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Lean statement

Finite-graph buffered Ising likelihood comparison

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.BufferedIsing.finite_graph_comparison

Lean statement file: BufferedIsing.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 218 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 finite_graph_comparison {n m : ℕ} (G : FiniteGraph n m)    (h : Fin n → ℝ) (I J F : Finset (Fin n))    (hIJ : Disjoint I J) (hIF : Disjoint I F) (hJF : Disjoint J F)    (i j : KeptVertex F) (hi : i.val ∈ I) (hj : j.val ∈ J)    (a τ : Spins n) (mix mix' : Spins n → ℝ)    (hmix : ∀ b, 0 ≤ mix b) (hmix' : ∀ b, 0 ≤ mix' b)    (hmixsum : ∑ b, mix b = 1) (hmix'sum : ∑ b, mix' b = 1) :    Real.exp (-4 * Real.artanh (qZero G I J F i j)) ≤        mixtureProb G h I J F a τ mix / mixtureProb G h I J F a τ mix' ∧      mixtureProb G h I J F a τ mix / mixtureProb G h I J F a τ mix' ≤        Real.exp (4 * Real.artanh (qZero G I J F i j)) := 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.

  • structure OAI.BufferedIsing.FiniteGraph line 11

    structure FiniteGraph (n m : ℕ) where  left : Fin m → Fin n  right : Fin m → Fin n  coupling : Fin m → ℝ  coupling_nonneg : ∀ e, 0 ≤ coupling e  no_loops : ∀ e, left e ≠ right e

  • abbreviation OAI.BufferedIsing.Spins line 18

    abbrev Spins (n : ℕ) := Fin n → Bool

  • definition OAI.BufferedIsing.spin line 20

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

  • definition OAI.BufferedIsing.energy line 22

    noncomputable def energy {n m : ℕ} (G : FiniteGraph n m)    (h : Fin n → ℝ) (σ : Spins n) : ℝ :=  (∑ e, G.coupling e * spin (σ (G.left e)) * spin (σ (G.right e))) +    ∑ v, h v * spin (σ v)

  • definition OAI.BufferedIsing.weight line 27

    noncomputable def weight {n m : ℕ} (G : FiniteGraph n m)    (h : Fin n → ℝ) (σ : Spins n) : ℝ := Real.exp (energy G h σ)

  • definition OAI.BufferedIsing.AgreesOn line 30

    def AgreesOn {n : ℕ} (S : Finset (Fin n)) (σ τ : Spins n) : Prop :=  ∀ v ∈ S, σ v = τ v

  • definition OAI.BufferedIsing.constrainedSum line 33

    noncomputable def constrainedSum {n m : ℕ} (G : FiniteGraph n m)    (h : Fin n → ℝ) (P : Spins n → Prop) : ℝ := by  classical  exact ∑ σ, if P σ then weight G h σ else 0

  • definition OAI.BufferedIsing.conditionalProb line 40

    noncomputable def conditionalProb {n m : ℕ} (G : FiniteGraph n m)    (h : Fin n → ℝ) (I J F : Finset (Fin n)) (a b τ : Spins n) : ℝ :=  constrainedSum G h (fun σ => AgreesOn I σ a ∧ AgreesOn J σ b ∧ AgreesOn F σ τ) /    constrainedSum G h (fun σ => AgreesOn J σ b ∧ AgreesOn F σ τ)

  • definition OAI.BufferedIsing.keptEdges line 46

    def keptEdges {n m : ℕ} (G : FiniteGraph n m) (F : Finset (Fin n)) : Finset (Fin m) :=  Finset.univ.filter (fun e => G.left e ∉ F ∧ G.right e ∉ F)

