`ComparatorChallenges/CubicTorus.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

19 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.CubicTorus.Cover line 11
Also declared: OAI/Geometry/CubicTorus/Model.lean:11 (same declaration text).
abbrev OAI.CubicTorus.Torus line 12
Also declared: OAI/Geometry/CubicTorus/Model.lean:12 (same declaration text).
def OAI.CubicTorus.quotientMap The quotient R³ → R³/Z³ in orthonormal coordinates line 15
Source documentation
The quotient R³ → R³/Z³ in orthonormal coordinates.
L4DEL explanation
The quotient R³ → R³/Z³ in orthonormal coordinates.
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/Geometry/CubicTorus/Model.lean:15 (same declaration text).
```
def quotientMap (x : Cover) : Torus :=
…
```
Mentions: OAI.CubicTorus.Cover, OAI.CubicTorus.Torus
def OAI.CubicTorus.fundamentalCube One unit-volume fundamental domain; its boundary is immaterial to integration line 19
Source documentation
One unit-volume fundamental domain; its boundary is immaterial to integration.
L4DEL explanation
One unit-volume fundamental domain; its boundary is immaterial to integration.
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/Geometry/CubicTorus/Model.lean:19 (same declaration text).
```
def fundamentalCube : Set Cover := {x | ∀ i, 0 ≤ x i ∧ x i < 1}
```
Mentions: OAI.CubicTorus.Cover
def OAI.CubicTorus.torusVolume The Riemannian volume measure of the unit flat torus line 22
Source documentation
The Riemannian volume measure of the unit flat torus.
L4DEL explanation
The Riemannian volume measure of the unit flat torus.
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/Geometry/CubicTorus/Model.lean:22 (same declaration text).
```
def torusVolume : Measure Torus :=
…
```
Mentions: OAI.CubicTorus.Torus
def OAI.CubicTorus.latticeVector An integer lattice vector in the Euclidean cover line 26
Source documentation
An integer lattice vector in the Euclidean cover.
L4DEL explanation
An integer lattice vector in the Euclidean cover.
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/Geometry/CubicTorus/Model.lean:26 (same declaration text).
```
def latticeVector (n : Fin 3 → ℤ) : Cover := WithLp.toLp 2 (fun i => (n i : ℝ))
```
Mentions: OAI.CubicTorus.Cover
def OAI.CubicTorus.AdmissibleField The smooth vector fields on the flat torus with pointwise length at most one, expressed as Z³-periodic fields on its Euclidean cover line 31
Source documentation
The smooth vector fields on the flat torus with pointwise length at most one, expressed as Z³-periodic fields on its Euclidean cover. The explicit ℕ∞ index is C^∞; top of WithTop ℕ∞ would instead mean analytic in this Mathlib.
L4DEL explanation
The smooth vector fields on the flat torus with pointwise length at most one, expressed as Z³-periodic fields on its Euclidean cover. The explicit ℕ∞ index is C^∞; top of WithTop ℕ∞ would instead mean analytic in this Mathlib.
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/Geometry/CubicTorus/Model.lean:31 (same declaration text).
```
def AdmissibleField (F : Cover → Cover) : Prop :=
…
```
Mentions: OAI.CubicTorus.Cover
def OAI.CubicTorus.divergence Euclidean divergence in the orthonormal coordinate frame line 37
Source documentation
Euclidean divergence in the orthonormal coordinate frame.
L4DEL explanation
Euclidean divergence in the orthonormal coordinate frame.
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/Geometry/CubicTorus/Model.lean:37 (same declaration text).
```
def divergence (F : Cover → Cover) (x : Cover) : ℝ :=
…
```
Mentions: OAI.CubicTorus.Cover
def OAI.CubicTorus.perimeter De Giorgi perimeter, allowed to be infinite; no regularity of E is imposed line 41
Source documentation
De Giorgi perimeter, allowed to be infinite; no regularity of E is imposed.
L4DEL explanation
De Giorgi perimeter, allowed to be infinite; no regularity of E is imposed.
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/Geometry/CubicTorus/Model.lean:41 (same declaration text).
```
def perimeter (E : Set Torus) : ℝ≥0∞ :=
…
```
Mentions: OAI.CubicTorus.Torus
def OAI.CubicTorus.FinitePerimeter Finite-perimeter measurable regions, including all null-set modifications line 46
Source documentation
Finite-perimeter measurable regions, including all null-set modifications.
L4DEL explanation
Finite-perimeter measurable regions, including all null-set modifications.
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/Geometry/CubicTorus/Model.lean:46 (same declaration text).
```
def FinitePerimeter (E : Set Torus) : Prop :=
…
```
Mentions: OAI.CubicTorus.Torus
def OAI.CubicTorus.IsMinimizerAt A global perimeter minimizer under the prescribed volume constraint line 50
Source documentation
A global perimeter minimizer under the prescribed volume constraint.
L4DEL explanation
A global perimeter minimizer under the prescribed volume constraint.
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/Geometry/CubicTorus/Model.lean:50 (same declaration text).
