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Lean statement

Brownian continuum-random-tree limit for finite FK maps

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.FKCRT.finite_fk_maps_converge_to_brownian_crt

Lean statement file: FKCRT.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 211 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.

def MainStatement : Prop :=  ∀ q : ℝ, 4 < q → ∃ c : ℝ, 0 < c ∧    ∀ (Ω : Type) (mΩ : MeasurableSpace Ω) (P : @Measure Ω mΩ),      @IsProbabilityMeasure Ω mΩ P →      ∀ B : Fin 3 → ℝ≥0 → Ω → ℝ,        (∀ i, @IsBrownianReal Ω mΩ (B i) P) →        @iIndepFun Ω (Fin 3) mΩ (fun _ => ℝ≥0 → ℝ) (fun _ => inferInstance)          (fun i ω t => B i t ω) P →        ∀ F : MetricMeasureData → ℝ, IsGHPTest F →          Tendsto (fun n : ℕ => finiteExpectation q (c / Real.sqrt (n + 1)) n F)            atTop (𝓝 (@brownianExpectation Ω mΩ P B F))theorem finite_fk_maps_converge_to_brownian_crt : MainStatement := 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.FKCRT.Dart line 36

    abbrev Dart (n : ℕ) := Fin (2 * (n + 1))

  • abbreviation OAI.FKCRT.Orbit line 38

    abbrev Orbit {n : ℕ} (s : Equiv.Perm (Dart n)) :=  Quotient (Equiv.Perm.SameCycle.setoid s)

  • definition OAI.FKCRT.orbit line 41

    def orbit {n : ℕ} (s : Equiv.Perm (Dart n)) (d : Dart n) : Orbit s :=  Quotient.mk _ d

  • definition OAI.FKCRT.spanningGraph line 49

    def spanningGraph {n : ℕ} (a s : Equiv.Perm (Dart n)) (A : Set (Orbit a)) :    SimpleGraph (Orbit s) :=  SimpleGraph.fromRel fun v w =>    ∃ d : Dart n, orbit a d ∈ A ∧ orbit s d = v ∧ orbit s (a d) = w

  • definition OAI.FKCRT.MapConditions line 55

    def MapConditions {n : ℕ} (P : Equiv.Perm (Dart n) × Equiv.Perm (Dart n)) : Prop :=  Function.Involutive P.1 ∧ (∀ d, P.1 d ≠ d) ∧  (spanningGraph P.1 P.2 Set.univ).Connected ∧  Nat.card (Orbit P.2) + Nat.card (Orbit (P.2 * P.1)) = n + 3

  • abbreviation OAI.FKCRT.LabelledMap line 60

    abbrev LabelledMap (n : ℕ) :=  {P : Equiv.Perm (Dart n) × Equiv.Perm (Dart n) // MapConditions P}

  • definition OAI.FKCRT.MapIso line 64

    def MapIso {n : ℕ} (M N : LabelledMap n) : Prop :=  ∃ r : Equiv.Perm (Dart n), r 0 = 0 ∧    (∀ d, N.val.1 (r d) = r (M.val.1 d)) ∧    (∀ d, N.val.2 (r d) = r (M.val.2 d))

  • abbreviation OAI.FKCRT.RootedMap line 69

    abbrev RootedMap (n : ℕ) := Quotient (Relation.EqvGen.setoid (@MapIso n))

  • definition OAI.FKCRT.reversal line 73

    def reversal {n : ℕ} (M : RootedMap n) : Equiv.Perm (Dart n) := M.out.val.1

  • definition OAI.FKCRT.rotation line 75

    def rotation {n : ℕ} (M : RootedMap n) : Equiv.Perm (Dart n) := M.out.val.2

  • abbreviation OAI.FKCRT.Vertex line 77

    abbrev Vertex {n : ℕ} (M : RootedMap n) := Orbit (rotation M)

  • abbreviation OAI.FKCRT.Edge line 78

    abbrev Edge {n : ℕ} (M : RootedMap n) := Orbit (reversal M)