  • abbreviation OAI.BufferedIsing.KeptVertex line 49

    abbrev KeptVertex {n : ℕ} (F : Finset (Fin n)) := {v : Fin n // v ∉ F}

  • definition OAI.BufferedIsing.comparisonRel line 53

    def comparisonRel {n m : ℕ} (G : FiniteGraph n m)    (I J F : Finset (Fin n)) (ω : Finset (Fin m)) (u v : KeptVertex F) : Prop :=  (u.val ∈ I ∧ v.val ∈ I) ∨ (u.val ∈ J ∧ v.val ∈ J) ∨    ∃ e ∈ ω, (G.left e = u.val ∧ G.right e = v.val) ∨      (G.left e = v.val ∧ G.right e = u.val)

  • definition OAI.BufferedIsing.comparisonSetoid line 60

    def comparisonSetoid {n m : ℕ} (G : FiniteGraph n m)    (I J F : Finset (Fin n)) (ω : Finset (Fin m)) : Setoid (KeptVertex F) :=  Relation.EqvGen.setoid (comparisonRel G I J F ω)

  • definition OAI.BufferedIsing.clusterCount line 64

    noncomputable def clusterCount {n m : ℕ} (G : FiniteGraph n m)    (I J F : Finset (Fin n)) (ω : Finset (Fin m)) : ℕ :=  Nat.card (Quotient (comparisonSetoid G I J F ω))

  • definition OAI.BufferedIsing.fkWeight line 72

    noncomputable def fkWeight {n m : ℕ} (G : FiniteGraph n m)    (I J F : Finset (Fin n)) (ω : Finset (Fin m)) : ℝ := by  classical  exact (2 : ℝ) ^ clusterCount G I J F ω *    ∏ e ∈ keptEdges G F,      if e ∈ ω then 1 - Real.exp (-2 * G.coupling e) else Real.exp (-2 * G.coupling e)

  • definition OAI.BufferedIsing.qZero line 81

    noncomputable def qZero {n m : ℕ} (G : FiniteGraph n m)    (I J F : Finset (Fin n)) (i j : KeptVertex F) : ℝ := by  classical  exact (∑ ω ∈ (keptEdges G F).powerset,    if (comparisonSetoid G I J F ω).r i j then fkWeight G I J F ω else 0) /    ∑ ω ∈ (keptEdges G F).powerset, fkWeight G I J F ω

  • definition OAI.BufferedIsing.mixtureProb line 91

    noncomputable def mixtureProb {n m : ℕ} (G : FiniteGraph n m)    (h : Fin n → ℝ) (I J F : Finset (Fin n)) (a τ : Spins n)    (mix : Spins n → ℝ) : ℝ :=  ∑ b, mix b * conditionalProb G h I J F a b τ

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: arbitrary, comparison, conditional, configurations, finite, graph, laws, mixture and one more.

Statements in Buffered comparison and stopping-band resolution in critical Ising come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Buffered comparison and stopping-band resolution in critical Ising

Theorem (Finite-graph comparison)

For every two configurations b,b' on J and every configuration a on I,

e^{-4\operatorname{arctanh}q_0} \leq \frac{\mu(\sigma_I=a\mid\sigma_J=b,\sigma_F)} {\mu(\sigma_I=a\mid\sigma_J=b',\sigma_F)} \leq e^{4\operatorname{arctanh}q_0}.

Either conditional law may be replaced by an arbitrary mixture of these laws. The bound is symmetric in the two terminal sets.

Section 1, Introduction Read it in the paper TeX source, line 33
Show the TeX
For every two configurations $b,b'$ on $J$ and every configuration $a$
on $I$,
\[
 e^{-4\operatorname{arctanh}q_0}
 \leq
 \frac{\mu(\sigma_I=a\mid\sigma_J=b,\sigma_F)}
      {\mu(\sigma_I=a\mid\sigma_J=b',\sigma_F)}
 \leq e^{4\operatorname{arctanh}q_0}.
\]
Either conditional law may be replaced by an arbitrary mixture of
these laws. The bound is symmetric in the two terminal sets.

