`ComparatorChallenges/DaugavetModuli.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, 3 contain sorry, 0 declared as axioms

def OAI.ExactModuli.Cofinite Closed linear subspaces of finite codimension line 17
Source documentation
Closed linear subspaces of finite codimension.
L4DEL explanation
Closed linear subspaces of finite codimension.
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/Analysis/Daugavet/Model.lean:16 (same declaration text).
```
def Cofinite (F : Submodule ℝ X) : Prop :=
…
```
def OAI.ExactModuli.averagedIncrement Averaged increment for directions of norm at least one line 21
Source documentation
Averaged increment for directions of norm at least one.
L4DEL explanation
Averaged increment for directions of norm at least one.
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/Analysis/Daugavet/Model.lean:20 (same declaration text).
```
def averagedIncrement (N : Seminorm ℝ X) (x : X) (t : ℝ) (y : X) : ℝ :=
…
```
def OAI.ExactModuli.H line 24
Also declared: OAI/Analysis/Daugavet/Model.lean:23 (same declaration text).
def OAI.ExactModuli.D line 28
Also declared: OAI/Analysis/Daugavet/Model.lean:27 (same declaration text).
def OAI.ExactModuli.averagedModulus line 32
Also declared: OAI/Analysis/Daugavet/Model.lean:31 (same declaration text).
def OAI.ExactModuli.oneSidedModulus line 35
Also declared: OAI/Analysis/Daugavet/Model.lean:34 (same declaration text).
def OAI.ExactModuli.EquivalentNorm Two-sided comparison makes this seminorm a genuine norm equivalent to the original line 39
Source documentation
Two-sided comparison makes this seminorm a genuine norm equivalent to the original.
L4DEL explanation
Two-sided comparison makes this seminorm a genuine norm equivalent to the original.
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/Analysis/Daugavet/Model.lean:38 (same declaration text).
```
def EquivalentNorm (N : Seminorm ℝ X) : Prop :=
…
```
def OAI.ExactModuli.AUC line 42
Also declared: OAI/Analysis/Daugavet/Model.lean:41 (same declaration text).
def OAI.ExactModuli.Daugavet Every rank-one operator has the form ell.smulRight v line 46
Source documentation
Every rank-one operator has the form ell.smulRight v.
L4DEL explanation
Every rank-one operator has the form ell.smulRight v.
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/Analysis/Daugavet/Model.lean:45 (same declaration text).
```
def Daugavet (X : Type uX) [NormedAddCommGroup X] [NormedSpace ℝ X] : Prop :=
…
```
def OAI.ExactModuli.measureDistance Convergence-in-measure distance on L1 equivalence classes line 53
Source documentation
Convergence-in-measure distance on L1 equivalence classes.
L4DEL explanation
Convergence-in-measure distance on L1 equivalence classes.
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/Analysis/Daugavet/Model.lean:52 (same declaration text).
```
def measureDistance (f g : Lp ℝ 1 μ) : ℝ :=
…
```
def OAI.ExactModuli.MeasurePrecompactBall Finite covers of the entire unit ball in convergence-in-measure distance line 57
Source documentation
Finite covers of the entire unit ball in convergence-in-measure distance.
L4DEL explanation
Finite covers of the entire unit ball in convergence-in-measure distance.
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/Analysis/Daugavet/Model.lean:56 (same declaration text).
```
def MeasurePrecompactBall (E : Submodule ℝ (Lp ℝ 1 μ)) : Prop :=
…
```
def OAI.ExactModuli.UnitSliceProperty Every nonempty unit-ball slice contains an almost antipodal unit vector line 62
Source documentation
Every nonempty unit-ball slice contains an almost antipodal unit vector.
L4DEL explanation
Every nonempty unit-ball slice contains an almost antipodal unit vector.
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/Analysis/Daugavet/Model.lean:61 (same declaration text).
```
def UnitSliceProperty (X : Type uX) [NormedAddCommGroup X] [NormedSpace ℝ X] : Prop :=
…
```
abbrev OAI.ExactModuli.KWOmega The countable product of unit intervals line 68
Source documentation
The countable product of unit intervals.
L4DEL explanation
The countable product of unit intervals.
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/Analysis/Daugavet/Model.lean:67 (same declaration text).
```
abbrev KWOmega := ℕ → unitInterval
```
def OAI.ExactModuli.kwMeasure line 70
Also declared: OAI/Analysis/Daugavet/Model.lean:69 (same declaration text).
theorem OAI.ExactModuli.exact_main Challenge statement: Exact asymptotic moduli and a Daugavet example sorry line 72
L4DEL explanation
This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content.