```
def IsMinimizerAt (V : ℝ) (E : Set Torus) : Prop :=
…
```
Mentions: OAI.CubicTorus.Torus
def OAI.CubicTorus.candidateProfile The candidate unit-torus isoperimetric profile line 56
Source documentation
The candidate unit-torus isoperimetric profile.
L4DEL explanation
The candidate unit-torus isoperimetric profile.
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/Geometry/CubicTorus/Model.lean:56 (same declaration text).
```
def candidateProfile (V : ℝ) : ℝ :=
…
```
def OAI.CubicTorus.canonicalBall Canonical centered ball line 62
Source documentation
Canonical centered ball. Its radius on the listed branch is below 1/2.
L4DEL explanation
Canonical centered ball. Its radius on the listed branch is below 1/2.
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/Geometry/CubicTorus/Model.lean:62 (same declaration text).
```
def canonicalBall (V : ℝ) : Set Torus :=
…
```
Mentions: OAI.CubicTorus.Torus
def OAI.CubicTorus.canonicalTube Canonical tube around the shortest closed geodesic in coordinate direction 0 line 66
Source documentation
Canonical tube around the shortest closed geodesic in coordinate direction 0.
L4DEL explanation
Canonical tube around the shortest closed geodesic in coordinate direction 0.
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/Geometry/CubicTorus/Model.lean:66 (same declaration text).
```
def canonicalTube (V : ℝ) : Set Torus :=
…
```
Mentions: OAI.CubicTorus.Torus
def OAI.CubicTorus.canonicalSlab Canonical slab of width V between two parallel coordinate tori line 70
Source documentation
Canonical slab of width V between two parallel coordinate tori.
L4DEL explanation
Canonical slab of width V between two parallel coordinate tori.
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/Geometry/CubicTorus/Model.lean:70 (same declaration text).
```
def canonicalSlab (V : ℝ) : Set Torus := {x | ‖x 0‖ < V / 2}
```
Mentions: OAI.CubicTorus.Torus
def OAI.CubicTorus.CongruentAE Equality of regions up to ambient flat isometry and a null set line 73
Source documentation
Equality of regions up to ambient flat isometry and a null set.
L4DEL explanation
Equality of regions up to ambient flat isometry and a null set.
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/Geometry/CubicTorus/Model.lean:73 (same declaration text).
```
def CongruentAE (E F : Set Torus) : Prop :=
…
```
Mentions: OAI.CubicTorus.Torus
def OAI.CubicTorus.SmallStandardRegion Exactly the closed phase ranges, so adjacent types are BOTH included at ties line 77
Source documentation
Exactly the closed phase ranges, so adjacent types are BOTH included at ties.
L4DEL explanation
Exactly the closed phase ranges, so adjacent types are BOTH included at ties.
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/Geometry/CubicTorus/Model.lean:77 (same declaration text).
```
def SmallStandardRegion (V : ℝ) (E : Set Torus) : Prop :=
…
```
Mentions: OAI.CubicTorus.Torus
def OAI.CubicTorus.StandardRegionAt The equality class at every interior volume, including complementation line 83
Source documentation
The equality class at every interior volume, including complementation.
L4DEL explanation
The equality class at every interior volume, including complementation.
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/Geometry/CubicTorus/Model.lean:83 (same declaration text).
```
def StandardRegionAt (V : ℝ) (E : Set Torus) : Prop :=
…
```
Mentions: OAI.CubicTorus.Torus
theorem OAI.CubicTorus.unit_cubic_isoperimetric Challenge statement: The isoperimetric profile of the cubic three-torus sorry line 88
Source documentation
Attainment, the sharp profile, and all equality cases at interior volumes.
L4DEL explanation
The formalization determines the isoperimetric profile and all minimizers in the unit cubic flat three-torus. At every volume $0<V<1$, a minimizer exists, and the minimizers, up to null-set changes, are exactly balls, circular tubes about shortest closed geodesics, coordinate slabs, and their complements in the appropriate volume ranges. The classification includes all equality cases at the transition volumes $4\pi/81$ and $1/\pi$, and every minimizer has the stated candidate-profile perimeter.
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.Geometry.CubicTorus.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Geometry/CubicTorus/Main.lean:1103 (same statement text; different proof or body). Qualify references as OAI.CubicTorus.unit_cubic_isoperimetric@path.
```
theorem unit_cubic_isoperimetric (V : ℝ) (hV : V ∈ Ioo 0 1) :
    (∃ E : Set Torus, IsMinimizerAt V E) ∧
    (∀ E : Set Torus, IsMinimizerAt V E →
      perimeter E = ENNReal.ofReal (candidateProfile V)) ∧
    (∀ E : Set Torus, IsMinimizerAt V E ↔ StandardRegionAt V E) := by
…
```
Mentions: OAI.CubicTorus.Torus, OAI.CubicTorus.IsMinimizerAt, OAI.CubicTorus.perimeter, OAI.CubicTorus.candidateProfile, OAI.CubicTorus.StandardRegionAt