`ComparatorChallenges/RelativeGeneration.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

12 declarations, 1 contain sorry, 0 declared as axioms

def OAI.RelativeGeneration.WOTClosed Weak operator closedness of a concrete unital star subalgebra line 15
Source documentation
Weak operator closedness of a concrete unital star subalgebra.
L4DEL explanation
Weak operator closedness of a concrete unital star subalgebra.
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/Analysis/RelativeGeneration/Topology.lean:15 (same declaration text).
```
def WOTClosed (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) : Prop :=
…
```
def OAI.RelativeGeneration.wstar The weakly closed unital star algebra generated by a set of operators line 19
Source documentation
The weakly closed unital star algebra generated by a set of operators.
L4DEL explanation
The weakly closed unital star algebra generated by a set of operators.
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/Analysis/RelativeGeneration/Topology.lean:19 (same declaration text).
```
def wstar (operators : Set (Hilbert →L[ℂ] Hilbert)) :
    StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert) :=
…
```
structure OAI.RelativeGeneration.IsII1Factor A nonzero, weakly closed, finite diffuse algebra with scalar center line 24
Source documentation
A nonzero, weakly closed, finite diffuse algebra with scalar center.
L4DEL explanation
A nonzero, weakly closed, finite diffuse algebra with scalar center.
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/Analysis/RelativeGeneration/Topology.lean:24 (same declaration text).
```
structure IsII1Factor (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) : Prop where
…
```
def OAI.RelativeGeneration.HasSeparablePredual A same-universe separable Banach predual, with the conjugate-linear dual convention line 37
Source documentation
A same-universe separable Banach predual, with the conjugate-linear dual convention. This imposes no separability on the Hilbert representation or the operator norm topology.
L4DEL explanation
A same-universe separable Banach predual, with the conjugate-linear dual convention. This imposes no separability on the Hilbert representation or the operator norm topology.
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/Analysis/RelativeGeneration/Topology.lean:37 (same declaration text).
```
def HasSeparablePredual (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) : Prop :=
…
```
abbrev OAI.RelativeGeneration.ultraweakTopology The ultraweak topology is generated by absolutely summable vector functionals line 43
Source documentation
The ultraweak topology is generated by absolutely summable vector functionals.
L4DEL explanation
The ultraweak topology is generated by absolutely summable vector functionals.
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/Analysis/RelativeGeneration/Topology.lean:43 (same declaration text).
```
abbrev ultraweakTopology (Hilbert : Type level) [NormedAddCommGroup Hilbert]
    [InnerProductSpace ℂ Hilbert] [CompleteSpace Hilbert] :
    TopologicalSpace (Hilbert →L[ℂ] Hilbert) :=
…
```
abbrev OAI.RelativeGeneration.algebraUltraweakTopology The concrete algebra carries the induced ultraweak topology line 53
Source documentation
The concrete algebra carries the induced ultraweak topology.
L4DEL explanation
The concrete algebra carries the induced ultraweak topology.
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/Analysis/RelativeGeneration/Topology.lean:53 (same declaration text).
```
abbrev algebraUltraweakTopology (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) :
    TopologicalSpace algebra :=
…
```
structure OAI.RelativeGeneration.IsNormalizedTrace A faithful normal normalized trace, not an additional hypothesis on the inclusion line 59
Source documentation
A faithful normal normalized trace, not an additional hypothesis on the inclusion.
L4DEL explanation
A faithful normal normalized trace, not an additional hypothesis on the inclusion.
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/Analysis/RelativeGeneration/Topology.lean:59 (same declaration text).
```
structure IsNormalizedTrace (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))
    (trace : algebra →L[ℂ] ℂ) : Prop where
…
```
def OAI.RelativeGeneration.traceDistance The normalized trace two-distance between genuine unitaries of the original algebra line 69
Source documentation
The normalized trace two-distance between genuine unitaries of the original algebra.
L4DEL explanation
The normalized trace two-distance between genuine unitaries of the original algebra.
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/Analysis/RelativeGeneration/Topology.lean:69 (same declaration text).
```
def traceDistance (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))
    (trace : algebra →L[ℂ] ℂ) (left right : unitary algebra) : ℝ :=
…
```
abbrev OAI.RelativeGeneration.traceTopology The topology generated by trace two-distance balls, not the inherited operator norm topology line 75
Source documentation
The topology generated by trace two-distance balls, not the inherited operator norm topology.
L4DEL explanation
The topology generated by trace two-distance balls, not the inherited operator norm topology.
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/Analysis/RelativeGeneration/Topology.lean:75 (same declaration text).
```
abbrev traceTopology (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))
    (trace : algebra →L[ℂ] ℂ) : TopologicalSpace (unitary algebra) :=
…
```
def OAI.RelativeGeneration.relativeGeneratorLocus The entire relative-generator locus in the original unitary group line 81
Source documentation
The entire relative-generator locus in the original unitary group.
L4DEL explanation
The entire relative-generator locus in the original unitary group.
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/Analysis/RelativeGeneration/Topology.lean:125 (same declaration text).
```
def relativeGeneratorLocus (small large : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) :
    Set (unitary large) :=
…
```
def OAI.RelativeGeneration.MainTarget Documented declaration: MainTarget line 90
Source documentation
Every irreducible inclusion of concrete type II₁ factors with separable predual of the larger factor has a dense Gδ full relative-generator locus in the normalized trace two-topology. The normalized faithful normal trace is asserted to exist; no representation, transport, cyclicity, or perturbation premise is imposed.
L4DEL explanation
Every irreducible inclusion of concrete type II₁ factors with separable predual of the larger factor has a dense Gδ full relative-generator locus in the normalized trace two-topology. The normalized faithful normal trace is asserted to exist; no representation, transport, cyclicity, or perturbation premise 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/Analysis/RelativeGeneration/Topology.lean:134 (same declaration text).
```
def MainTarget : Prop :=
…
```
theorem OAI.RelativeGeneration.main_theorem Challenge statement: Dense relative generators for irreducible inclusions of finite factors sorry line 103
L4DEL explanation
This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content.
1. The linked formalization note covers 2 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.Analysis.RelativeGeneration.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/RelativeGeneration/Main.lean:92 (same statement text; different proof or body). Qualify references as OAI.RelativeGeneration.main_theorem@path.
```
theorem main_theorem : MainTarget.{level} := by
…
```