`ComparatorChallenges/BassTorsionFree.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, 1 contain sorry, 0 declared as axioms

Traces and augmentation rank on the idempotent presentation of algebraic K₀.

def OAI.BassTrace.classCoefficient line 11
Distinct definitions of this name: ComparatorChallenges/BassTrace.lean:19 (same statement text; different proof or body), OAI/RingTheory/BassTrace/TraceCoefficients.lean:9 (same statement text; different proof or body). Qualify references as OAI.BassTrace.classCoefficient@path.
theorem OAI.BassTrace.classCoefficient_zero line 18
Also declared: OAI/RingTheory/BassTrace/TraceCoefficients.lean:20 (same declaration text).
theorem OAI.BassTrace.classCoefficient_single line 22
Also declared: OAI/RingTheory/BassTrace/TraceCoefficients.lean:24 (same declaration text).
theorem OAI.BassTrace.classCoefficient_add line 27
Also declared: OAI/RingTheory/BassTrace/TraceCoefficients.lean:29 (same declaration text).
theorem OAI.BassTrace.classCoefficient_smul line 36
Also declared: OAI/RingTheory/BassTrace/TraceCoefficients.lean:38 (same declaration text).
theorem OAI.BassTrace.classCoefficient_sum line 46
Also declared: OAI/RingTheory/BassTrace/TraceCoefficients.lean:48 (same declaration text).
theorem OAI.BassTrace.classCoefficient_mul_comm line 54
Also declared: OAI/RingTheory/BassTrace/TraceCoefficients.lean:56 (same declaration text).
theorem OAI.BassTrace.classCoefficient_matrix_mul_comm line 73
Also declared: OAI/RingTheory/BassTrace/TraceCoefficients.lean:75 (same declaration text).
structure OAI.BassTrace.K0.Idempotent line 101
Also declared: ComparatorChallenges/BassTrace.lean:30 (same declaration text), OAI/RingTheory/BassTrace/Definitions.lean:19 (same declaration text).
def OAI.BassTrace.K0.directSum line 108
Also declared: ComparatorChallenges/BassTrace.lean:37 (same declaration text), OAI/RingTheory/BassTrace/Definitions.lean:26 (same declaration text).
def OAI.BassTrace.K0.relationGenerators line 115
Also declared: ComparatorChallenges/BassTrace.lean:44 (same declaration text), OAI/RingTheory/BassTrace/Definitions.lean:33 (same declaration text).
def OAI.BassTrace.K0.relations line 121
Also declared: ComparatorChallenges/BassTrace.lean:50 (same declaration text), OAI/RingTheory/BassTrace/Definitions.lean:39 (same declaration text).
instance OAI.BassTrace.K0.relations_normal line 124
Also declared: ComparatorChallenges/BassTrace.lean:53 (same declaration text), OAI/RingTheory/BassTrace/Definitions.lean:42 (same declaration text).
abbrev OAI.BassTrace.K0.Group line 126
Distinct definitions of this name: ComparatorChallenges/BassTrace.lean:55 (same statement text; different proof or body), OAI/RingTheory/BassTrace/Definitions.lean:44 (same statement text; different proof or body). Qualify references as OAI.BassTrace.K0.Group@path.
instance OAI.BassTrace.K0.groupRingRelations_normal line 130
Also declared: ComparatorChallenges/BassTrace.lean:61 (same declaration text), OAI/RingTheory/BassTrace/Definitions.lean:50 (same declaration text).
def OAI.BassTrace.K0.rawCoefficient line 132
Also declared: ComparatorChallenges/BassTrace.lean:63 (same declaration text), OAI/RingTheory/BassTrace/Definitions.lean:52 (same declaration text).
lemma OAI.BassTrace.K0.trace_reindex line 155
Also declared: OAI/RingTheory/BassTrace/MatrixK0.lean:18 (same declaration text).