  • definition OAI.FKCRT.graph line 80

    def graph {n : ℕ} (M : RootedMap n) (A : Set (Edge M)) : SimpleGraph (Vertex M) :=  spanningGraph (reversal M) (rotation M) A

  • definition OAI.FKCRT.clusterCount line 84

    def clusterCount {n : ℕ} (M : RootedMap n) (A : Finset (Edge M)) : ℕ :=  Nat.card (graph M (A : Set (Edge M))).ConnectedComponent

  • definition OAI.FKCRT.fkWeight line 88

    def fkWeight {n : ℕ} (q : ℝ) (M : RootedMap n) (A : Finset (Edge M)) : ℝ :=  Real.rpow q ((clusterCount M A : ℝ) +    ((A.card : ℝ) - (Nat.card (Vertex M) : ℝ)) / 2)

  • definition OAI.FKCRT.partition line 92

    def partition (q : ℝ) (n : ℕ) : ℝ :=  ∑ M : RootedMap n, ∑ A : Finset (Edge M), fkWeight q M A

  • structure OAI.FKCRT.MetricMeasureData line 96

    structure MetricMeasureData where  Carrier : Type  distance : Carrier → Carrier → ℝ  measurableSpace : MeasurableSpace Carrier  mass : @Measure Carrier measurableSpace

  • definition OAI.FKCRT.MetricMeasureData.Valid line 103

    def MetricMeasureData.Valid (X : MetricMeasureData) : Prop :=  ∃ m : MetricSpace X.Carrier,    letI := m    (∀ x y, dist x y = X.distance x y) ∧    CompactSpace X.Carrier ∧ X.measurableSpace = borel X.Carrier ∧    X.mass Set.univ = 1

  • definition OAI.FKCRT.degreeMass line 111

    def degreeMass {n : ℕ} (M : RootedMap n) :    @Measure (Vertex M) ⊤ :=  letI : MeasurableSpace (Vertex M) := ⊤  ∑ d : Dart n, ENNReal.ofReal (1 / (2 * (n + 1) : ℝ)) •    Measure.dirac (orbit (rotation M) d)

  • definition OAI.FKCRT.mapSpace line 118

    def mapSpace {n : ℕ} (scale : ℝ) (M : RootedMap n) : MetricMeasureData where  Carrier := Vertex M  distance v w := scale * ((graph M Set.univ).dist v w : ℝ)  measurableSpace := ⊤  mass := degreeMass M

  • definition OAI.FKCRT.Amalgamates line 125

    def Amalgamates (X Y : MetricMeasureData)    (m : PseudoMetricSpace (X.Carrier ⊕ Y.Carrier)) : Prop :=  letI := m  (∀ x x', dist (Sum.inl x : X.Carrier ⊕ Y.Carrier) (Sum.inl x') = X.distance x x') ∧  (∀ y y', dist (Sum.inr y : X.Carrier ⊕ Y.Carrier) (Sum.inr y') = Y.distance y y')

  • definition OAI.FKCRT.ambientCost line 131

    def ambientCost (X Y : MetricMeasureData)    (m : PseudoMetricSpace (X.Carrier ⊕ Y.Carrier)) : ℝ≥0∞ :=
    Show the remaining 9 lines
      letI := m  letI : MeasurableSpace (X.Carrier ⊕ Y.Carrier) := borel _  max (Metric.hausdorffEDist (Set.range (Sum.inl : X.Carrier → X.Carrier ⊕ Y.Carrier))    (Set.range (Sum.inr : Y.Carrier → X.Carrier ⊕ Y.Carrier)))    (levyProkhorovEDist      (@Measure.map X.Carrier (X.Carrier ⊕ Y.Carrier) X.measurableSpace        (borel _) Sum.inl X.mass)      (@Measure.map Y.Carrier (X.Carrier ⊕ Y.Carrier) Y.measurableSpace        (borel _) Sum.inr Y.mass))

  • definition OAI.FKCRT.ghpEDist line 144

    def ghpEDist (X Y : MetricMeasureData) : ℝ≥0∞ :=  ⨅ m : {m : PseudoMetricSpace (X.Carrier ⊕ Y.Carrier) // Amalgamates X Y m},    ambientCost X Y m.val

  • definition OAI.FKCRT.IsGHPTest line 150

    def IsGHPTest (F : MetricMeasureData → ℝ) : Prop :=  (∃ C : ℝ, ∀ X, |F X| ≤ C) ∧  ∀ X, X.Valid → ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧    ∀ Y, Y.Valid → ghpEDist X Y < ENNReal.ofReal δ → |F Y - F X| < ε

7 more definitions are reached the same way; the full file lists them.