Other possible paper matches, closest wording first

  • Corollary 2.6 of Buffered comparison and stopping-band resolution in critical Ising. Words of the Lean statement, its names split into words, that also appear in the theorem: vertices, missing, edges, pinning, graph, agree and eleven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: deleted, compared, conditioning, allowed, buffered and boundary.
  • Theorem 9.1 of Buffered comparison and stopping-band resolution in critical Ising. Words of the Lean statement, its names split into words, that also appear in the theorem: kept, component, counting, full, retaining, list and 13 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: resulting, stopping, state, bands, ratio, buffer and one more.
  • Proposition 5.1 of Buffered comparison and stopping-band resolution in critical Ising. Words of the Lean statement, its names split into words, that also appear in the theorem: parallel, sides, full, arbitrary, mixture, number and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: comparable, differ, ratio, allowed and buffer.
  • Proposition 2.1 of SLE3 universality for weak finite-range Ising interactions. Words of the Lean statement, its names split into words, that also appear in the theorem: disjoint, terminal, graph, vertex, arbitrary, mixture and 14 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: outside, ising, lie, ratio and buffered.
  • Corollary 9.3 of Buffered comparison and stopping-band resolution in critical Ising. Words of the Lean statement, its names split into words, that also appear in the theorem: closure, countable, remain, list, separated, prescribed and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: approximation, lie, band, allow and buffered.
  • Proposition 6.3 of SLE3 universality for weak finite-range Ising interactions. Words of the Lean statement, its names split into words, that also appear in the theorem: separately, components, pinned, vertex, remains, arbitrary and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: conditioned and outside.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

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).

  • Accepted Their term: “accepted”.

    Re-run through Comparator with Lean’s kernel: accepted (the check took 76 s; peak memory 6.2 GiB).

    Reported 10 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 218’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: The only statement is a technical finite-graph inequality: finite_graph_comparison : exp(-4 * artanh (qZero ..)) ≤ mixtureProb .. mix / mixtureProb .. mix' ∧ ... ≤ exp(4 * artanh (qZero ..)) for Ising with arbitrary fields and boundary mixtures (docs: 'The paper's finite stopping-band approximation theorem is outside this selected statement'). None of the headline results (universality of critical bulk spin/energy limits under weak multispin perturbations, SLE_3 interface limits, weak iid bond disorder) is stated.

  • 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: None of the headline's three claims (bulk conformal universality, SLE₃ interface limits for contour perturbations, quenched SLE₃ for weak random bonds) is stated. The family's only challenge is a finite-graph Ising likelihood-ratio inequality — a proof ingredient from one of the family's six manuscripts ('Buffered comparison and stopping-band resolution in critical Ising'). It neither implies the headline nor is implied by it, and it is not among the supporting results declared in ComparatorChallenges/README.md.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalized result compares conditional Ising probabilities on finite graphs when two boundary mixtures differ but a common set of spins is pinned. If $q_0$ is the associated zero-field Fortuin–Kasteleyn connection probability after deleting the pinned vertices, the likelihood ratio lies between $\exp(-4\mathrm{artanh}\,q_0)$ and $\exp(4\mathrm{artanh}\,q_0)$. The statement allows arbitrary external fields and arbitrary probability mixtures satisfying the stated disjointness conditions. The paper's finite stopping-band approximation theorem is outside this selected statement.”

