`ComparatorChallenges/ThreeTorus.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

13 declarations, 1 contain sorry, 0 declared as axioms

The torus is (ℝ/ℤ)³ with normalized product Haar measure. Smoothness means local smooth lifts on the universal cover, for both the map and its inverse.

The almost-everywhere graph conditions are independent of representatives of L² classes. The real generator's bilateral Koopman orbit spans the entire complex mean-zero L² space. The spectral model is the full circle L² space with multiplication by the first Fourier character.

abbrev OAI.ThreeTorus.Torus line 21
Also declared: OAI/Dynamics/ThreeTorus/Basic.lean:14 (same declaration text).
abbrev OAI.ThreeTorus.Cover line 22
Also declared: OAI/Dynamics/ThreeTorus/Basic.lean:15 (same declaration text).
def OAI.ThreeTorus.project line 24
Also declared: OAI/Dynamics/ThreeTorus/Basic.lean:17 (same declaration text).
def OAI.ThreeTorus.Smooth line 26
Also declared: OAI/Dynamics/ThreeTorus/Basic.lean:19 (same declaration text).
def OAI.ThreeTorus.μ line 31
Also declared: OAI/Dynamics/ThreeTorus/Basic.lean:24 (same declaration text).
instance OAI.ThreeTorus.(instance at line 33) line 33
abbrev OAI.ThreeTorus.H line 35
Also declared: OAI/Dynamics/ThreeTorus/Basic.lean:28 (same declaration text).
def OAI.ThreeTorus.meanZero The full complex subspace defined by zero Lebesgue integral line 38
Source documentation
The full complex subspace defined by zero Lebesgue integral.
L4DEL explanation
The full complex subspace defined by zero Lebesgue integral.
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/Dynamics/ThreeTorus/Basic.lean:31 (same declaration text).
```
def meanZero : Submodule ℂ H where
…
```
Mentions: OAI.ThreeTorus.H
abbrev OAI.ThreeTorus.H₀ line 61
Also declared: OAI/Dynamics/ThreeTorus/Basic.lean:54 (same declaration text).
abbrev OAI.ThreeTorus.CircleH line 62
Also declared: OAI/Dynamics/ThreeTorus/Basic.lean:55 (same declaration text).
def OAI.ThreeTorus.IsReal line 64
Also declared: OAI/Dynamics/ThreeTorus/Basic.lean:57 (same declaration text).
def OAI.ThreeTorus.MainConclusion Complete simple Lebesgue spectrum, including the stated consequences line 67
Source documentation
Complete simple Lebesgue spectrum, including the stated consequences.
L4DEL explanation
Complete simple Lebesgue spectrum, including the stated consequences.
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/Dynamics/ThreeTorus/Basic.lean:60 (same declaration text).
```
def MainConclusion : Prop :=
…
```
theorem OAI.ThreeTorus.simple_lebesgue_spectrum Challenge statement: A smooth probability-preserving system with simple Lebesgue spectrum sorry line 82
L4DEL explanation
a C∞ diffeomorphism of the three-torus that preserves standard volume and whose Koopman operator has simple Lebesgue spectrum
— OpenAI simple-Lebesgue-spectrum manuscript (abstract)
there is a conservative, ergodic, σ-finite measure preserving transformation with simple Lebesgue spectrum
— el Houcein el Abdalaoui (2015), arXiv v1 abstract
There is a smooth volume-preserving diffeomorphism of the three-torus whose Koopman operator has simple Lebesgue spectrum on the entire complex mean-zero L² space. One real-valued vector generates a bilateral orthonormal basis under all integer iterates.
1. The probability space is the three-torus (R/Z)³ with normalized product Haar measure. H₀ is the complex L² subspace of functions with integral zero, removing the unavoidable invariant constants.
2. MainConclusion supplies a torus permutation T whose map and inverse are locally C∞ lifts, which preserves volume, together with a real-valued f and a sequence v(n)=f∘Tⁿ forming a dense orthonormal spanning family.
3. The same conclusion states ergodicity and gives a complex-linear isometric equivalence W to circle L² under which composition by T becomes multiplication by the first Fourier character. This is the precise simple-Lebesgue spectral model.
4. Earlier work cited here includes an infinite-measure solution; this target is the stronger smooth compact probability-preserving form and should not be conflated with it. The comparator body is sorry, paired with OAI.Dynamics.ThreeTorus.Main.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI simple-Lebesgue-spectrum manuscript (abstract), Pinned OpenAI formalization note 144, el Houcein el Abdalaoui (2015), arXiv v1 abstract
Distinct definitions of this name: OAI/Dynamics/ThreeTorus/Main.lean:17 (same statement text; different proof or body). Qualify references as OAI.ThreeTorus.simple_lebesgue_spectrum@path.

Koopman operator: the operator on functions given here by composition with the measure-preserving transformation
simple Lebesgue spectrum: unitary equivalence to one copy of multiplication by the circle coordinate, rather than multiple spectral copies
```
theorem simple_lebesgue_spectrum : MainConclusion := by
…
```
Mentions: OAI.ThreeTorus.MainConclusion