`ComparatorChallenges/EinsteinFour.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

35 declarations, 3 contain sorry, 0 declared as axioms

abbrev OAI.EinsteinFour.FourSpace line 12
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:14 (same declaration text).
abbrev OAI.EinsteinFour.I4 line 13
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:15 (same declaration text).
def OAI.EinsteinFour.coordinateVector line 15
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:18 (same declaration text).
abbrev OAI.EinsteinFour.RoundSphere4 line 17
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:21 (same declaration text).
abbrev OAI.EinsteinFour.ComplexProjective2 line 19
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:24 (same declaration text).
abbrev OAI.EinsteinFour.RealProjective4 line 21
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:27 (same declaration text).
def OAI.EinsteinFour.roundDistance line 23
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:30 (same declaration text).
def OAI.EinsteinFour.fubiniStudyDistance line 26
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:35 (same declaration text).
def OAI.EinsteinFour.realProjectiveDistance line 29
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:40 (same declaration text).
abbrev OAI.EinsteinFour.SmoothMetric line 37
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:49 (same declaration text).
def OAI.EinsteinFour.metricInChart line 42
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:56 (same declaration text).
def OAI.EinsteinFour.partialDeriv line 48
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:63 (same declaration text).
def OAI.EinsteinFour.christoffel line 51
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:68 (same declaration text).
def OAI.EinsteinFour.curvatureComponent line 58
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:77 (same declaration text).
def OAI.EinsteinFour.ricciInChart line 66
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:86 (same declaration text).
def OAI.EinsteinFour.sectionalNumerator line 70
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:92 (same declaration text).
def OAI.EinsteinFour.IsEinstein line 76
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:99 (same declaration text).
def OAI.EinsteinFour.HasPositiveSectionalCurvature line 81
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:105 (same declaration text).
def OAI.EinsteinFour.scaledEDistance line 85
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:111 (same declaration text).
def OAI.EinsteinFour.ScaledIsometricTo line 89
Also declared: OAI/Geometry/Manifold/ProjectiveSphere.lean:117 (same declaration text).
abbrev OAI.PositiveEinsteinFour.Geometry.Mat4 line 107
Also declared: OAI/Geometry/EinsteinFour/WeylGeometry.lean:20 (same declaration text).
def OAI.PositiveEinsteinFour.Geometry.chartMetric line 109
Also declared: OAI/Geometry/EinsteinFour/WeylGeometry.lean:23 (same declaration text).
structure OAI.PositiveEinsteinFour.Geometry.OrthonormalFrame line 112
Also declared: OAI/Geometry/EinsteinFour/WeylGeometry.lean:31 (same declaration text).
def OAI.PositiveEinsteinFour.Geometry.OrthonormalFrame.map line 116
Distinct definitions of this name: OAI/Geometry/EinsteinFour/WeylGeometry.lean:46 (different declaration text). Qualify references as OAI.PositiveEinsteinFour.Geometry.OrthonormalFrame.map@path.
def OAI.PositiveEinsteinFour.Geometry.OrthonormalFrame.vector line 121
Distinct definitions of this name: OAI/Geometry/EinsteinFour/WeylGeometry.lean:59 (different declaration text). Qualify references as OAI.PositiveEinsteinFour.Geometry.OrthonormalFrame.vector@path.
structure OAI.PositiveEinsteinFour.Geometry.FrameBundle line 125
Also declared: OAI/Geometry/EinsteinFour/FrameBundle.lean:19 (same declaration text).
def OAI.PositiveEinsteinFour.Geometry.frameRealization line 129
Distinct definitions of this name: OAI/Geometry/EinsteinFour/FrameBundle.lean:23 (same statement text; different proof or body). Qualify references as OAI.PositiveEinsteinFour.Geometry.frameRealization@path.
instance OAI.PositiveEinsteinFour.Geometry.(instance at line 135) line 135
structure OAI.PositiveEinsteinFour.Geometry.FrameOrientation An orientation is a clopen choice of determinant sign in every frame fiber line 139
Source documentation
An orientation is a clopen choice of determinant sign in every frame fiber.
L4DEL explanation
An orientation is a clopen choice of determinant sign in every frame fiber.
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/EinsteinFour/FrameOrientation.lean:20 (same declaration text).
```
structure FrameOrientation (g : SmoothMetric M) where
…
```
structure OAI.PositiveEinsteinFour.ClosedOrientedMetricFour A connected closed oriented smooth Riemannian four-manifold line 156
Source documentation
A connected closed oriented smooth Riemannian four-manifold.