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: isometry, prokhorov, modulo, preserving, hausdorff, spaces, real, metric and four more. Found in the paper’s title, which its main theorem states: brownian, tree, maps, limits and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: gromov, 1/2, topology, constant and positive.

Statements in Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four

Theorem

For every fixed real q>4, there is a deterministic constant c(q)\in(0,\infty) such that, as n\to\infty through all positive integers,

\big(V(M_n),c(q)n^{-1/2}d_n,\mu_n\big) \ \Longrightarrow\ (\mathcal T_e,d_e,\mu_e)

in the Gromov–Hausdorff–Prokhorov topology on compact metric probability spaces modulo measure-preserving isometry.

Section 1, Introduction Read it in the paper TeX source, line 100
Show the TeX
For every fixed real $q>4$, there is a deterministic constant
$c(q)\in(0,\infty)$ such that, as $n\to\infty$ through all positive
integers,
\[
 \big(V(M_n),c(q)n^{-1/2}d_n,\mu_n\big)
 \ \Longrightarrow\ (\mathcal T_e,d_e,\mu_e)
\]
in the Gromov--Hausdorff--Prokhorov topology on compact metric probability
spaces modulo measure-preserving isometry.

Other possible paper matches, closest wording first

  • Theorem 1.1 of Metric-measure limits of subcritical FK and spanning-tree planar maps. Words of the Lean statement, its names split into words, that also appear in the theorem: decoration, spanning, isometry, prokhorov, specified, modulo and 13 more. Found in the paper’s title, which its main theorem states: maps. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: gromov, convergence, topology and positive.
  • Proposition 5.1 of Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four. Words of the Lean statement, its names split into words, that also appear in the theorem: standard, sigma, excursion, sqrt, brownian, tree and two more.
  • Proposition 2.1 of A Linear Clock for Random Walk on Tree-Weighted Planar Maps. Words of the Lean statement, its names split into words, that also appear in the theorem: distribution, spanning, edges, modulo, rooted, oriented and 22 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: 1/2 and continuum.
  • Theorem 13.1 of Metric-measure limits of subcritical FK and spanning-tree planar maps. Words of the Lean statement, its names split into words, that also appear in the theorem: spanning, equivalence, edges, embedding, gives, root and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: planar.
  • Theorem 1.2 of The critical Liouville quantum sphere and geometric limits of FK maps at q=4. Words of the Lean statement, its names split into words, that also appear in the theorem: thm, weight, fk, max, hausdorff, distance and nine more. Found in the paper’s title, which its main theorem states: maps and limits. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: convergence, critical, continuum, topology and positive.
  • Theorem 1.1 of Spectral convergence for critical FK–Ising planar maps. Words of the Lean statement, its names split into words, that also appear in the theorem: isometry, closure, prokhorov, hausdorff, eq, spaces and seven more. Found in the paper’s title, which its main theorem states: fk and maps. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: gromov, convergence, continuum, topology, constants and positive.

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 211’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: Only the q > 4 Brownian-CRT limit is stated: MainStatement : ∀ q > 4, ∃ c > 0, ... Tendsto (fun n => finiteExpectation q (c / √(n+1)) n F) atTop (𝓝 (brownianExpectation P B F)) for every GHP-test F, with the standard excursion realized as the norm of three independent Mathlib IsBrownianReal bridges. Not stated: the 0 < q <= 4 convergence to Liouville quantum gravity spheres, the Liouville Brownian motion limits of random walks, and the FK-Ising spectral/heat-trace convergence (docs: 'For every fixed q>4 ...').

