`ComparatorChallenges/NuclearUltrapower.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

30 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.NuclearUltrapower.Dyadic line 12
Source comment
The dyadic coefficient ring is exactly Z[1/2].

Also declared: OAI/Analysis/NuclearUltrapower/Model.lean:12 (same declaration text).
```
abbrev Dyadic := Localization.Away (2 : ℤ)
```
abbrev OAI.NuclearUltrapower.Vector3 line 13
Also declared: OAI/Analysis/NuclearUltrapower/Model.lean:13 (same declaration text).
abbrev OAI.NuclearUltrapower.SL3Z line 14
Also declared: OAI/Analysis/NuclearUltrapower/Model.lean:14 (same declaration text).
def OAI.NuclearUltrapower.dyadicTwo line 16
Also declared: OAI/Analysis/NuclearUltrapower/Model.lean:16 (same declaration text).
def OAI.NuclearUltrapower.matrixAction line 19
Also declared: OAI/Analysis/NuclearUltrapower/Model.lean:19 (same declaration text).
def OAI.NuclearUltrapower.dilationAction line 23
Also declared: OAI/Analysis/NuclearUltrapower/Model.lean:23 (same declaration text).
def OAI.NuclearUltrapower.groupAction line 27
Also declared: OAI/Analysis/NuclearUltrapower/Model.lean:27 (same declaration text).
abbrev OAI.NuclearUltrapower.G The exact group of the semidirect-product construction line 39
Source documentation
The exact group of the semidirect-product construction.
L4DEL explanation
The exact group of the semidirect-product construction.
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/NuclearUltrapower/Model.lean:39 (same declaration text).
```
abbrev G := SemidirectProduct (Multiplicative Vector3)
…
```
Mentions: OAI.NuclearUltrapower.Vector3
abbrev OAI.NuclearUltrapower.BoundedSeq Bounded sequences, with their supremum norm line 46
Source documentation
Bounded sequences, with their supremum norm. N is discrete.
L4DEL explanation
Bounded sequences, with their supremum norm. N is discrete.
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/NuclearUltrapower/Model.lean:46 (same declaration text).
```
abbrev BoundedSeq := BoundedContinuousFunction ℕ B
```
def OAI.NuclearUltrapower.ultraRel Equality modulo I_omega(B) line 49
Source documentation
Equality modulo I_omega(B). This is equivalent to norm-null difference.
L4DEL explanation
Equality modulo I_omega(B). This is equivalent to norm-null difference.
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/NuclearUltrapower/Model.lean:49 (same declaration text).
```
def ultraRel (x y : BoundedSeq B) : Prop :=
…
```
Mentions: OAI.NuclearUltrapower.BoundedSeq
def OAI.NuclearUltrapower.ultraCon The ring congruence defining the norm (not algebraic) ultrapower line 53
Source documentation
The ring congruence defining the norm (not algebraic) ultrapower.
L4DEL explanation
The ring congruence defining the norm (not algebraic) ultrapower.
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/NuclearUltrapower/Model.lean:53 (same declaration text).
```
def ultraCon : RingCon (BoundedSeq B) where
…
```
Mentions: OAI.NuclearUltrapower.BoundedSeq
abbrev OAI.NuclearUltrapower.NormUltrapower The exact quotient l-infinity(N,B)/I_omega(B) as a complex star algebra line 74
Source documentation
The exact quotient l-infinity(N,B)/I_omega(B) as a complex star algebra. The quotient identifies sequences whose difference tends to zero along the ultrafilter.
L4DEL explanation
The exact quotient l-infinity(N,B)/I_omega(B) as a complex star algebra. The quotient identifies sequences whose difference tends to zero along the ultrafilter.
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/NuclearUltrapower/Model.lean:74 (same declaration text).
