Finiteness of the Ising-perceptron variational value
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Theorem selected for verification:OAI.IsingPerceptron.variationalValue_real_of_continuous
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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.
defstepFunction (h : FieldStep) (u : ℝ) : ℝ := ∑ i : Fin h.partition.levels,if h.partition.cut i.castSucc < u ∧ u < h.partition.cut i.succ then h.value i else0
defuniformPattern (f : ℝ → ℝ) (q : OverlapPath) (n : ℕ) : ℝ :=let k := n + 1let levelVal := fun i => if i < k then cellAverage q k i else1let increments := List.ofFn (fun i : Fin (k + 1) => (levelVal i - (if i.val = 0then0else levelVal (i.val - 1)), (i : ℝ) / k)) increments.foldr (fun sd U => gaussianTransform sd.1 sd.2 U) f 0
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: borel, bounded, continuous, converges, finite, holds, ising, perceptron and three more.
Statements in The free energy of the Ising random perceptron come first because the release’s name for this Lean statement shares words with that paper’s title.
Theorem 2.3 of 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.
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.
Other possible paper matches, closest wording first
Theorem 3.5 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: partition, field, paths, step, functional and continuous. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: extension, supporting and bounded.
Theorem 8.5 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: patterns, measure, uniform, continuous, gaussian, value and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: borel, pressure and bounded.
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: path, variational, step, entropy, order and value. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
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, uniform, variational and continuous. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: borel and finite.
Proposition 3.2 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: partitions, paths, step, order and valued. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bounded.
Proposition 7.2 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: field, entropy and continuous. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: extension and bounded.
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Reported Comparator runs
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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…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 505 s; peak memory 6.5 GiB).
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Do family 222’s Lean statements express the advertised result?
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Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
Finds part of it stated, or a narrower or weaker statementTheir 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.
Finds part of it stated, or a narrower or weaker statementTheir 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.
OpenAI’s own note on what this Lean statement covers
“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.”
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
94 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: 26 to its own declarations, 34 to Mathlib’s documentation. 14 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.
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.
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,
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,
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}
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.
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
13 local vocabulary declarations
Lines of Lean, without comments or blank lines
54
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Finiteness of the Ising-perceptron variational value
Family 222: Perceptron free energies and microscopic jamming exponents
Candidate pairing
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.
The free energy of the Ising random perceptron
TeX label thm:main, build/sections/model.tex, line 88
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.
Recorded comparisons with the paper
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Contributed Comparator run reports
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