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

113 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: 53 to its own declarations, 62 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.

import Mathlibnamespace OAIopen scoped BigOperators Classicalnamespace BufferedIsing/-- A finite graph given by its edge list. Zero couplings can encode missing edges.Edges are indexed separately so parallel edges are counted separately. -/structure FiniteGraph (n m : ℕ) where  left : Fin m → Fin n  right : Fin m → Fin n  coupling : Fin m → ℝ  coupling_nonneg : ∀ e, 0 ≤ coupling e  no_loops : ∀ e, left e ≠ right eabbrev Spins (n : ℕ) := Fin n → Booldef spin (b : Bool) : ℝ := if b then 1 else -1noncomputable def energy {n m : ℕ} (G : FiniteGraph n m)    (h : Fin n → ℝ) (σ : Spins n) : ℝ :=  (∑ e, G.coupling e * spin (σ (G.left e)) * spin (σ (G.right e))) +    ∑ v, h v * spin (σ v)noncomputable def weight {n m : ℕ} (G : FiniteGraph n m)    (h : Fin n → ℝ) (σ : Spins n) : ℝ := Real.exp (energy G h σ)def AgreesOn {n : ℕ} (S : Finset (Fin n)) (σ τ : Spins n) : Prop :=  ∀ v ∈ S, σ v = τ vnoncomputable def constrainedSum {n m : ℕ} (G : FiniteGraph n m)    (h : Fin n → ℝ) (P : Spins n → Prop) : ℝ := by  classical  exact ∑ σ, if P σ then weight G h σ else 0/-- Conditional probability of σ_I=a, with σ_J=b and σ_F=τ prescribed.All sums are finite and all fields are finite real numbers. -/noncomputable def conditionalProb {n m : ℕ} (G : FiniteGraph n m)    (h : Fin n → ℝ) (I J F : Finset (Fin n)) (a b τ : Spins n) : ℝ :=  constrainedSum G h (fun σ => AgreesOn I σ a ∧ AgreesOn J σ b ∧ AgreesOn F σ τ) /    constrainedSum G h (fun σ => AgreesOn J σ b ∧ AgreesOn F σ τ)/-- Edges remaining after DELETION, not pinning, of F. -/def keptEdges {n m : ℕ} (G : FiniteGraph n m) (F : Finset (Fin n)) : Finset (Fin m) :=  Finset.univ.filter (fun e => G.left e ∉ F ∧ G.right e ∉ F)abbrev KeptVertex {n : ℕ} (F : Finset (Fin n)) := {v : Fin n // v ∉ F}/-- Open-edge adjacency together with the two separate terminal identifications.No virtual identification of common pins is present: those vertices are absent. -/def comparisonRel {n m : ℕ} (G : FiniteGraph n m)    (I J F : Finset (Fin n)) (ω : Finset (Fin m)) (u v : KeptVertex F) : Prop :=  (u.val ∈ I ∧ v.val ∈ I) ∨ (u.val ∈ J ∧ v.val ∈ J) ∨    ∃ e ∈ ω, (G.left e = u.val ∧ G.right e = v.val) ∨      (G.left e = v.val ∧ G.right e = u.val)/-- The equivalence closure implements components of the contracted open graph. -/def comparisonSetoid {n m : ℕ} (G : FiniteGraph n m)    (I J F : Finset (Fin n)) (ω : Finset (Fin m)) : Setoid (KeptVertex F) :=  Relation.EqvGen.setoid (comparisonRel G I J F ω)noncomputable def clusterCount {n m : ℕ} (G : FiniteGraph n m)    (I J F : Finset (Fin n)) (ω : Finset (Fin m)) : ℕ :=  Nat.card (Quotient (comparisonSetoid G I J F ω))/-- Zero-field FK weight with q=2 and p_e=1-exp(-2 K_e).Parallel edges may be retained: the measure after summing them is the measurefor the summed parallel coupling. Loops created by contraction do not changecomponents and cancel from connection probabilities. -/noncomputable def fkWeight {n m : ℕ} (G : FiniteGraph n m)    (I J F : Finset (Fin n)) (ω : Finset (Fin m)) : ℝ := by  classical  exact (2 : ℝ) ^ clusterCount G I J F ω *    ∏ e ∈ keptEdges G F,      if e ∈ ω then 1 - Real.exp (-2 * G.coupling e) else Real.exp (-2 * G.coupling e)/-- Choose any i in I and j in J. The terminal connection is independent ofthese choices because I and J are each identified. -/noncomputable def qZero {n m : ℕ} (G : FiniteGraph n m)    (I J F : Finset (Fin n)) (i j : KeptVertex F) : ℝ := by  classical  exact (∑ ω ∈ (keptEdges G F).powerset,    if (comparisonSetoid G I J F ω).r i j then fkWeight G I J F ω else 0) /    ∑ ω ∈ (keptEdges G F).powerset, fkWeight G I J F ω/-- Any probability mixture of source prescriptions. Full configurationsindex the mixture, but only their restrictions to J are used. Thus this isexactly the class of all mixtures on {-1,+1}^J. -/noncomputable def mixtureProb {n m : ℕ} (G : FiniteGraph n m)    (h : Fin n → ℝ) (I J F : Finset (Fin n)) (a τ : Spins n)    (mix : Spins n → ℝ) : ℝ :=  ∑ b, mix b * conditionalProb G h I J F a b τ/-- Finite-graph comparison for arbitrary probability mixtures on both sides.Dirac mixtures recover the conditional laws. -/theorem finite_graph_comparison {n m : ℕ} (G : FiniteGraph n m)    (h : Fin n → ℝ) (I J F : Finset (Fin n))    (hIJ : Disjoint I J) (hIF : Disjoint I F) (hJF : Disjoint J F)    (i j : KeptVertex F) (hi : i.val ∈ I) (hj : j.val ∈ J)    (a τ : Spins n) (mix mix' : Spins n → ℝ)    (hmix : ∀ b, 0 ≤ mix b) (hmix' : ∀ b, 0 ≤ mix' b)    (hmixsum : ∑ b, mix b = 1) (hmix'sum : ∑ b, mix' b = 1) :    Real.exp (-4 * Real.artanh (qZero G I J F i j)) ≤        mixtureProb G h I J F a τ mix / mixtureProb G h I J F a τ mix' ∧      mixtureProb G h I J F a τ mix / mixtureProb G h I J F a τ mix' ≤        Real.exp (4 * Real.artanh (qZero G I J F i j)) := by  sorryend BufferedIsingend 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.