lemma OAI.BassTrace.K0.trace_directSum line 161
Also declared: OAI/RingTheory/BassTrace/MatrixK0.lean:24 (same declaration text).
lemma OAI.BassTrace.K0.relations_in_kernel line 168
Also declared: OAI/RingTheory/BassTrace/MatrixK0.lean:31 (same declaration text).
def OAI.BassTrace.K0.hsCoefficient line 181
Also declared: OAI/RingTheory/BassTrace/MatrixK0.lean:44 (same declaration text).
def OAI.TorsionFreeBass.TorsionFree A group is torsion-free when its only finite-order element is the identity line 196
Source documentation
A group is torsion-free when its only finite-order element is the identity. This is the usual group-theoretic condition, not the stronger unique-roots condition.
L4DEL explanation
A group is torsion-free when its only finite-order element is the identity. This is the usual group-theoretic condition, not the stronger unique-roots condition.
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/RingTheory/BassTrace/TraceRank.lean:13 (same declaration text).
```
def TorsionFree (G : Type*) [Group G] : Prop := ∀ g : G, IsOfFinOrder g → g = 1
…
```
def OAI.TorsionFreeBass.augmentation The sum of coefficients, or augmentation, of the complex group algebra line 201
Source documentation
The sum of coefficients, or augmentation, of the complex group algebra.
L4DEL explanation
The sum of coefficients, or augmentation, of the complex group 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/RingTheory/BassTrace/TraceRank.lean:18 (same declaration text).
```
def augmentation : MonoidAlgebra ℂ G →+* ℂ :=
…
```
def OAI.TorsionFreeBass.canonicalTrace The canonical trace is the coefficient of the identity line 205
Source documentation
The canonical trace is the coefficient of the identity.
L4DEL explanation
The canonical trace is the coefficient of the identity.
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/RingTheory/BassTrace/TraceRank.lean:26 (same declaration text).
```
def canonicalTrace (a : MonoidAlgebra ℂ G) : ℂ := a.coeff 1
```
def OAI.TorsionFreeBass.matrixTrace The matrix trace obtained by applying the canonical trace to each diagonal entry line 208
Source documentation
The matrix trace obtained by applying the canonical trace to each diagonal entry.
L4DEL explanation
The matrix trace obtained by applying the canonical trace to each diagonal entry.
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/RingTheory/BassTrace/TraceRank.lean:29 (same declaration text).
```
def matrixTrace {q : ℕ} (e : Matrix (Fin q) (Fin q) (MonoidAlgebra ℂ G)) : ℂ :=
…
```
def OAI.TorsionFreeBass.augmentedMatrix Entrywise augmentation line 212
Source documentation
Entrywise augmentation.
L4DEL explanation
Entrywise augmentation.
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/RingTheory/BassTrace/TraceRank.lean:33 (same declaration text).
```
def augmentedMatrix {m n : ℕ} (e : Matrix (Fin m) (Fin n) (MonoidAlgebra ℂ G)) :
    Matrix (Fin m) (Fin n) ℂ := e.map augmentation
```
Mentions: OAI.TorsionFreeBass.augmentation
lemma OAI.TorsionFreeBass.augmentedMatrix_mul line 215
Also declared: OAI/RingTheory/BassTrace/TraceRank.lean:45 (same declaration text).
lemma OAI.TorsionFreeBass.augmentedMatrix_trace line 221
Also declared: OAI/RingTheory/BassTrace/TraceRank.lean:51 (same declaration text).
lemma OAI.TorsionFreeBass.idempotent_matrix_trace_eq_rank line 226
Distinct definitions of this name: OAI/RingTheory/BassTrace/TraceRank.lean:56 (same statement text; different proof or body). Qualify references as OAI.TorsionFreeBass.idempotent_matrix_trace_eq_rank@path.
def OAI.TorsionFreeBass.rawAugmentationRank The augmentation rank on the free abelian group of idempotents, before imposing isomorphism and direct-sum relations line 237
Source documentation
The augmentation rank on the free abelian group of idempotents, before imposing isomorphism and direct-sum relations.