    Definitions the group asks readers to compare with the paper: Entire topology is hand-rolled: MetricMeasureData, ghpEDist := ⨅ amalgamating pseudometrics on X ⊕ Y (Hausdorff/Lévy-Prokhorov), IsGHPTest F := (∃ C, ∀ X, |F X| ≤ C) ∧ ∀ X valid, ∀ ε, ∃ δ, ∀ Y valid, ghpEDist X Y < δ → |F Y - F X| < ε (convergence in distribution defined by bounded tests continuous at valid points); maps are pairs of dart permutations with fkWeight q M A := q^(k(A) + (|A| - |V|)/2); treeSpace is a quotient of unitInterval by excursionDist. Plausible and documented in the file header, but none of it is Mathlib's notion.

  • 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: LQG-sphere limits for 0<q<=4 (and the transition), random-walk convergence to Liouville Brownian motion for FK–Ising and tree-weighted maps, and FK–Ising spectral convergence; six of the seven manuscripts have no Lean statement

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“For every fixed $q>4$, the formalization proves the Brownian continuum-random-tree limit for critical finite Fortuin–Kasteleyn planar maps. After rescaling graph distances in an $n$-edge map by a constant depending on $q$ times $n^{-1/2}$ and using normalized degree measure, the metric-measure space converges in distribution to the Brownian continuum random tree in the Gromov–Hausdorff–Prokhorov topology. The convergence holds through all positive integer sizes.”

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

214 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: 94 to its own declarations, 114 to Mathlib’s documentation. 33 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name 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.