Buffered comparison and stopping-band resolution in critical Ising

Theorem (Finite-graph comparison)

For every two configurations b,b' on J and every configuration a on I,

e^{-4\operatorname{arctanh}q_0} \leq \frac{\mu(\sigma_I=a\mid\sigma_J=b,\sigma_F)} {\mu(\sigma_I=a\mid\sigma_J=b',\sigma_F)} \leq e^{4\operatorname{arctanh}q_0}.

Either conditional law may be replaced by an arbitrary mixture of these laws. The bound is symmetric in the two terminal sets.

Section 1, Introduction Read it in the paper TeX source, line 33
Show the TeX
For every two configurations $b,b'$ on $J$ and every configuration $a$
on $I$,
\[
 e^{-4\operatorname{arctanh}q_0}
 \leq
 \frac{\mu(\sigma_I=a\mid\sigma_J=b,\sigma_F)}
      {\mu(\sigma_I=a\mid\sigma_J=b',\sigma_F)}
 \leq e^{4\operatorname{arctanh}q_0}.
\]
Either conditional law may be replaced by an arbitrary mixture of
these laws. The bound is symmetric in the two terminal sets.
Theorem (Finite conditional resolution)

Fix the preceding band geometry and use the critical plus-boundary law in a larger reference square. For every \epsilon,\eta>0, one can choose contour levels in any prescribed separated subwindows, a finite descriptor Z^\delta\in\{0,1,\ldots,m\}, and regular signed barriers \Gamma_1,\ldots,\Gamma_m with the following properties for all sufficiently fine meshes.

