`ComparatorChallenges/ArtinCAT0.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.

CAT: Cartan-Alexandrov-Toponogov
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

14 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.ArtinCAT0.Letters line 9
Also declared: OAI/GroupTheory/ArtinCAT0/Model.lean:9 (same declaration text).
abbrev OAI.ArtinCAT0.Label line 10
Also declared: OAI/GroupTheory/ArtinCAT0/Model.lean:10 (same declaration text).
def OAI.ArtinCAT0.blockLetter line 12
Also declared: OAI/GroupTheory/ArtinCAT0/Model.lean:12 (same declaration text).
def OAI.ArtinCAT0.Adjacent line 17
Also declared: OAI/GroupTheory/ArtinCAT0/Model.lean:17 (same declaration text).
def OAI.ArtinCAT0.Distant line 21
Also declared: OAI/GroupTheory/ArtinCAT0/Model.lean:21 (same declaration text).
def OAI.ArtinCAT0.Sentinel line 25
Also declared: OAI/GroupTheory/ArtinCAT0/Model.lean:25 (same declaration text).
def OAI.ArtinCAT0.explicitMatrix line 29
Also declared: OAI/GroupTheory/ArtinCAT0/Model.lean:29 (same declaration text).
def OAI.ArtinCAT0.alternating line 34
Also declared: OAI/GroupTheory/ArtinCAT0/Model.lean:35 (same declaration text).
def OAI.ArtinCAT0.artinRelators line 38
Also declared: OAI/GroupTheory/ArtinCAT0/Model.lean:40 (same declaration text).
abbrev OAI.ArtinCAT0.ArtinGroup line 42
Also declared: OAI/GroupTheory/ArtinCAT0/Model.lean:44 (same declaration text).
def OAI.ArtinCAT0.Segment line 44
Also declared: OAI/GroupTheory/ArtinCAT0/Model.lean:48 (same declaration text).
def OAI.ArtinCAT0.CAT0 line 49
Also declared: OAI/GroupTheory/ArtinCAT0/Model.lean:58 (same declaration text).
def OAI.ArtinCAT0.GeometricAction line 57
Distinct definitions of this name: OAI/GroupTheory/ArtinCAT0/Model.lean:68 (same statement text; different proof or body). Qualify references as OAI.ArtinCAT0.GeometricAction@path.
theorem OAI.ArtinCAT0.main Challenge statement: An explicit Artin group with no geometric $\mathrm{CAT}(0)$ action sorry line 65
L4DEL explanation
We construct an Artin group on 116 generators that admits no proper, cocompact isometric action on a nonempty proper CAT(0) space.
— OpenAI Artin-CAT(0) manuscript (abstract)
One question which is still open for both spherical and non-spherical Artin groups is whether they are CAT(0) groups
— Ruth Charney, An introduction to right-angled Artin groups (2007)
An explicitly presented Artin group on 116 generators cannot act geometrically on any nonempty proper $\mathrm{CAT}(0)$ metric space, in any dimension.
1. explicitMatrix is proved symmetric, has diagonal entries one, and has every off-diagonal label in $\{2,3,\infty\}$. Finite labels impose alternating Artin relations; the label $\infty$ imposes no relation.
2. GeometricAction combines four requirements: the metric space satisfies the stated $\mathrm{CAT}(0)$ triangle inequality, every group element acts isometrically, the action is proper, and one compact set has translates covering the space.
3. The final conjunct quantifies over every metric space, proper-space structure, nonempty witness, and action of this exact Artin group. It therefore refutes the group-level CAT(0) conjecture, not merely one proposed metric on one standard complex.
4. Ruth Charney's survey recorded the proper-cocompact-isometric-action question as open for spherical and non-spherical Artin groups. The Lean target gives one explicit counterexample rather than classifying all Artin groups.
5. Trust boundary: this comparator declaration ends in by sorry. The pinned comparator configuration pairs it with OAI.GroupTheory.ArtinCAT0.Main; the site's Lean trust result, not this annotation, determines whether that separate solution module checks.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI Artin-CAT(0) manuscript (abstract), Pinned OpenAI formalization note 254, Ruth Charney, An introduction to right-angled Artin groups (2007)
Distinct definitions of this name: OAI/GroupTheory/ArtinCAT0/Main.lean:53 (same statement text; different proof or body). Qualify references as OAI.ArtinCAT0.main@path.

Artin group: a group presented by generators whose pairwise relations equate alternating words of lengths prescribed by a Coxeter matrix
geometric CAT(0) action: a proper and cocompact action by isometries on a nonempty proper CAT(0) metric space
```
theorem main :
    (∀ s t : Letters, explicitMatrix s t = explicitMatrix t s) ∧
    (∀ s : Letters, explicitMatrix s s = 1) ∧
    (∀ s t : Letters, s ≠ t →
      explicitMatrix s t = 2 ∨ explicitMatrix s t = 3 ∨ explicitMatrix s t = ⊤) ∧
    (∀ (X : Type u) [MetricSpace X] [ProperSpace X] [Nonempty X]
      [MulAction (ArtinGroup explicitMatrix) X],
      ¬ GeometricAction (ArtinGroup explicitMatrix) X) := by
…
```
Mentions: OAI.ArtinCAT0.Letters, OAI.ArtinCAT0.explicitMatrix, OAI.ArtinCAT0.ArtinGroup