import Mathlibnamespace OAI/-!Finite Fortuin–Kasteleyn maps and the Brownian continuum random tree.All combinatorial maps below have a distinguished dart 0 and are quotientedby orientation-preserving dart relabellings fixing that dart. A rotation systemwith Euler characteristic 2 encodes precisely a connected oriented-sphere map.No simplicity restriction is imposed. Vertices and edges are rotation andedge-reversal orbits, respectively. Every edge subset is included in the law.The index n here represents source size n+1, so the limit is through ALLpositive edge sizes, not a subsequence.MetricMeasureData packages the actual distance, measurable space and measure,without using arbitrary representatives of measure-preserving isometry classesas a global typeclass. Valid objects are compact metric probability spaces.The GHP extended distance is the infimum over common isometric embeddings,represented by pseudometrics on the disjoint union; zero cross distances simplyidentify points in the common ambient space. Tests are bounded functionscontinuous for that distance on valid objects. Thus GHP convergence is writtenin its bounded-continuous-test (weak distributional convergence) definition.A standard normalized Brownian excursion is represented in its standardthree-dimensional Bessel-bridge realization: the norm of three independentstandard Brownian bridges of duration 1. No rescaling of the excursion is used.-/noncomputable sectionopen Set Filter MeasureTheory ProbabilityTheoryopen scoped Topology BigOperators ENNReal NNRealnamespace FKCRTabbrev Dart (n : ℕ) := Fin (2 * (n + 1))abbrev Orbit {n : ℕ} (s : Equiv.Perm (Dart n)) :=  Quotient (Equiv.Perm.SameCycle.setoid s)def orbit {n : ℕ} (s : Equiv.Perm (Dart n)) (d : Dart n) : Orbit s :=  Quotient.mk _ dinstance {n : ℕ} (s : Equiv.Perm (Dart n)) : Fintype (Orbit s) := Fintype.ofFinite _/-- The spanning graph with precisely the edges in A, with loops omitted onlyfrom adjacency: they still exist as edge orbits, have FK weight, and contributetwo darts to the degree measure. Omitting loops does not change distance or k. -/def spanningGraph {n : ℕ} (a s : Equiv.Perm (Dart n)) (A : Set (Orbit a)) :    SimpleGraph (Orbit s) :=  SimpleGraph.fromRel fun v w =>    ∃ d : Dart n, orbit a d ∈ A ∧ orbit s d = v ∧ orbit s (a d) = w/-- Labelled connected orientable genus-zero rotation systems. -/def MapConditions {n : ℕ} (P : Equiv.Perm (Dart n) × Equiv.Perm (Dart n)) : Prop :=  Function.Involutive P.1 ∧ (∀ d, P.1 d ≠ d) ∧  (spanningGraph P.1 P.2 Set.univ).Connected ∧  Nat.card (Orbit P.2) + Nat.card (Orbit (P.2 * P.1)) = n + 3abbrev LabelledMap (n : ℕ) :=  {P : Equiv.Perm (Dart n) × Equiv.Perm (Dart n) // MapConditions P}/-- Root-preserving isomorphism of oriented maps (both dart permutations intertwine). -/def MapIso {n : ℕ} (M N : LabelledMap n) : Prop :=  ∃ r : Equiv.Perm (Dart n), r 0 = 0 ∧    (∀ d, N.val.1 (r d) = r (M.val.1 d)) ∧    (∀ d, N.val.2 (r d) = r (M.val.2 d))abbrev RootedMap (n : ℕ) := Quotient (Relation.EqvGen.setoid (@MapIso n))instance (n : ℕ) : Fintype (RootedMap n) := Fintype.ofFinite _def reversal {n : ℕ} (M : RootedMap n) : Equiv.Perm (Dart n) := M.out.val.1def rotation {n : ℕ} (M : RootedMap n) : Equiv.Perm (Dart n) := M.out.val.2abbrev Vertex {n : ℕ} (M : RootedMap n) := Orbit (rotation M)abbrev Edge {n : ℕ} (M : RootedMap n) := Orbit (reversal M)def graph {n : ℕ} (M : RootedMap n) (A : Set (Edge M)) : SimpleGraph (Vertex M) :=  spanningGraph (reversal M) (rotation M) A/-- The source k_M(A), including isolated vertices. -/def clusterCount {n : ℕ} (M : RootedMap n) (A : Finset (Edge M)) : ℕ :=  Nat.card (graph M (A : Set (Edge M))).ConnectedComponent/-- Exact decorated finite-map weight from eq:finite-law. -/def fkWeight {n : ℕ} (q : ℝ) (M : RootedMap n) (A : Finset (Edge M)) : ℝ :=  Real.rpow q ((clusterCount M A : ℝ) +    ((A.card : ℝ) - (Nat.card (Vertex M) : ℝ)) / 2)def partition (q : ℝ) (n : ℕ) : ℝ :=  ∑ M : RootedMap n, ∑ A : Finset (Edge M), fkWeight q M A/-- Unbundled presentation of a measured metric space; validity is specified below. -/structure MetricMeasureData where  Carrier : Type  distance : Carrier → Carrier → ℝ  measurableSpace : MeasurableSpace Carrier  mass : @Measure Carrier measurableSpace/-- Exactly compact metric probability spaces, with their Borel sigma algebras. -/def MetricMeasureData.Valid (X : MetricMeasureData) : Prop :=  ∃ m : MetricSpace X.Carrier,    letI := m    (∀ x y, dist x y = X.distance x y) ∧    CompactSpace X.Carrier ∧ X.measurableSpace = borel X.Carrier ∧    X.mass Set.univ = 1/-- Normalized degree measure is uniform dart mass pushed to vertices. -/def degreeMass {n : ℕ} (M : RootedMap n) :    @Measure (Vertex M) ⊤ :=  letI : MeasurableSpace (Vertex M) := ⊤  ∑ d : Dart n, ENNReal.ofReal (1 / (2 * (n + 1) : ℝ)) •    Measure.dirac (orbit (rotation M) d)/-- Root and decoration are not included in the metric probability space. -/def mapSpace {n : ℕ} (scale : ℝ) (M : RootedMap n) : MetricMeasureData where  Carrier := Vertex M  distance v w := scale * ((graph M Set.univ).dist v w : ℝ)  measurableSpace := ⊤  mass := degreeMass M/-- An ambient pseudometric on the disjoint union giving isometric embeddings. -/def Amalgamates (X Y : MetricMeasureData)    (m : PseudoMetricSpace (X.Carrier ⊕ Y.Carrier)) : Prop :=  letI := m  (∀ x x', dist (Sum.inl x : X.Carrier ⊕ Y.Carrier) (Sum.inl x') = X.distance x x') ∧  (∀ y y', dist (Sum.inr y : X.Carrier ⊕ Y.Carrier) (Sum.inr y') = Y.distance y y')def ambientCost (X Y : MetricMeasureData)    (m : PseudoMetricSpace (X.Carrier ⊕ Y.Carrier)) : ℝ≥0∞ :=  letI := m  letI : MeasurableSpace (X.Carrier ⊕ Y.Carrier) := borel _  max (Metric.hausdorffEDist (Set.range (Sum.inl : X.Carrier → X.Carrier ⊕ Y.Carrier))    (Set.range (Sum.inr : Y.Carrier → X.Carrier ⊕ Y.Carrier)))    (levyProkhorovEDist      (@Measure.map X.Carrier (X.Carrier ⊕ Y.Carrier) X.measurableSpace        (borel _) Sum.inl X.mass)      (@Measure.map Y.Carrier (X.Carrier ⊕ Y.Carrier) Y.measurableSpace        (borel _) Sum.inr Y.mass))/-- The common-isometric-embedding definition of the source GHP distance. -/def ghpEDist (X Y : MetricMeasureData) : ℝ≥0∞ :=  ⨅ m : {m : PseudoMetricSpace (X.Carrier ⊕ Y.Carrier) // Amalgamates X Y m},    ambientCost X Y m.val/-- Bounded continuous tests on compact metric probability spaces modulomeasure-preserving isometry, expressed on presentations. -/def IsGHPTest (F : MetricMeasureData → ℝ) : Prop :=  (∃ C : ℝ, ∀ X, |F X| ≤ C) ∧  ∀ X, X.Valid → ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧    ∀ Y, Y.Valid → ghpEDist X Y < ENNReal.ofReal δ → |F Y - F X| < ε/-- Source excursion pseudometric, with the closed interval minimum. -/def excursionDist (e : unitInterval → ℝ) (s t : unitInterval) : ℝ :=  e s + e t - 2 * sInf (e '' Set.uIcc s t)abbrev ExcursionTree (e : unitInterval → ℝ) :=  Quotient (Relation.EqvGen.setoid (fun s t => excursionDist e s t = 0))def treeProjection (e : unitInterval → ℝ) (t : unitInterval) : ExcursionTree e :=  Quotient.mk _ t/-- The quotient by d_e=0, its induced metric, and pushforward of uniform time.The equivalence closure is redundant for continuous e; the induced distanceis independent of representatives, as required by the pseudometric proof. -/def treeSpace (e : unitInterval → ℝ) : MetricMeasureData :=  letI : TopologicalSpace (ExcursionTree e) :=    TopologicalSpace.coinduced (treeProjection e) inferInstance  letI : MeasurableSpace (ExcursionTree e) := borel _  { Carrier := ExcursionTree e    distance := fun x y => excursionDist e x.out y.out    measurableSpace := borel _    mass := Measure.map (treeProjection e) (volume : Measure unitInterval) }/-- Standard normalized excursion, realized as the 3D Bessel bridge. -/def standardExcursion {Ω : Type} (B : Fin 3 → ℝ≥0 → Ω → ℝ) (ω : Ω)    (t : unitInterval) : ℝ :=  Real.sqrt (∑ i : Fin 3, (B i ⟨(t : ℝ), t.property.1⟩ ω -    (t : ℝ) * B i 1 ω) ^ 2)/-- Exact expectation under the finite law, with n+1 edges. -/def finiteExpectation (q scale : ℝ) (n : ℕ) (F : MetricMeasureData → ℝ) : ℝ :=  ∑ M : RootedMap n, ∑ A : Finset (Edge M),    (fkWeight q M A / partition q n) * F (mapSpace scale M)def brownianExpectation {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω)    (B : Fin 3 → ℝ≥0 → Ω → ℝ) (F : MetricMeasureData → ℝ) : ℝ :=  ∫ ω, F (treeSpace (standardExcursion B ω)) ∂P/-- thm:main. The quantifiers over Brownian realizations express that the limitis the standard excursion-tree law, not a coupling-specific object. -/def MainStatement : Prop :=  ∀ q : ℝ, 4 < q → ∃ c : ℝ, 0 < c ∧    ∀ (Ω : Type) (mΩ : MeasurableSpace Ω) (P : @Measure Ω mΩ),      @IsProbabilityMeasure Ω mΩ P →      ∀ B : Fin 3 → ℝ≥0 → Ω → ℝ,        (∀ i, @IsBrownianReal Ω mΩ (B i) P) →        @iIndepFun Ω (Fin 3) mΩ (fun _ => ℝ≥0 → ℝ) (fun _ => inferInstance)          (fun i ω t => B i t ω) P →        ∀ F : MetricMeasureData → ℝ, IsGHPTest F →          Tendsto (fun n : ℕ => finiteExpectation q (c / Real.sqrt (n + 1)) n F)            atTop (𝓝 (@brownianExpectation Ω mΩ P B F))/-- Finite FK maps converge in GHP distribution to the standard Brownian CRT. -/theorem finite_fk_maps_converge_to_brownian_crt : MainStatement := by  sorryend FKCRTendend OAI
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Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four

