`ComparatorChallenges/BufferedIsing.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 120 curated AI-written annotations and 1273 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 9 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. Those 9 annotations were revised on 2026-10-14 to bind all of them to exact page-one quotations from 6 pinned OpenAI manuscripts, the matching pinned formalization notes, and named historical primary sources; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. The 3 manuscript-grounded matrix-multiplication annotations were composed from pinned OpenAI manuscripts and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Six additional target-specific annotations for Catalan's constant, the irrationality exponent of pi, dyadic Erdos similarity, Unique Games, quintic Lienard systems, and Yau nodal growth were composed from pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for Seymour's second-neighborhood conjecture, Cannon's conjecture, Koebe circle-domain uniformization, and circle-domain rigidity were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for simple Lebesgue spectrum on the three-torus, Rokhlin multiple mixing, the nine-dimensional Borsuk counterexample, and Tingley's sphere-isometry problem were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the Abhyankar-Sathaye counterexample, the circulant Hadamard conjecture, the Eilenberg-Ganea counterexample, and short Egyptian fractions were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and quotations from Sathaye, Turyn with Storer, Eilenberg with Ganea, and Erdos in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the complete Crouzeix inequality, unrestricted Euclidean five-coloring, the general Mahler conjecture, and Thompson F nonamenability were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, the matching formalization notes, and linked historical sources by Crouzeix, de Grey, Mahler, Cannon, Floyd, and Parry; no human review is claimed. Four additional target-specific annotations for algorithmic strong thin trees, Barnette's Hamiltonian-cycle conjecture, the explicit Artin CAT(0) counterexample, and generalized amplitude-damping capacity were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, matching formalization notes, and linked historical sources by Anari with Goddyn, Arts with Hopman and Timmermans, Charney, and Khatri with Sharma and Wilde; no human review is claimed. Fourteen solution-level annotations for Seymour's second-neighborhood theorem were added on 2026-10-14 from the pinned 15-page manuscript, covering the exact-neighborhood model, minimal-counterexample reduction, subset deficit, transitive blowup, pruning, extremal contradiction, and proof-bearing final theorem. The subset and blowup annotations also bind exact quotations from Tyler Seacrest (2019) and Brantner, Brockman, Kay, and Snively (2009); no human review is claimed. Thirty-seven declaration-level annotations were added on 2026-10-14 for the rest of ComparatorChallenges/AlgorithmicThinTrees.lean, covering its multigraph semantics, stack-machine cost model, explicit and binary-multiplicity encodings, output certificate, and complete algorithmic proposition. They are bound to the pinned OpenAI manuscript and linked earlier thin-tree sources by Luis Goddyn, Nima Anari, and Shayan Oveis Gharan; the annotation explicitly preserves that the concluding comparator theorem still has body by sorry, and no human review is claimed. Thirty-one supporting declarations in ComparatorChallenges/UniqueGamesTheorem.lean were upgraded or added on 2026-10-14, completing curated coverage of all thirty-two declarations in that file. The annotations cover the finite game model, Boolean formula and parser, binary-vector translations, explicit output serialization, finite-alphabet machine contract, and gap theorem; they distinguish manuscript-backed mathematics from Lean-specific canonical parsing and unary game output encoding, bind exact OpenAI manuscript quotations, and credit Subhash Khot through his 2010 survey. The comparator theorem remains explicitly labeled as having body by sorry; no human review is claimed. The source-derived generator was hardened on 2026-10-14 so a formalization note covering multiple comparator results no longer copies its combined Scope prose onto each declaration; those targets now use declaration-specific source comments when available and otherwise remain explicitly bounded to the exact Lean statement. The remaining source-derived annotations were generated mechanically from pinned formalization notes and source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the source prose.

Credit: OpenAI's pinned repository says an unreleased internal model produced the manuscripts and supporting proof artifacts. The collection builds on the mathematics community's accumulated problems, results, language, and formal libraries; each manuscript's BibTeX and bibliography are the authority for individual scholarly credit.

Project sources: Pinned openai/math documentation, Pinned formalization catalogue