1. The linked formalization note covers 7 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.Analysis.Daugavet.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/Daugavet/ExactModuli.lean:42 (same statement text; different proof or body). Qualify references as OAI.ExactModuli.exact_main@path.
```
theorem exact_main {Ω : Type u_1} [MeasurableSpace Ω] (μ : Measure Ω)
    [IsProbabilityMeasure μ] (E : Submodule ℝ (Lp ℝ 1 μ))
    (_hclosed : IsClosed (E : Set (Lp ℝ 1 μ))) (hdim : ¬ FiniteDimensional ℝ E)
    (hprecompact : MeasurePrecompactBall μ E) (hdaugavet : Daugavet E) :
    (∀ (x : E), ‖x‖ = 1 → ∀ t : ℝ, 0 < t →
      H (normSeminorm ℝ E) x t = max (t / 2) (t - 1) ∧
      D (normSeminorm ℝ E) x t = max 0 (t - 2)) ∧
    (∀ t : ℝ, 0 < t →
      averagedModulus (normSeminorm ℝ E) t = max (t / 2) (t - 1) ∧
      oneSidedModulus (normSeminorm ℝ E) t = max 0 (t - 2)) ∧
    (∀ N : Seminorm ℝ E, EquivalentNorm N → ¬ AUC N) := by
…
```
Mentions: OAI.ExactModuli.MeasurePrecompactBall, OAI.ExactModuli.Daugavet, OAI.ExactModuli.H, OAI.ExactModuli.D, OAI.ExactModuli.averagedModulus, OAI.ExactModuli.oneSidedModulus, OAI.ExactModuli.EquivalentNorm, OAI.ExactModuli.AUC
theorem OAI.ExactModuli.kadets_werner_existence Challenge statement: Exact asymptotic moduli and a Daugavet example sorry line 85
L4DEL explanation
This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content.
1. The linked formalization note covers 7 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.Analysis.Daugavet.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/Daugavet/Main.lean:11 (same statement text; different proof or body). Qualify references as OAI.ExactModuli.kadets_werner_existence@path.
```
theorem kadets_werner_existence :
    ∃ E : Submodule ℝ (Lp ℝ 1 kwMeasure),
      IsClosed (E : Set (Lp ℝ 1 kwMeasure)) ∧ (¬ FiniteDimensional ℝ E) ∧
      MeasurePrecompactBall kwMeasure E ∧ UnitSliceProperty E ∧ Daugavet E := by
…
```
Mentions: OAI.ExactModuli.kwMeasure, OAI.ExactModuli.MeasurePrecompactBall, OAI.ExactModuli.UnitSliceProperty, OAI.ExactModuli.Daugavet
theorem OAI.ExactModuli.exact_example Challenge statement: Exact asymptotic moduli and a Daugavet example sorry line 91
L4DEL explanation
This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content.
1. The linked formalization note covers 7 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.Analysis.Daugavet.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/Daugavet/Main.lean:22 (same statement text; different proof or body). Qualify references as OAI.ExactModuli.exact_example@path.
```
theorem exact_example :
    ∃ E : Submodule ℝ (Lp ℝ 1 kwMeasure),
      IsClosed (E : Set (Lp ℝ 1 kwMeasure)) ∧ (¬ FiniteDimensional ℝ E) ∧
      MeasurePrecompactBall kwMeasure E ∧ UnitSliceProperty E ∧ Daugavet E ∧
      (∀ (x : E), ‖x‖ = 1 → ∀ t : ℝ, 0 < t →
        H (normSeminorm ℝ E) x t = max (t / 2) (t - 1) ∧
        D (normSeminorm ℝ E) x t = max 0 (t - 2)) ∧
      (∀ t : ℝ, 0 < t →
        averagedModulus (normSeminorm ℝ E) t = max (t / 2) (t - 1) ∧
        oneSidedModulus (normSeminorm ℝ E) t = max 0 (t - 2)) ∧
      (∀ N : Seminorm ℝ E, EquivalentNorm N → ¬ AUC N) := by
…
```
Mentions: OAI.ExactModuli.kwMeasure, OAI.ExactModuli.MeasurePrecompactBall, OAI.ExactModuli.UnitSliceProperty, OAI.ExactModuli.Daugavet, OAI.ExactModuli.H, OAI.ExactModuli.D, OAI.ExactModuli.averagedModulus, OAI.ExactModuli.oneSidedModulus, OAI.ExactModuli.EquivalentNorm, OAI.ExactModuli.AUC