`ComparatorChallenges/ArnoldCounterexample.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

20 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.ArnoldCounterexample.CV line 10
Also declared: OAI/Geometry/Arnold/Model.lean:10 (same declaration text).
def OAI.ArnoldCounterexample.quadricPolynomial line 12
Also declared: OAI/Geometry/Arnold/Model.lean:12 (same declaration text).
def OAI.ArnoldCounterexample.normSquared line 14
Also declared: OAI/Geometry/Arnold/Model.lean:14 (same declaration text).
def OAI.ArnoldCounterexample.IsUnitQuadric line 16
Also declared: OAI/Geometry/Arnold/Model.lean:16 (same declaration text).
abbrev OAI.ArnoldCounterexample.UnitQuadric line 19
Also declared: OAI/Geometry/Arnold/Model.lean:19 (same declaration text).
def OAI.ArnoldCounterexample.Q3 The literal projective quadric sum z_j^2 = 0 in CP^4 line 22
Source documentation
The literal projective quadric sum z_j^2 = 0 in CP^4.
L4DEL explanation
The literal projective quadric sum z_j^2 = 0 in CP^4.
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/Arnold/Model.lean:27 (same declaration text).
```
def Q3 := {q : Projectivization ℂ CV //
…
```
Mentions: OAI.ArnoldCounterexample.CV
def OAI.ArnoldCounterexample.project line 26
Also declared: OAI/Geometry/Arnold/Model.lean:31 (same declaration text).
def OAI.ArnoldCounterexample.Horizontal Horizontal lifts of the real tangent of Q3 at [z] line 34
Source documentation
Horizontal lifts of the real tangent of Q3 at [z].
L4DEL explanation
Horizontal lifts of the real tangent of Q3 at [z].
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/Arnold/Model.lean:39 (same declaration text).
```
def Horizontal (z v : CV) : Prop :=
…
```
Mentions: OAI.ArnoldCounterexample.CV
def OAI.ArnoldCounterexample.omega The restricted Fubini--Study form, evaluated on horizontal lifts line 38
Source documentation
The restricted Fubini--Study form, evaluated on horizontal lifts.
L4DEL explanation
The restricted Fubini--Study form, evaluated on horizontal lifts.
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/Arnold/Model.lean:43 (same declaration text).
```
def omega (v w : CV) : ℝ := 2 * (∑ j, star (v j) * w j).im
```
Mentions: OAI.ArnoldCounterexample.CV
def OAI.ArnoldCounterexample.LocalFunctionLift A local smooth extension of the pullback of h to the unit isotropic cone line 41
Source documentation
A local smooth extension of the pullback of h to the unit isotropic cone.
L4DEL explanation
A local smooth extension of the pullback of h to the unit isotropic cone.
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/Arnold/Model.lean:46 (same declaration text).
```
def LocalFunctionLift (h : Q3 → ℝ) (z : UnitQuadric)
    (U : Set CV) (g : CV → ℝ) : Prop :=
…
```
Mentions: OAI.ArnoldCounterexample.Q3, OAI.ArnoldCounterexample.UnitQuadric, OAI.ArnoldCounterexample.CV
def OAI.ArnoldCounterexample.IsSmoothFunction line 46
Also declared: OAI/Geometry/Arnold/Model.lean:51 (same declaration text).
structure OAI.ArnoldCounterexample.SmoothFunction line 49
Also declared: OAI/Geometry/Arnold/Model.lean:54 (same declaration text).
def OAI.ArnoldCounterexample.IsCritical dh_q = 0, expressed using all local lifts, independent of normal extensions line 54
Source documentation
dh_q = 0, expressed using all local lifts, independent of normal extensions.
L4DEL explanation
dh_q = 0, expressed using all local lifts, independent of normal extensions.
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/Arnold/Model.lean:59 (same declaration text).
