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Perceptron free energies and microscopic jamming exponents

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Determines finite-temperature variational free energies for Gaussian Ising perceptrons with bounded Borel log-potentials and Gaussian spherical perceptrons with bounded continuous potentials, at every positive pattern density. A spherical extension treats bi-orthogonally invariant disorder with compact limiting singular-value distributions and no outliers. At margin −1, the quadratic-penalty spherical model has a sharp feasibility threshold and limiting gap and force laws, with system size, zero temperature, and critical density taken in that order. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments

The paper states

Theorem 2.2

In The spherical perceptron with bi-orthogonally invariant disorder, 24 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Seven other possible matches are listed below the card, among them Theorem 2.3 (Gaussian Ising perceptron) in The free energy of the Ising random perceptron and Theorem 3.1 (Gaussian formula and global penalties) in the same paper.

Fix \alpha>0, M=\lfloor\alpha N\rfloor, and f\in C_b(\R). Suppose the law of A_N is invariant under left and right orthogonal multiplication and, in probability,

\frac1{r_N}\sum_{r=1}^{r_N}\delta_{s_{N,r}}\Longrightarrow\nu, \qquad \max_{r\le r_N}\operatorname{dist}(s_{N,r},\supp\nu)\longrightarrow0,

where \nu is a deterministic compactly supported probability measure on [0,\infty). Then F_N(A_N,f) converges in probability and in mean to a deterministic value F_{\alpha,\nu}(f).

[3] Here is its formula. Approximate the singular slots in operator norm by finitely many positive values s_1,\ldots,s_h whose multiplicities satisfy n_j/M\to p_j>0. Thus \sum_jp_j=\min(1,1/\alpha). Put v_j=\alpha p_j, and, if \alpha<1, add an input null block with v_0=1-\alpha. Let \mathcal I=\{1,\ldots,h\} when \alpha\ge1 and \mathcal I=\{0,1,\ldots,h\} otherwise. The variable \rho_i is the fraction of the input squared norm assigned to block i. The value for this finite approximation is

\begin{gathered}\sup_{\substack{\rho_i>0\\\sum_{i\in\mathcal I}\rho_i=1}} \left\{ \frac12\sum_{i\in\mathcal I}v_i\log\frac{\rho_i}{v_i} +\alpha P_p^f\left(\left(\frac{s_j^2\rho_j}{\alpha}\right)_{j=1}^h\right) \right\}.\end{gathered}

The limit of (2.3) as the spectral mesh tends to zero is F_{\alpha,\nu}(f). Blocks with zero limiting proportion are not used in (2.3); the no-outlier assumption ensures the required approximations with positive limiting proportions.

No Lean statement in the release is paired with this theorem.

Seven more possible matches

  • Theorem 2.3 (Gaussian Ising perceptron) in The free energy of the Ising random perceptron, 24 September 2026.
  • Theorem 3.1 (Gaussian formula and global penalties) in The spherical perceptron with bi-orthogonally invariant disorder, 24 September 2026.
  • Theorem 3.5 (Concavity and a supporting field) in The free energy of the Ising random perceptron, 24 September 2026.
  • Theorem 2.1 (Angular formula) in The spherical perceptron with bi-orthogonally invariant disorder, 24 September 2026.
  • Theorem 1.1 (Microscopic criticality and mixture universality) in Microscopic jamming in the negative spherical perceptron, 24 September 2026.
  • Theorem 4.2 (Positivity and ultrametricity) in The free energy of the Ising random perceptron, 24 September 2026.
  • Theorem 1 in The free energy of the spherical random perceptron, 24 September 2026.
Reported proof verification
Three Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
Papers and explanations
Four current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

The free energy of the Ising random perceptron

24 September 2026, current edition.

Read it herePDF

We determine the limiting free energy of the Ising perceptron with independent Gaussian patterns for every bounded Borel log-potential, at every fixed positive temperature and pattern density. We give an explicit variational formula for the limit and prove convergence in expectation and probability.

