounded.
Theorem 8.3 of The free energy of the Ising random perceptron. Words of the Lean statement, its names split into words, that also appear in the theorem: real, expectation, borel, variational and probability. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite, formula, continuous and limit.
Theorem 3.1 of The spherical perceptron with bi-orthogonally invariant disorder. Words of the Lean statement, its names split into words, that also appear in the theorem: counts, expectation, pressure, log, gaussian and probability. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: formula, continuous, converges and bounded.
Proposition 4.5 of Microscopic jamming in the negative spherical perceptron. Words of the Lean statement, its names split into words, that also appear in the theorem: spin, entropy, pressure and gaussian. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: inverse.
Proposition A.4 of The spherical perceptron with bi-orthogonally invariant disorder. Words of the Lean statement, its names split into words, that also appear in the theorem: borel, measure, pressure, intervals, value and probability. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: formula, continuous, converges, limit and bounded.

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 222’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 part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Stated: the spherical Gaussian perceptron formula SphericalPerceptronFreeEnergy.main : ∀ α β > 0, ∀ φ : ℝ →ᵇ ℝ, ∃ p, variationalValue P α β φ = p ∧ expectedPressure → p ∧ pressure → p in probability, plus a supporting spherical_linear_field_formula. For the Ising perceptron only well-definedness is stated: variationalValue_real_of_continuous : ∃ p : ℝ, variationalValue α f = (p : EReal) (docs: 'supporting well-definedness result. It does not assert convergence of the finite-system pressure or ... all bounded Borel log-potentials'). Not stated: Ising free-energy convergence, the bi-orthogonally invariant spherical extension, and the margin -1 jamming gap/force laws.
Definitions the group asks readers to compare with the paper: Hand-written variational 'free energies': Ising variationalValue α f := ⨅ q : OverlapPath, (α * patternFunctional f q : ℝ) + isingEntropy q with patternFunctional f q := Filter.limUnder atTop (uniformPattern f q) (junk value if the limit does not exist) and isingEntropy q := ⨆ h : FieldStep, ... in EReal; spherical variationalValue P α β φ := ⨅ m : Trial, α * controlValue P (β • φ) m + entropy m via Brownian stochastic control (Progressive, usualBrownianSigma). Finiteness of a limUnder-defined inf is a weak guarantee.
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: Ising: only finiteness of the variational value (no convergence; continuous potentials only) — declared. The bi-orthogonally invariant spherical model and the margin −1 jamming threshold, gap and force laws are not stated, and the scope note is silent on both.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves the variational formula for the spherical random perceptron's limiting pressure for every positive pattern density and inverse temperature and every bounded continuous single-pattern potential. It shows that the variational value is finite and that the pressure converges to it both in expectation and in probability.”
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

111 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: 75 to its own declarations, 65 to Mathlib’s documentation. 25 declared names also link to the same declaration in OpenAI’s library. 2 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main 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 24 of the file’s 27 other declarations; with the theorem itself they span 68 lines. 6 of them are named in the statement itself.

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

abbrev Timeline 11library
abbrev BrownianPathline 12in the statementlibrary
abbrev Spinline 17library
def patternCountline 19library
abbrev Patternsline 20library
def patternLawline 22in the statementlibrary
def unitSphereLawline 25library
def sphereLawline 29library
def patternFieldline 32library
def hamiltonianline 36library
def pressureline 40in the statementlibrary
def expectedPressureline 43in the statementlibrary
structure Trialline 46library
def timeLawline 55library
def tailIntegralline 57library
def entropyline 60library
def brownianEvalline 64in the statementlibrary
def nullSigmaline 66library
def usualBrownianSigmaline 69library
def Progressiveline 75library
def controlCostline 81library
def controlPayoffline 85library
def controlValueline 90library
def variationalValueline 94in the statementlibrary
theorem mainline 97selected theoremlibrary

Other Lean rules that may affect the statement 3

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 MeasurableSpace BrownianPath line 14
instance BorelSpace BrownianPath line 15
instance CoeFun Trial (fun _ => Time → ℝ) line 53