```
def IsCritical (h : Q3 → ℝ) (q : Q3) : Prop :=
…
```
Mentions: OAI.ArnoldCounterexample.Q3
def OAI.ArnoldCounterexample.criticalSet line 59
Also declared: OAI/Geometry/Arnold/Model.lean:64 (same declaration text).
def OAI.ArnoldCounterexample.criticalNumber The minimum critical number, including degenerate functions (ENat allows infinity) line 62
Source documentation
The minimum critical number, including degenerate functions (ENat allows infinity).
L4DEL explanation
The minimum critical number, including degenerate functions (ENat allows infinity).
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/Arnold/Model.lean:67 (same declaration text).
```
def criticalNumber : ℕ∞ := ⨅ h : SmoothFunction, (criticalSet h).encard
```
Mentions: OAI.ArnoldCounterexample.SmoothFunction, OAI.ArnoldCounterexample.criticalSet
structure OAI.ArnoldCounterexample.HamiltonianPath An explicit smooth lift of a Hamiltonian isotopy, with inverse and generator line 66
Source documentation
An explicit smooth lift of a Hamiltonian isotopy, with inverse and generator. The Hamiltonian equation uses i_X omega = dH.
L4DEL explanation
An explicit smooth lift of a Hamiltonian isotopy, with inverse and generator. The Hamiltonian equation uses i_X omega = dH.
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/Arnold/Model.lean:71 (same declaration text).
```
structure HamiltonianPath (φ : Q3 ≃ Q3) where
…
```
Mentions: OAI.ArnoldCounterexample.Q3
def OAI.ArnoldCounterexample.IsHamiltonian line 95
Also declared: OAI/Geometry/Arnold/Model.lean:100 (same declaration text).
def OAI.ArnoldCounterexample.fixedSet line 97
Also declared: OAI/Geometry/Arnold/Model.lean:102 (same declaration text).
def OAI.ArnoldCounterexample.HasDegenerateFixedPoint Eigenvalue 1 in the derivative on the quotient tangent, not an ambient spurious vertical eigenvector line 101
Source documentation
Eigenvalue 1 in the derivative on the quotient tangent, not an ambient spurious vertical eigenvector. v is nonzero and horizontal.
L4DEL explanation
Eigenvalue 1 in the derivative on the quotient tangent, not an ambient spurious vertical eigenvector. v is nonzero and horizontal.
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/Arnold/Model.lean:106 (same declaration text).
```
def HasDegenerateFixedPoint (φ : Q3 ≃ Q3) : Prop :=
…
```
Mentions: OAI.ArnoldCounterexample.Q3
theorem OAI.ArnoldCounterexample.main Challenge statement: Counterexamples to stable-Morse and strong Arnold fixed-point bounds sorry line 111
Source documentation
The Hamiltonian counterexample, with the complete fixed set, the lower bound for all smooth functions, and a genuine nonzero quotient-tangent eigenvector.
L4DEL explanation
The critical-number form of Arnold's fixed-point conjecture predicts at least as many fixed points as the minimum number of critical points of a smooth function. The formalized counterexample is a Hamiltonian diffeomorphism of the complex quadric threefold with exactly three fixed points, while every smooth function on that manifold has at least four critical points. At least one fixed point is degenerate. The separate nondegenerate Morse-number construction is not included.
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.Arnold.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Geometry/Arnold/Main.lean:14 (same statement text; different proof or body). Qualify references as OAI.ArnoldCounterexample.main@path.
```
theorem main :
    ∃ φ : Q3 ≃ Q3, IsHamiltonian φ ∧ (fixedSet φ).encard = 3 ∧
      (3 : ℕ∞) < 4 ∧ 4 ≤ criticalNumber ∧ HasDegenerateFixedPoint φ := by
…
```
Mentions: OAI.ArnoldCounterexample.Q3, OAI.ArnoldCounterexample.IsHamiltonian, OAI.ArnoldCounterexample.fixedSet, OAI.ArnoldCounterexample.criticalNumber, OAI.ArnoldCounterexample.HasDegenerateFixedPoint