Microscopic jamming in the negative spherical perceptron

24 September 2026, current edition.

Read it herePDF

We prove a sharp feasibility threshold and limiting gap and force laws for the spherical perceptron with margin −1 and quadratic penalty. The limits are taken successively in system size, inverse temperature, and density approaching the threshold from above. They agree for Gaussian coordinates and for the equal mixture of centered Gaussian coordinates with variances \(1-\varepsilon\) and \(1+\varepsilon\), for every sufficiently small fixed ε. The contact-removed gap cumulative law and the mean-one force cumulative law satisfy \(\displaystyle G_J(u)=u^{1-\gamma+o(1)},\qquad F_J(s)=s^{1+\theta+o(1)},\) as \(u\downarrow0\) and \(s\downarrow0\), with \(\gamma=(2+\theta)^{-1}\), \(0.4126930\lt \gamma\lt 0.4126934\), and \(0.4231063\lt \theta\lt 0.4231088\). A finite numerical certificate for these exponent intervals, together with its mathematical error bounds, is included.

The spherical perceptron with bi-orthogonally invariant disorder

24 September 2026, current edition.

Read it herePDF

We determine the limiting free energy of a spherical perceptron with a bounded continuous activation and a bi-orthogonally invariant disorder matrix. The singular values may have any compact limiting distribution, provided there are no outliers. We give an explicit variational formula for this limit.

The free energy of the spherical random perceptron

24 September 2026, current edition.

Read it herePDF

We prove an exact variational formula for the limiting pressure of the spherical random perceptron with an arbitrary bounded continuous single-pattern potential. The formula holds at every fixed positive density and inverse temperature, with convergence in expectation and in probability.

Three Lean statements in this family

Choose one to see it beside the paper theorem it is paired with.

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

IsingFiniteness

The paper states

Theorem 2.3 (Gaussian Ising perceptron)

In The free energy of the Ising random perceptron, 24 September 2026

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.

The words they share: borel, bounded, continuous, converges, finite, holds, ising, perceptron, probability, value, variational.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Finiteness of the Ising-perceptron variational value

IsingFiniteness.lean, the release lists it as a supporting statement.

theorem variationalValue_real_of_continuous {α : ℝ} (hα : 0 ≤ α)    {f : ℝ → ℝ} (hf : Continuous f) (hbounded : ∃ K : ℝ, ∀ x, |f x| ≤ K) :    ∃ p : ℝ, variationalValue α f = (p : EReal) := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.
  • Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

PerceptronFreeEnergy

The paper states

Theorem 8.5 (Random-sign patterns)

In The free energy of the Ising random perceptron, 24 September 2026

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 2.3. In particular, this holds for every bounded continuous activation.

The words they share: patterns, expected, borel, measure, pressure, gaussian, value, probability, random, continuous, converge, bounded.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Variational formula for spherical-perceptron pressure

PerceptronFreeEnergy.lean, the release lists it as a supporting statement.

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

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

SphericalField

The paper states

Proposition 4 (Spherical linear-field formula)

In The free energy of the spherical random perceptron, 24 September 2026

For fixed interval weights, uniformly on compact sets of nonnegative nondecreasing step field values,

l_n(h)\ \longrightarrow\ \min_{q_0\le\cdots\le q_{k-1}\,;\ 0\le q_i<1} \left\{S(q)-\sum_i w_i h_i q_i\right\}

as n\to\infty. Furthermore, for each fixed 0\le B<1, if 0\le q(u)\le B a.e. is nondecreasing and

2h(u)=A_q(q(u)),\qquad A_q(r):=\int_0^r\frac{dt}{D_q(t)^2}\quad(0\le r<1),

then

l_n(h)=S(q)-\int_0^1 hq+o(1),

where the error tends to zero uniformly over these q.