```
abbrev NormUltrapower := (ultraCon B ω).Quotient
```
Mentions: OAI.NuclearUltrapower.ultraCon
instance OAI.NuclearUltrapower.(instance at line 76) line 76
instance OAI.NuclearUltrapower.(instance at line 81) line 81
instance OAI.NuclearUltrapower.(instance at line 97) line 97
def OAI.NuclearUltrapower.IsFree Freeness: no finite subset of N belongs to the ultrafilter line 104
Source documentation
Freeness: no finite subset of N belongs to the ultrafilter.
L4DEL explanation
Freeness: no finite subset of N belongs to the ultrafilter.
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/NuclearUltrapower/Model.lean:104 (same declaration text).
```
def IsFree (ω : Ultrafilter ℕ) : Prop :=
…
```
structure OAI.NuclearUltrapower.ComplexHilbert A bundled complex Hilbert space, without any separability assumption line 110
Source documentation
A bundled complex Hilbert space, without any separability assumption.
L4DEL explanation
A bundled complex Hilbert space, without any separability assumption.
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/NuclearUltrapower/Model.lean:110 (same declaration text).
```
structure ComplexHilbert where
…
```
instance OAI.NuclearUltrapower.(instance at line 117) line 117
def OAI.NuclearUltrapower.spatialEval Spatial evaluation of the algebraic tensor product on separate Hilbert spaces line 124
Source documentation
Spatial evaluation of the algebraic tensor product on separate Hilbert spaces. Its operator norm is unchanged on the Hilbert completion.
L4DEL explanation
Spatial evaluation of the algebraic tensor product on separate Hilbert spaces. Its operator norm is unchanged on the Hilbert completion.
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/NuclearUltrapower/Model.lean:124 (same declaration text).
```
def spatialEval (H K : ComplexHilbert.{w})
    (ρ : B →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (σ : C →⋆ₙₐ[ℂ] (K →L[ℂ] K)) :
    B ⊗[ℂ] C →ₗ[ℂ] ((H ⊗[ℂ] K) →L[ℂ] (H ⊗[ℂ] K)) :=
…
```
def OAI.NuclearUltrapower.commutingEval Evaluation on one Hilbert space; used only when the two ranges commute line 134
Source documentation
Evaluation on one Hilbert space; used only when the two ranges commute.
L4DEL explanation
Evaluation on one Hilbert space; used only when the two ranges commute.
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/NuclearUltrapower/Model.lean:134 (same declaration text).
```
def commutingEval (H : ComplexHilbert.{w})
    (ρ : B →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (σ : C →⋆ₙₐ[ℂ] (H →L[ℂ] H)) :
    B ⊗[ℂ] C →ₗ[ℂ] (H →L[ℂ] H) :=
…
```
def OAI.NuclearUltrapower.minTensorNorm The minimal C*-tensor norm: supremum over spatial representations line 145
Source documentation
The minimal C*-tensor norm: supremum over spatial representations.
L4DEL explanation
The minimal C*-tensor norm: supremum over spatial representations.
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/NuclearUltrapower/Model.lean:145 (same declaration text).
```
def minTensorNorm (x : B ⊗[ℂ] C) : ℝ :=
…
```
def OAI.NuclearUltrapower.maxTensorNorm The maximal C*-tensor norm: supremum over commuting representations line 151
Source documentation
The maximal C*-tensor norm: supremum over commuting representations.
L4DEL explanation
The maximal C*-tensor norm: supremum over commuting representations.
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/NuclearUltrapower/Model.lean:151 (same declaration text).
```
def maxTensorNorm (x : B ⊗[ℂ] C) : ℝ :=
…
```
def OAI.NuclearUltrapower.IsNuclear Nuclearity in the tensor-norm convention of the tensor-norm formulation line 158
Source documentation
Nuclearity in the tensor-norm convention of the tensor-norm formulation. The second C*-algebra need not be unital, nonzero, or separable.
L4DEL explanation
Nuclearity in the tensor-norm convention of the tensor-norm formulation. The second C*-algebra need not be unital, nonzero, or separable.
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/NuclearUltrapower/Model.lean:158 (same declaration text).