L4DEL explanation
The augmentation rank on the free abelian group of idempotents, before imposing isomorphism and direct-sum relations.
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/RingTheory/BassTrace/TraceRank.lean:100 (same declaration text).
```
def rawAugmentationRank : FreeAbelianGroup (Idempotent (MonoidAlgebra ℂ G)) →+ ℤ :=
…
```
lemma OAI.TorsionFreeBass.rawAugmentationRank_of line 240
Also declared: OAI/RingTheory/BassTrace/TraceRank.lean:103 (same declaration text).
lemma OAI.TorsionFreeBass.augmentationRank_relations line 248
Also declared: OAI/RingTheory/BassTrace/TraceRank.lean:111 (same declaration text).
def OAI.TorsionFreeBass.augmentationRank The integer-valued augmentation rank on the usual idempotent presentation of algebraic K₀: on a generator it is the complex dimension of the augmented image line 269
Source documentation
The integer-valued augmentation rank on the usual idempotent presentation of algebraic K₀: on a generator it is the complex dimension of the augmented image.
L4DEL explanation
The integer-valued augmentation rank on the usual idempotent presentation of algebraic K₀: on a generator it is the complex dimension of the augmented image.
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/RingTheory/BassTrace/TraceRank.lean:132 (same statement text; different proof or body). Qualify references as OAI.TorsionFreeBass.augmentationRank@path.
```
def augmentationRank : BassTrace.K0.Group (MonoidAlgebra ℂ G) →+ ℤ :=
…
```
lemma OAI.TorsionFreeBass.traceK0_relations line 274
Also declared: OAI/RingTheory/BassTrace/TraceRank.lean:141 (same declaration text).
def OAI.TorsionFreeBass.traceK0 Coefficient at the identity class of the Hattori--Stallings trace on K₀ line 293
Source documentation
Coefficient at the identity class of the Hattori--Stallings trace on K₀.
L4DEL explanation
Coefficient at the identity class of the Hattori--Stallings trace on K₀.
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/RingTheory/BassTrace/TraceRank.lean:160 (same declaration text).
```
def traceK0 : BassTrace.K0.Group (MonoidAlgebra ℂ G) →+ ℂ :=
…
```
theorem OAI.TorsionFreeBass.torsion_free_corollary Challenge statement: Trace rank and characteristic-zero Kaplansky idempotents for torsion-free groups sorry line 299
Source documentation
Trace/rank, equality and range on algebraic K₀, and characteristic-zero scalar idempotents for torsion-free groups.
L4DEL explanation
Trace/rank, equality and range on algebraic K₀, and characteristic-zero scalar idempotents for torsion-free groups.
1. This explanation is a condensed rendering of this declaration's own doc comment at the pinned source commit.
2. The linked formalization note covers 2 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
3. The repository's comparator configuration selects OAI.RingTheory.BassTrace.TorsionFree as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/RingTheory/BassTrace/TorsionFree.lean:50 (same statement text; different proof or body). Qualify references as OAI.TorsionFreeBass.torsion_free_corollary@path.
```
theorem torsion_free_corollary (G : Type*) [Group G] (hG : TorsionFree G) :
    ((∀ (q : ℕ), 1 ≤ q →
      ∀ e : Matrix (Fin q) (Fin q) (MonoidAlgebra ℂ G), e ^ 2 = e →
        matrixTrace e = ((augmentedMatrix e).rank : ℂ)) ∧
      traceK0 (G := G) = (Int.castAddHom ℂ).comp augmentationRank ∧
…
```
Mentions: OAI.TorsionFreeBass.TorsionFree, OAI.TorsionFreeBass.matrixTrace, OAI.TorsionFreeBass.augmentedMatrix, OAI.TorsionFreeBass.traceK0