The words they share: hq, linear, weights, field, compact, spherical, step, uniformly, min, formula, interval, values, zero.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Spherical linear-field dual formula and approximation

SphericalField.lean, the release lists it as a supporting statement.

def SphericalLinearFieldFormula : Prop :=  (∀ (k : ℕ) (w : Fin (k+1) → ℝ) (hw : ∀ i, 0 < w i) (hw1 : ∑ i, w i=1),    (∀ h : Fin (k+1) → ℝ, Monotone h → 0 ≤ h 0 →      ∃ q : Fin (k+1) → Time, Monotone q ∧ (q (Fin.last k):ℝ) < 1 ∧        IsLeast (finiteSphericalDualValues w h (fun i => (hw i).le) hw1)          (finiteSphericalDualObjective w h (fun i => (hw i).le) hw1 q)) ∧    (∀ K : Set (Fin (k+1) → ℝ), IsCompact K →      (∀ h ∈ K, Monotone h ∧ 0 ≤ h 0) →      TendstoUniformlyOn (fun n h => finiteSphericalFieldValue n k w h)        (fun h => sInf (finiteSphericalDualValues w h (fun i => (hw i).le) hw1)) atTop K)) ∧  (∀ (B : ℝ) (hB0 : 0 ≤ B) (hB1 : B < 1) (ε : ℝ), 0 < ε →    ∀ᶠ n : ℕ in atTop, ∀ (q : Time → Time) (hq : Monotone q),      (∀ᵐ u ∂timeLaw, (q u:ℝ) ≤ B) →      entropy (quantileTrial q) < ∞ ∧      (∀ᵐ u ∂timeLaw, 2*stationaryQuantileField q hq B hB0 hB1 u=quantileA q (q u)) ∧      |boundedSphericalFieldValue (stationaryQuantileField q hq B hB0 hB1) n-

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

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

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).

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.

“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.

Their method: 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.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).

The gap they name: 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.

“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The linked formalization proves finiteness of the variational value used for the Ising random perceptron. For every nonnegative pattern density and every bounded continuous log-potential, the infimum over admissible overlap paths is a finite real number.

This is a supporting well-definedness result. It does not assert convergence of the finite-system pressure or the paper’s extension to all bounded Borel log-potentials. 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. The linked spherical-field result supplies a supporting dual formula: finite hierarchy field values converge uniformly on compact parameter sets to an attained dual minimum, and the corresponding bounded stationary-field approximation holds for monotone overlap quantiles bounded away from one.

AI-drafted comparisons, with no recorded human review

  • Finiteness of the Ising-perceptron variational value IsingFiniteness

    Unreviewed AI draft: states a supporting result. States that the variational value for the Ising perceptron is a finite real number, a well-definedness result. Convergence of the pressure, the paper’s main theorem, is not asserted.

  • Variational formula for spherical-perceptron pressure PerceptronFreeEnergy

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the spherical paper: the limiting pressure of the spherical perceptron equals the variational formula for every positive density and inverse temperature and every bounded continuous potential.

  • Spherical linear-field dual formula and approximation SphericalField

    Unreviewed AI draft: states a supporting result. States a supporting dual formula for spherical linear fields and a bounded stationary-field approximation, not the pressure formula itself.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Finiteness of the Ising-perceptron variational value IsingFiniteness

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 505 s; peak memory 6.5 GiB).11 October 2026Original report

Variational formula for spherical-perceptron pressure PerceptronFreeEnergy

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Spherical linear-field dual formula and approximation SphericalField

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Other proofs and exact calculations

These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.

  • Microscopic jamming in the negative spherical perceptron, 24 September 2026

    Reported by
    cert-machine
    Method
    Exact computation
    Their verdict
    “REFUSED” The exact code did not decide it; cert-machine records this as refused, which is no evidence against the claim.
    What it covers
    “a finite component: stage (I) of the certificate — the two final spectral tests (.2936535023 < lambda(.2936533) < .2936535043, .2936533037 < lambda(.2936535) < .2936533057, hence lambda(a) - a changes sign on (.2936533, .2936535)) by own exact integer certificates of the profile and of the Riccati shooting, the numeric steps of the three error lemmas, and the exact exponent arithmetic from the bracket to the printed gamma and theta bounds. NOT stage (II) (the six interval routines: the [0, .410] -> [.285, .303] narrowing and the contraction that make the root unique and equal to the selected coefficient), NOT the theory; hence not the headline”
    Original report
    Their report

What these papers cite in the release, and what cites them

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  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

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