import Mathlibnamespace OAInoncomputable sectionopen MeasureTheory ProbabilityTheory Filter Setopen scoped ENNReal NNReal Topology BigOperators BoundedContinuousFunctionnamespace SphericalPerceptronFreeEnergyabbrev Time := unitIntervalabbrev BrownianPath := C(ℝ≥0, ℝ)instance : MeasurableSpace BrownianPath := borel BrownianPathinstance : BorelSpace BrownianPath := ⟨rfl⟩abbrev Spin (N : ℕ) := EuclideanSpace ℝ (Fin N)def patternCount (α : ℝ) (N : ℕ) : ℕ := ⌊α * (N : ℝ)⌋₊abbrev Patterns (α : ℝ) (N : ℕ) := Fin (patternCount α N) → Fin N → ℝdef patternLaw (α : ℝ) (N : ℕ) : Measure (Patterns α N) :=  Measure.pi fun _ => Measure.pi fun _ => gaussianReal 0 1def unitSphereLaw (N : ℕ) : Measure (Metric.sphere (0 : Spin N) 1) :=  let s := (volume : Measure (Spin N)).toSphere  (s Set.univ)⁻¹ • sdef sphereLaw (N : ℕ) : Measure (Spin N) :=  (unitSphereLaw N).map fun x => Real.sqrt (N : ℝ) • x.valdef patternField {α : ℝ} {N : ℕ} (g : Patterns α N)    (a : Fin (patternCount α N)) (x : Spin N) : ℝ :=  (∑ i : Fin N, g a i * x i) / Real.sqrt (N : ℝ)def hamiltonian (α : ℝ) (φ : ℝ →ᵇ ℝ) (N : ℕ) (g : Patterns α N)    (x : Spin N) : ℝ :=  ∑ a : Fin (patternCount α N), φ (patternField g a x)def pressure (α β : ℝ) (φ : ℝ →ᵇ ℝ) (N : ℕ) (g : Patterns α N) : ℝ :=  Real.log (∫ x, Real.exp (β * hamiltonian α φ N g x) ∂sphereLaw N) / (N : ℝ)def expectedPressure (α β : ℝ) (φ : ℝ →ᵇ ℝ) (N : ℕ) : ℝ :=  ∫ g, pressure α β φ N g ∂patternLaw α Nstructure Trial where  toFun : Time → ℝ  monotone : Monotone toFun  measurable : Measurable toFun  nonneg : ∀ t, 0 ≤ toFun t  le_one : ∀ t, toFun t ≤ 1instance : CoeFun Trial (fun _ => Time → ℝ) := ⟨Trial.toFun⟩def timeLaw : Measure Time := Measure.comap Subtype.val volumedef tailIntegral (m : Trial) (t : Time) : ℝ :=  ∫ s in Set.Ici t, m s ∂timeLawdef entropy (m : Trial) : ℝ≥0∞ :=  (∫⁻ t, (ENNReal.ofReal (tailIntegral m t))⁻¹ -    (ENNReal.ofReal (1 - (t : ℝ)))⁻¹ ∂timeLaw) / 2def brownianEval (t : ℝ≥0) (ω : BrownianPath) : ℝ := ω t@[instance_reducible] def nullSigma (P : Measure BrownianPath) : MeasurableSpace BrownianPath :=  MeasurableSpace.generateFrom {s | P s = 0}@[instance_reducible] def usualBrownianSigma (P : Measure BrownianPath) (t : Time) :    MeasurableSpace BrownianPath :=  ⨅ s : {s : ℝ≥0 // (t : ℝ) < s},    (⨆ r : {r : ℝ≥0 // r ≤ s.val},      MeasurableSpace.comap (brownianEval r.val) (borel ℝ)) ⊔ nullSigma Pdef Progressive (P : Measure BrownianPath) (v : Time → BrownianPath → ℝ) : Prop :=  ∀ t : Time,    @Measurable (Set.Iic t × BrownianPath) ℝ      (MeasurableSpace.prod inferInstance (usualBrownianSigma P t)) (borel ℝ)      (fun p => v p.1.val p.2)def controlCost (P : Measure BrownianPath) (m : Trial)    (v : Time → BrownianPath → ℝ) : ℝ≥0∞ :=  ∫⁻ ω, ∫⁻ t, ENNReal.ofReal (m t * v t ω ^ 2) ∂timeLaw ∂Pdef controlPayoff (P : Measure BrownianPath) (f : ℝ →ᵇ ℝ) (m : Trial)    (v : Time → BrownianPath → ℝ) : ℝ :=  (∫ ω, f (brownianEval 1 ω + ∫ t, m t * v t ω ∂timeLaw) ∂P) -    (controlCost P m v).toReal / 2def controlValue (P : Measure BrownianPath) (f : ℝ →ᵇ ℝ) (m : Trial) : ℝ :=  sSup {a | ∃ v : Time → BrownianPath → ℝ,    Progressive P v ∧ controlCost P m v < ∞ ∧ a = controlPayoff P f m v}def variationalValue (P : Measure BrownianPath) (α β : ℝ) (φ : ℝ →ᵇ ℝ) : EReal :=  ⨅ m : Trial, ((α * controlValue P (β • φ) m : ℝ) : EReal) + (entropy m).toERealtheorem main (P : Measure BrownianPath) [IsProbabilityMeasure P]    (hB : IsBrownianReal brownianEval P)    (α β : ℝ) (hα : 0 < α) (hβ : 0 < β) (φ : ℝ →ᵇ ℝ) :    ∃ p : ℝ, variationalValue P α β φ = (p : EReal) ∧      Tendsto (fun N : ℕ => expectedPressure α β φ (N + 1)) atTop (𝓝 p) ∧      ∀ ε : ℝ, 0 < ε →        Tendsto (fun N : ℕ => patternLaw α (N + 1)          {g | ε < |pressure α β φ (N + 1) g - p|}) atTop (𝓝 0) := by  sorryend SphericalPerceptronFreeEnergyendend 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 free energy of the Ising random perceptron

