`ComparatorChallenges/CriticalSK.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

31 declarations, 3 contain sorry, 0 declared as axioms

Critical SK equilibrium starts, covariance, and mixing lower bounds

The zero-field model has independent Gaussian edge couplings of variance 1 / n. Continuous heat bath updates each site at rate one; a discrete attempt chooses a site uniformly. Distances and mixing times use total variation threshold 1 / 4.

abbrev OAI.CriticalSK.Spin line 20
Also declared: ComparatorChallenges/CriticalSKMixing.lean:11 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:26 (same declaration text).
abbrev OAI.CriticalSK.Edge line 22
Also declared: ComparatorChallenges/CriticalSKMixing.lean:13 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:28 (same declaration text).
abbrev OAI.CriticalSK.Disorder line 24
Also declared: ComparatorChallenges/CriticalSKMixing.lean:15 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:30 (same declaration text).
def OAI.CriticalSK.spinValue line 26
Also declared: ComparatorChallenges/CriticalSKMixing.lean:17 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:32 (same declaration text).
def OAI.CriticalSK.hamiltonian line 28
Also declared: ComparatorChallenges/CriticalSKMixing.lean:19 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:34 (same declaration text).
def OAI.CriticalSK.partition line 31
Also declared: ComparatorChallenges/CriticalSKMixing.lean:22 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:37 (same declaration text).
def OAI.CriticalSK.gibbs line 34
Also declared: ComparatorChallenges/CriticalSKMixing.lean:25 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:40 (same declaration text).
def OAI.CriticalSK.disorderLaw line 37
Also declared: ComparatorChallenges/CriticalSKMixing.lean:28 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:43 (same declaration text).
def OAI.CriticalSK.sameExcept line 40
Also declared: ComparatorChallenges/CriticalSKMixing.lean:31 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:46 (same declaration text).
instance OAI.CriticalSK.(instance at line 43) line 43
def OAI.CriticalSK.siteKernel line 46
Also declared: ComparatorChallenges/CriticalSKMixing.lean:37 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:52 (same declaration text).
def OAI.CriticalSK.discreteKernel line 51
Also declared: ComparatorChallenges/CriticalSKMixing.lean:42 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:57 (same declaration text).
def OAI.CriticalSK.generator line 54
Also declared: ComparatorChallenges/CriticalSKMixing.lean:45 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:60 (same declaration text).
def OAI.CriticalSK.continuousKernel line 57
Also declared: ComparatorChallenges/CriticalSKMixing.lean:48 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:63 (same declaration text).
def OAI.CriticalSK.totalVariation line 61
Also declared: ComparatorChallenges/CriticalSKMixing.lean:52 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:67 (same declaration text).
def OAI.CriticalSK.continuousDistance line 64
Also declared: ComparatorChallenges/CriticalSKMixing.lean:55 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:70 (same declaration text).
def OAI.CriticalSK.discreteDistance line 67
Also declared: ComparatorChallenges/CriticalSKMixing.lean:58 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:73 (same declaration text).
def OAI.CriticalSK.continuousMixingTime line 70
Also declared: ComparatorChallenges/CriticalSKMixing.lean:61 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:76 (same declaration text).
def OAI.CriticalSK.discreteMixingTime line 73
Also declared: ComparatorChallenges/CriticalSKMixing.lean:64 (same declaration text), OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:79 (same declaration text).
def OAI.CriticalSK.mean line 76
Also declared: OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:82 (same declaration text).
def OAI.CriticalSK.variance line 79
Also declared: OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:85 (same declaration text).
def OAI.CriticalSK.dirichlet line 82
Also declared: OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:88 (same declaration text).
def OAI.CriticalSK.linearObservable line 85
Also declared: OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:91 (same declaration text).
def OAI.CriticalSK.covariance line 88
Also declared: OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:94 (same declaration text).
def OAI.CriticalSK.covarianceNorm line 93
Also declared: OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:99 (same declaration text).
def OAI.CriticalSK.linearRayleigh The supremum of variance divided by the rate-one Dirichlet form over nonzero linear spin observables line 98
Source documentation
The supremum of variance divided by the rate-one Dirichlet form over nonzero linear spin observables.
L4DEL explanation
The supremum of variance divided by the rate-one Dirichlet form over nonzero linear spin observables.
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/MathematicalPhysics/CriticalSK/SpinModel.lean:102 (same declaration text).
```
def linearRayleigh {n : ℕ} (W : Disorder n) : ℝ :=
…
```
Mentions: OAI.CriticalSK.Disorder
def OAI.CriticalSK.continuousGoodMass Gibbs mass of realized initial states still farther than 1 / 4 from equilibrium line 103
Source documentation
Gibbs mass of realized initial states still farther than 1 / 4 from equilibrium.
L4DEL explanation