17 declarations, 1 contain sorry, 0 declared as axioms

structure OAI.BufferedIsing.FiniteGraph A finite graph given by its edge list line 11
Source documentation
A finite graph given by its edge list. Zero couplings can encode missing edges. Edges are indexed separately so parallel edges are counted separately.
L4DEL explanation
A finite graph given by its edge list. Zero couplings can encode missing edges. Edges are indexed separately so parallel edges are counted separately.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Probability/Ising/Basic.lean:11 (same declaration text).
```
structure FiniteGraph (n m : ℕ) where
…
```
abbrev OAI.BufferedIsing.Spins line 18
Also declared: OAI/Probability/Ising/Basic.lean:18 (same declaration text).
def OAI.BufferedIsing.spin line 20
Also declared: OAI/Probability/Ising/Basic.lean:20 (same declaration text).
def OAI.BufferedIsing.energy line 22
Also declared: OAI/Probability/Ising/Basic.lean:22 (same declaration text).
def OAI.BufferedIsing.weight line 27
Also declared: OAI/Probability/Ising/Basic.lean:27 (same declaration text).
def OAI.BufferedIsing.AgreesOn line 30
Also declared: OAI/Probability/Ising/Basic.lean:30 (same declaration text).
def OAI.BufferedIsing.constrainedSum line 33
Also declared: OAI/Probability/Ising/Basic.lean:33 (same declaration text).
def OAI.BufferedIsing.conditionalProb Conditional probability of σ_I=a, with σ_J=b and σ_F=τ prescribed line 40
Source documentation
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.
L4DEL explanation
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.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Probability/Ising/Basic.lean:40 (same declaration text).
```
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 :=
…
```
Mentions: OAI.BufferedIsing.FiniteGraph, OAI.BufferedIsing.Spins
def OAI.BufferedIsing.keptEdges Edges remaining after DELETION, not pinning, of F line 46
Source documentation
Edges remaining after DELETION, not pinning, of F.
L4DEL explanation
Edges remaining after DELETION, not pinning, of F.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Probability/Ising/Basic.lean:46 (same declaration text).
```
def keptEdges {n m : ℕ} (G : FiniteGraph n m) (F : Finset (Fin n)) : Finset (Fin m) :=
…
```
Mentions: OAI.BufferedIsing.FiniteGraph
abbrev OAI.BufferedIsing.KeptVertex line 49
Also declared: OAI/Probability/Ising/Basic.lean:49 (same declaration text).
def OAI.BufferedIsing.comparisonRel Open-edge adjacency together with the two separate terminal identifications line 53
Source documentation
Open-edge adjacency together with the two separate terminal identifications. No virtual identification of common pins is present: those vertices are absent.
L4DEL explanation
Open-edge adjacency together with the two separate terminal identifications. No virtual identification of common pins is present: those vertices are absent.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Probability/Ising/Basic.lean:53 (same declaration text).
```
def comparisonRel {n m : ℕ} (G : FiniteGraph n m)
    (I J F : Finset (Fin n)) (ω : Finset (Fin m)) (u v : KeptVertex F) : Prop :=
…
```
Mentions: OAI.BufferedIsing.FiniteGraph, OAI.BufferedIsing.KeptVertex
def OAI.BufferedIsing.comparisonSetoid The equivalence closure implements components of the contracted open graph line 60
Source documentation
The equivalence closure implements components of the contracted open graph.
L4DEL explanation
The equivalence closure implements components of the contracted open graph.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Probability/Ising/Basic.lean:60 (same declaration text).
```
def comparisonSetoid {n m : ℕ} (G : FiniteGraph n m)
    (I J F : Finset (Fin n)) (ω : Finset (Fin m)) : Setoid (KeptVertex F) :=
…
```
Mentions: OAI.BufferedIsing.FiniteGraph, OAI.BufferedIsing.KeptVertex
def OAI.BufferedIsing.clusterCount line 64
Also declared: OAI/Probability/Ising/Basic.lean:64 (same declaration text).
def OAI.BufferedIsing.fkWeight Zero-field FK weight with q=2 and p_e=1-exp(-2 K_e) line 72
Source documentation
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 measure for the summed parallel coupling. Loops created by contraction do not change components and cancel from connection probabilities.
L4DEL explanation
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 measure for the summed parallel coupling. Loops created by contraction do not change components and cancel from connection probabilities.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Probability/Ising/Basic.lean:72 (same declaration text).
```
noncomputable def fkWeight {n m : ℕ} (G : FiniteGraph n m)
    (I J F : Finset (Fin n)) (ω : Finset (Fin m)) : ℝ := by
…
```
Mentions: OAI.BufferedIsing.FiniteGraph
def OAI.BufferedIsing.qZero Choose any i in I and j in J line 81
Source documentation
Choose any i in I and j in J. The terminal connection is independent of these choices because I and J are each identified.
L4DEL explanation
Choose any i in I and j in J. The terminal connection is independent of these choices because I and J are each identified.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Probability/Ising/Basic.lean:81 (same declaration text).
```
noncomputable def qZero {n m : ℕ} (G : FiniteGraph n m)
    (I J F : Finset (Fin n)) (i j : KeptVertex F) : ℝ := by
…
```
Mentions: OAI.BufferedIsing.FiniteGraph, OAI.BufferedIsing.KeptVertex
def OAI.BufferedIsing.mixtureProb Any probability mixture of source prescriptions line 91
Source documentation
Any probability mixture of source prescriptions. Full configurations index the mixture, but only their restrictions to J are used. Thus this is exactly the class of all mixtures on {-1,+1}^J.
L4DEL explanation
Any probability mixture of source prescriptions. Full configurations index the mixture, but only their restrictions to J are used. Thus this is exactly the class of all mixtures on {-1,+1}^J.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Probability/Ising/Basic.lean:91 (same declaration text).
```
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 :=
…
```
Mentions: OAI.BufferedIsing.FiniteGraph, OAI.BufferedIsing.Spins
theorem OAI.BufferedIsing.finite_graph_comparison Challenge statement: Conformal universality for weakly interacting and random-bond Ising models sorry line 98
Source documentation
Finite-graph comparison for arbitrary probability mixtures on both sides. Dirac mixtures recover the conditional laws.
L4DEL explanation
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.
1. This is an exact comparator target linked by the pinned repository's formalization note for this result.
2. The repository's comparator configuration selects OAI.Probability.Ising.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Probability/Ising/Main.lean:72 (same statement text; different proof or body). Qualify references as OAI.BufferedIsing.finite_graph_comparison@path.

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
```
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
…
```
Mentions: OAI.BufferedIsing.FiniteGraph, OAI.BufferedIsing.KeptVertex, OAI.BufferedIsing.Spins, OAI.BufferedIsing.qZero, OAI.BufferedIsing.mixtureProb