The exceptional state 0 has probability at most \epsilon. For every transcript in a good state i\in\{1,\ldots,m\}, insert the plus pins of \Gamma_i and then its minus pins. The sum of the two resulting total-variation changes on the target is at most \eta. The final target-side law Q_{i,\delta} is the same for every transcript in that state. Consequently this conclusion holds simultaneously for all bounded target observables, including mesh-dependent observables. For any fixed finite list of bands the descriptors have convergent joint laws.

The finite list of barriers and its positive geometric clearances are chosen before the mesh threshold. If another law has band density at most K times the reference density, its exceptional probability is at most K\epsilon.

Section 1, Introduction Read it in the paper TeX source, line 94
Show the TeX
Fix the preceding band geometry and use the critical plus-boundary law
in a larger reference square. For every $\epsilon,\eta>0$, one can
choose contour levels in any prescribed separated subwindows,
a finite descriptor $Z^\delta\in\{0,1,\ldots,m\}$, and regular
signed barriers $\Gamma_1,\ldots,\Gamma_m$ with the following properties
for all sufficiently fine meshes.

The exceptional state $0$ has probability at most $\epsilon$. For every
transcript in a good state $i\in\{1,\ldots,m\}$, insert the plus pins of $\Gamma_i$ and
then its minus pins. The sum of the two resulting total-variation
changes on the target is at most $\eta$. The final target-side law
$Q_{i,\delta}$ is the same for every transcript in that state.
Consequently this conclusion holds simultaneously for all bounded target
observables, including mesh-dependent observables. For any fixed finite
list of bands the descriptors have convergent joint laws.

The finite list of barriers and its positive geometric clearances are
chosen before the mesh threshold. If another law has band density at
most $K$ times the reference density, its exceptional probability is
at most $K\epsilon$.
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 218 link it.
Solution module
OAI.Probability.Ising.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
66
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

Finite-graph buffered Ising likelihood comparison

Family 218: Conformal universality for weakly interacting and random-bond Ising models

Candidate pairing

Theorem (Finite-graph comparison)

For every two configurations b,b' on J and every configuration a on I,

e^{-4\operatorname{arctanh}q_0} \leq \frac{\mu(\sigma_I=a\mid\sigma_J=b,\sigma_F)} {\mu(\sigma_I=a\mid\sigma_J=b',\sigma_F)} \leq e^{4\operatorname{arctanh}q_0}.

Either conditional law may be replaced by an arbitrary mixture of these laws. The bound is symmetric in the two terminal sets.

Buffered comparison and stopping-band resolution in critical Ising

TeX label in:comparison, build/sections/introduction.tex, line 33

The Lean statement says

theorem finite_graph_comparison {n m : ℕ} (G : FiniteGraph n m)    (h : Fin n → ℝ) (I J F : Finset (Fin n))    (hIJ : Disjoint I J) (hIF : Disjoint I F) (hJF : Disjoint J F)    (i j : KeptVertex F) (hi : i.val ∈ I) (hj : j.val ∈ J)    (a τ : Spins n) (mix mix' : Spins n → ℝ)    (hmix : ∀ b, 0 ≤ mix b) (hmix' : ∀ b, 0 ≤ mix' b)    (hmixsum : ∑ b, mix b = 1) (hmix'sum : ∑ b, mix' b = 1) :    Real.exp (-4 * Real.artanh (qZero G I J F i j)) ≤        mixtureProb G h I J F a τ mix / mixtureProb G h I J F a τ mix' ∧      mixtureProb G h I J F a τ mix / mixtureProb G h I J F a τ mix' ≤        Real.exp (4 * Real.artanh (qZero G I J F i j)) := 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.

Atlas selected this possible match because the two statements share words. No review of whether they mean the same thing is recorded.three independent groups report that Comparator accepted runs for this file at the release version shown here.Statement review: none recorded

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

Recorded comparisons with the paper

Atlas records no contributed comparison of this statement with the paper.

Contributed Comparator run reports

Atlas records no contributed report of running Comparator for this statement file.

Questions to resolve when comparing this statement with the paper

  • Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.

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