```
def IsNuclear (B : Type u) [NonUnitalCStarAlgebra B] : Prop :=
…
```
structure OAI.NuclearUltrapower.UnitalCStar A nonzero unital complex C*-algebra line 164
Source documentation
A nonzero unital complex C*-algebra.
L4DEL explanation
A nonzero unital complex C*-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/NuclearUltrapower/Model.lean:164 (same declaration text).
```
structure UnitalCStar where
…
```
instance OAI.NuclearUltrapower.(instance at line 169) line 169
def OAI.NuclearUltrapower.IsFullGroupAlgebra The universal property defining the full, not the reduced, group C*-algebra line 174
Source documentation
The universal property defining the full, not the reduced, group C*-algebra. The dense generation clause makes the intended completion explicit. This is definition data, not a hypothesis silently supplied to the main claim.
L4DEL explanation
The universal property defining the full, not the reduced, group C*-algebra. The dense generation clause makes the intended completion explicit. This is definition data, not a hypothesis silently supplied to the main claim.
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/NuclearUltrapower/Model.lean:174 (same declaration text).
```
def IsFullGroupAlgebra (A : UnitalCStar.{u}) (ι : G →* unitary A) : Prop :=
…
```
Mentions: OAI.NuclearUltrapower.G
def OAI.NuclearUltrapower.CuntzTwoRelations The two Cuntz relations, including unit preservation explicitly line 180
Source documentation
The two Cuntz relations, including unit preservation explicitly.
L4DEL explanation
The two Cuntz relations, including unit preservation explicitly.
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/NuclearUltrapower/Model.lean:180 (same declaration text).
```
def CuntzTwoRelations {D : Type u} [CStarAlgebra D] (s : Fin 2 → D) : Prop :=
…
```
def OAI.NuclearUltrapower.IsCuntzTwoAlgebra The universal nonzero unital C*-algebra on the two Cuntz isometries line 184
Source documentation
The universal nonzero unital C*-algebra on the two Cuntz isometries.
L4DEL explanation
The universal nonzero unital C*-algebra on the two Cuntz isometries.
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/NuclearUltrapower/Model.lean:184 (same declaration text).
```
def IsCuntzTwoAlgebra (O : UnitalCStar.{u}) (s : Fin 2 → O) : Prop :=
…
```
def OAI.NuclearUltrapower.MainStatementWithCuntzCorollary The manuscript's main statement including its explicitly stated O_2 corollary line 191
Source documentation
The manuscript's main statement including its explicitly stated O_2 corollary. Both universal C*-algebras must be constructed to prove this.
L4DEL explanation
The manuscript's main statement including its explicitly stated O_2 corollary. Both universal C*-algebras must be constructed to prove this.
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/NuclearUltrapower/Model.lean:191 (same declaration text).
```
def MainStatementWithCuntzCorollary : Prop :=
…
```
theorem OAI.NuclearUltrapower.main_no_embedding Challenge statement: Kirchberg's $`\mathcal O_2`$ norm-ultrapower embedding problem sorry line 208
Source documentation
The full dyadic group C*-algebra has no nuclear norm-ultrapower embedding, including the corollary for the universal Cuntz algebra on two isometries.
L4DEL explanation
Kirchberg's norm-ultrapower embedding problem asks whether separable $C^*$-algebras embed into norm ultrapowers of nuclear algebras. The formalization constructs an explicit separable unital full group $C^*$-algebra that has no unital embedding into $B^\omega$ for any nonzero unital nuclear $C^*$-algebra $B$ and any free ultrafilter $\omega$ on $\mathbb N$. In particular, it does not embed unitally into an ultrapower of the Cuntz algebra $\mathcal O_2$.
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.Analysis.NuclearUltrapower.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/NuclearUltrapower/Main.lean:14 (same statement text; different proof or body). Qualify references as OAI.NuclearUltrapower.main_no_embedding@path.
```
theorem main_no_embedding : MainStatementWithCuntzCorollary.{0} := by
…
```