Theorem

For every fixed real q>4, there is a deterministic constant c(q)\in(0,\infty) such that, as n\to\infty through all positive integers,

\big(V(M_n),c(q)n^{-1/2}d_n,\mu_n\big) \ \Longrightarrow\ (\mathcal T_e,d_e,\mu_e)

in the Gromov–Hausdorff–Prokhorov topology on compact metric probability spaces modulo measure-preserving isometry.

Section 1, Introduction Read it in the paper TeX source, line 100
Show the TeX
For every fixed real $q>4$, there is a deterministic constant
$c(q)\in(0,\infty)$ such that, as $n\to\infty$ through all positive
integers,
\[
 \big(V(M_n),c(q)n^{-1/2}d_n,\mu_n\big)
 \ \Longrightarrow\ (\mathcal T_e,d_e,\mu_e)
\]
in the Gromov--Hausdorff--Prokhorov topology on compact metric probability
spaces modulo measure-preserving isometry.
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 211 link it.
Solution module
OAI.Probability.FKMaps.Convergence
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
34 local vocabulary declarations
Lines of Lean, without comments or blank lines
116
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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Plectis Atlas

Brownian continuum-random-tree limit for finite FK maps

Family 211: The geometric phase diagram, diffusion, and spectra of random planar maps

Candidate pairing

Theorem

For every fixed real q>4, there is a deterministic constant c(q)\in(0,\infty) such that, as n\to\infty through all positive integers,

\big(V(M_n),c(q)n^{-1/2}d_n,\mu_n\big) \ \Longrightarrow\ (\mathcal T_e,d_e,\mu_e)

in the Gromov–Hausdorff–Prokhorov topology on compact metric probability spaces modulo measure-preserving isometry.

Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four

TeX label thm:main, build/main.tex, line 100

The Lean statement says

def MainStatement : Prop :=  ∀ q : ℝ, 4 < q → ∃ c : ℝ, 0 < c ∧    ∀ (Ω : Type) (mΩ : MeasurableSpace Ω) (P : @Measure Ω mΩ),      @IsProbabilityMeasure Ω mΩ P →      ∀ B : Fin 3 → ℝ≥0 → Ω → ℝ,        (∀ i, @IsBrownianReal Ω mΩ (B i) P) →        @iIndepFun Ω (Fin 3) mΩ (fun _ => ℝ≥0 → ℝ) (fun _ => inferInstance)          (fun i ω t => B i t ω) P →        ∀ F : MetricMeasureData → ℝ, IsGHPTest F →          Tendsto (fun n : ℕ => finiteExpectation q (c / Real.sqrt (n + 1)) n F)            atTop (𝓝 (@brownianExpectation Ω mΩ P B F))theorem finite_fk_maps_converge_to_brownian_crt : MainStatement := by  sorry

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

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

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