Theorem (Gaussian Ising perceptron)

For every \alpha>0 and every bounded Borel f:\mathbb R\to\mathbb R,

\lim_{N\to\infty}\mathbb E p_N(f) =\mathcal P_f(\alpha) :=\inf_{q\in\mathcal Q}\{\alpha V_f(q)+S_{\mathrm I}(q)\}.

Moreover, p_N(f) converges to \mathcal P_f(\alpha) in probability. The variational value is finite. In particular, the conclusion holds for f=\beta\phi at every \beta>0 and every bounded continuous activation \phi, without symmetry, concavity, or a smallness restriction.

Section 2, The model and the variational formula Read it in the paper TeX source, line 88

Show the TeX

For every $\alpha>0$ and every bounded Borel $f:\mathbb R\to\mathbb R$,
\begin{equation}
 \lim_{N\to\infty}\mathbb E p_N(f)
 =\mathcal P_f(\alpha)
 :=\inf_{q\in\mathcal Q}\{\alpha V_f(q)+S_{\mathrm I}(q)\}.
\end{equation}
Moreover, $p_N(f)$ converges to $\mathcal P_f(\alpha)$ in probability.
The variational value is finite. In particular, the conclusion holds for
$f=\beta\phi$ at every $\beta>0$ and every bounded continuous activation
$\phi$, without symmetry, concavity, or a smallness restriction.

Theorem (Concavity and a supporting field)

The functional \ell is concave on nonnegative nondecreasing step paths. For any two such paths, represented on a common partition,

\ell(h')\le\ell(h) -\frac12\int_0^1(h'(u)-h(u))\rho_h(u)\,du.

Concavity also holds for its continuous extension to bounded nonnegative nondecreasing paths.

Section 3, Variations, order, and the Ising entropy Read it in the paper TeX source, line 292

Show the TeX

The functional $\ell$ is concave on nonnegative nondecreasing
step paths. For any two such paths, represented on a common
partition,
\begin{equation}

 \ell(h')\le\ell(h)
      -\frac12\int_0^1(h'(u)-h(u))\rho_h(u)\,du.
\end{equation}
Concavity also holds for its continuous extension to bounded
nonnegative nondecreasing paths.

Theorem (Positivity and ultrametricity)

Suppose B is a bounded, symmetric, weakly exchangeable Gram array with constant diagonal D and the Ghirlanda–Guerra property. Then, almost surely, for distinct indices,

B_{\ell m}\geq0, \qquad B_{12}\geq\min\{B_{13},B_{23}\}.

Section 4, Overlap arrays and probability cascades Read it in the paper TeX source, line 68

Show the TeX

Suppose $B$ is a bounded, symmetric, weakly exchangeable Gram array with
constant diagonal $D$ and the Ghirlanda--Guerra property.  Then, almost
surely, for distinct indices,
\begin{equation}
 B_{\ell m}\geq0,
 \qquad
 B_{12}\geq\min\{B_{13},B_{23}\}.
 
\end{equation}

The free energy of the spherical random perceptron

Theorem

For every \alpha,\beta>0 and every \phi\in C_b(\R;\R),

\lim_{N\to\infty}\E p_N =\inf_{m\in\U}\{\alpha V_{\beta\phi}(m)+S(m)\} =:\mathcal P(\alpha,\beta,\phi), \qquad p_N\xrightarrow{\mathbb P}\mathcal P(\alpha,\beta,\phi).

Trials of infinite entropy have value +\infty. No condition that m=1 near 1 is imposed.

Section 1, Introduction and statement Read it in the paper TeX source, line 72

Show the TeX

For every $\alpha,\beta>0$ and every $\phi\in C_b(\R;\R)$,
\[
 \lim_{N\to\infty}\E p_N
 =\inf_{m\in\U}\{\alpha V_{\beta\phi}(m)+S(m)\}
 =:\mathcal P(\alpha,\beta,\phi),
 \qquad
 p_N\xrightarrow{\mathbb P}\mathcal P(\alpha,\beta,\phi).
\]
Trials of infinite entropy have value $+\infty$. No condition that $m=1$ near
$1$ is imposed.

Configuration and file facts

File: lean/ComparatorChallenges/PerceptronFreeEnergy.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 222 link it.
Solution module: OAI.Probability.Perceptron.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: 27 local vocabulary declarations
Lines of Lean, without comments or blank lines: 71
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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