L4DEL explanation
A connected closed oriented smooth Riemannian four-manifold.
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)
Distinct definitions of this name: OAI/Geometry/EinsteinFour/ConditionalUnoriented.lean:23 (same statement text; different proof or body). Qualify references as OAI.PositiveEinsteinFour.ClosedOrientedMetricFour@path.
```
structure ClosedOrientedMetricFour where
…
```
def OAI.PositiveEinsteinFour.Classified Positive rescaling identifies the metric with one of the three standard models line 175
Source documentation
Positive rescaling identifies the metric with one of the three standard models.
L4DEL explanation
Positive rescaling identifies the metric with one of the three standard models.
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/EinsteinFour/Statement.lean:22 (same declaration text).
```
def Classified (g : SmoothMetric M) : Prop :=
…
```
def OAI.PositiveEinsteinFour.ClassificationOn line 181
Also declared: OAI/Geometry/EinsteinFour/Statement.lean:30 (same declaration text).
theorem OAI.PositiveEinsteinFour.orientedClassification Challenge statement: Nonnegative-curvature Einstein classification and an L2 topological gap sorry line 190
Source documentation
The oriented classification has the round sphere and Fubini--Study models.
L4DEL explanation
The formalization classifies connected smooth closed Einstein four-manifolds with strictly positive sectional curvature. Up to positive scaling and isometry, every such manifold is the round four-sphere, the complex projective plane with its Fubini–Study metric, or round real projective four-space. No orientability hypothesis is required. In the oriented case, only the sphere and complex projective plane occur.
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.EinsteinFour.Classification as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Geometry/EinsteinFour/ActualModelIdentification.lean:63 (same statement text; different proof or body). Qualify references as OAI.PositiveEinsteinFour.orientedClassification@path.
```
theorem orientedClassification (X : ClosedOrientedMetricFour.{u})
    (hEinstein : IsEinstein X.metric) (hpositive : HasPositiveSectionalCurvature X.metric) :
    ∃ a : ℝ, 0 < a ∧ (ScaledIsometricTo X.metric a roundDistance ∨
      ScaledIsometricTo X.metric a fubiniStudyDistance) := by
…
```
theorem OAI.PositiveEinsteinFour.classificationOn Challenge statement: Nonnegative-curvature Einstein classification and an L2 topological gap sorry line 202
L4DEL explanation
The formalization classifies connected smooth closed Einstein four-manifolds with strictly positive sectional curvature. Up to positive scaling and isometry, every such manifold is the round four-sphere, the complex projective plane with its Fubini–Study metric, or round real projective four-space. No orientability hypothesis is required. In the oriented case, only the sphere and complex projective plane occur.
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.EinsteinFour.Classification as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Geometry/EinsteinFour/ActualModelIdentification.lean:78 (same statement text; different proof or body). Qualify references as OAI.PositiveEinsteinFour.classificationOn@path.
```
theorem classificationOn : ClassificationOn M := by
…
```
Mentions: OAI.PositiveEinsteinFour.ClassificationOn
theorem OAI.PositiveEinsteinFour.classification Challenge statement: Nonnegative-curvature Einstein classification and an L2 topological gap sorry line 208
Source documentation
The connected closed classification includes the real projective quotient.
L4DEL explanation
The formalization classifies connected smooth closed Einstein four-manifolds with strictly positive sectional curvature. Up to positive scaling and isometry, every such manifold is the round four-sphere, the complex projective plane with its Fubini–Study metric, or round real projective four-space. No orientability hypothesis is required. In the oriented case, only the sphere and complex projective plane occur.
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.EinsteinFour.Classification as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Geometry/EinsteinFour/Classification.lean:13 (same statement text; different proof or body). Qualify references as OAI.PositiveEinsteinFour.classification@path.
```
theorem classification (M : Type*) [TopologicalSpace M] [ChartedSpace FourSpace M]
    [IsManifold I4 ∞ M] [T2Space M] [SecondCountableTopology M]
    [CompactSpace M] [ConnectedSpace M] (g : SmoothMetric M)
    (hEinstein : IsEinstein g) (hpositive : HasPositiveSectionalCurvature g) :
    ∃ a : ℝ, 0 < a ∧
      (ScaledIsometricTo g a roundDistance ∨
        ScaledIsometricTo g a fubiniStudyDistance ∨
        ScaledIsometricTo g a realProjectiveDistance) := by
…
```