Gibbs mass of realized initial states still farther than 1 / 4 from equilibrium.
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/MathematicalPhysics/CriticalSK/SpinModel.lean:106 (same declaration text).
```
def continuousGoodMass {n : ℕ} (W : Disorder n) (t : ℝ) : ℝ :=
…
```
Mentions: OAI.CriticalSK.Disorder
def OAI.CriticalSK.discreteGoodMass Gibbs mass of realized initial states still farther than 1 / 4 after k attempts line 107
Source documentation
Gibbs mass of realized initial states still farther than 1 / 4 after k attempts.
L4DEL explanation
Gibbs mass of realized initial states still farther than 1 / 4 after k attempts.
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)
Distinct definitions of this name: OAI/MathematicalPhysics/CriticalSK/SpinModel.lean:109 (same statement text; different proof or body). Qualify references as OAI.CriticalSK.discreteGoodMass@path.
```
def discreteGoodMass {n : ℕ} (W : Disorder n) (k : ℕ) : ℝ :=
…
```
Mentions: OAI.CriticalSK.Disorder
theorem OAI.CriticalSK.realized_equilibrium_initial_states Challenge statement: Equilibrium starts, covariance, and mixing lower bounds sorry line 112
Source documentation
For deterministic nonnegative times o(n^(2/3)) and attempt counts o(n^(5/3)), the Gibbs mass of obstructed initial states tends to one in disorder probability.
L4DEL explanation
For deterministic nonnegative times o(n^(2/3)) and attempt counts o(n^(5/3)), the Gibbs mass of obstructed initial states tends to one in disorder probability.
1. This explanation is a condensed rendering of this declaration's own doc comment at the pinned source commit.
2. The linked formalization note covers 5 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
3. The repository's comparator configuration selects OAI.MathematicalPhysics.CriticalSK.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/MathematicalPhysics/CriticalSK/Equilibrium.lean:103 (same statement text; different proof or body). Qualify references as OAI.CriticalSK.realized_equilibrium_initial_states@path.
```
theorem realized_equilibrium_initial_states :
    (∀ t : ℕ → ℝ, (∀ n, 0 ≤ t n) →
      Tendsto (fun n => t n / (n : ℝ) ^ ((2 : ℝ) / 3)) atTop (𝓝 0) →
      ∀ δ : ℝ, 0 < δ →
        Tendsto (fun n => (disorderLaw n {W |
          δ ≤ |continuousGoodMass W (t n) - 1|}).toReal) atTop (𝓝 0)) ∧
    (∀ k : ℕ → ℕ,
      Tendsto (fun n => (k n : ℝ) / (n : ℝ) ^ ((5 : ℝ) / 3)) atTop (𝓝 0) →
      ∀ δ : ℝ, 0 < δ →
        Tendsto (fun n => (disorderLaw n {W |
          δ ≤ |discreteGoodMass W (k n) - 1|}).toReal) atTop (𝓝 0)) := by
…
```
Mentions: OAI.CriticalSK.disorderLaw, OAI.CriticalSK.continuousGoodMass
theorem OAI.CriticalSK.covariance_and_linear_tests Challenge statement: Equilibrium starts, covariance, and mixing lower bounds sorry line 127
Source documentation
Both the covariance operator norm and the linear inverse-Rayleigh supremum lie between n^(2/3) / M n and M n * n^(2/3) with probability tending to one.
L4DEL explanation
Both the covariance operator norm and the linear inverse-Rayleigh supremum lie between n^(2/3) / M n and M n * n^(2/3) with probability tending to one.
1. This explanation is a condensed rendering of this declaration's own doc comment at the pinned source commit.
2. The linked formalization note covers 5 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
3. The repository's comparator configuration selects OAI.MathematicalPhysics.CriticalSK.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/MathematicalPhysics/CriticalSK/Rayleigh.lean:607 (same statement text; different proof or body). Qualify references as OAI.CriticalSK.covariance_and_linear_tests@path.
```
theorem covariance_and_linear_tests (M : ℕ → ℝ)
    (hMpos : ∀ n, 0 < M n) (hM : Tendsto M atTop atTop) :
    Tendsto (fun n : ℕ => (disorderLaw n {W |
      (n : ℝ) ^ ((2 : ℝ) / 3) / M n ≤ covarianceNorm W ∧
      covarianceNorm W ≤ M n * (n : ℝ) ^ ((2 : ℝ) / 3) ∧
      (n : ℝ) ^ ((2 : ℝ) / 3) / M n ≤ linearRayleigh W ∧
      linearRayleigh W ≤ M n * (n : ℝ) ^ ((2 : ℝ) / 3)}).toReal) atTop (𝓝 1) := by
…
```
Mentions: OAI.CriticalSK.disorderLaw, OAI.CriticalSK.covarianceNorm, OAI.CriticalSK.linearRayleigh
theorem OAI.CriticalSK.critical_mixing_lower_bounds Challenge statement: Equilibrium starts, covariance, and mixing lower bounds sorry line 138
Source documentation
Worst-start mixing takes at least n^(2/3-ε) continuous time and n^(5/3-ε) discrete attempts with probability tending to one.
L4DEL explanation
Worst-start mixing takes at least n^(2/3-ε) continuous time and n^(5/3-ε) discrete attempts with probability tending to one.
1. This explanation is a condensed rendering of this declaration's own doc comment at the pinned source commit.
2. The linked formalization note covers 5 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
3. The repository's comparator configuration selects OAI.MathematicalPhysics.CriticalSK.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/MathematicalPhysics/CriticalSK/Mixing.lean:284 (different declaration text). Qualify references as OAI.CriticalSK.critical_mixing_lower_bounds@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 critical_mixing_lower_bounds (ε : ℝ) (hε : 0 < ε) :
    Tendsto (fun n => (disorderLaw n).real {W |
      (n : ℝ) ^ (2 / 3 - ε) ≤ continuousMixingTime W}) atTop (𝓝 1) ∧
    Tendsto (fun n => (disorderLaw n).real {W |
      (n : ℝ) ^ (5 / 3 - ε) ≤ (discreteMixingTime W : ℝ)}) atTop (𝓝 1) := by
…
```
Mentions: OAI.CriticalSK.disorderLaw, OAI.CriticalSK.continuousMixingTime, OAI.CriticalSK.discreteMixingTime