`ComparatorChallenges/MatrixMultiplication.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

21 declarations, 3 contain sorry, 0 declared as axioms

The three main complex matrix multiplication bounds, checked independently.

inductive OAI.MatrixMultiplication.Arithmetic.Gate line 13
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:11 (same declaration text).
def OAI.MatrixMultiplication.Arithmetic.Gate.eval line 24
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:22 (same declaration text).
def OAI.MatrixMultiplication.Arithmetic.Gate.cost line 32
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:30 (same declaration text).
inductive OAI.MatrixMultiplication.Arithmetic.Program line 41
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:39 (same declaration text).
def OAI.MatrixMultiplication.Arithmetic.Program.eval line 49
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:47 (same declaration text).
def OAI.MatrixMultiplication.Arithmetic.Program.cost line 54
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:52 (same declaration text).
abbrev OAI.MatrixMultiplication.Arithmetic.MatrixInput line 60
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:58 (same declaration text).
def OAI.MatrixMultiplication.Arithmetic.matrixInputs line 62
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:60 (same declaration text).
structure OAI.MatrixMultiplication.Arithmetic.MatrixAlgorithm line 68
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:66 (same declaration text).
def OAI.MatrixMultiplication.Arithmetic.MatrixAlgorithm.eval line 77
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:75 (same declaration text).
def OAI.MatrixMultiplication.Arithmetic.MatrixAlgorithm.Correct line 82
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:80 (same declaration text).
def OAI.MatrixMultiplication.Arithmetic.MatrixAlgorithm.cost line 85
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:83 (same declaration text).
def OAI.MatrixMultiplication.Arithmetic.AdmissibleExponent Positive slack, one uniform constant, and a correct program at every size line 90
Source documentation
Positive slack, one uniform constant, and a correct program at every size.
L4DEL explanation
Positive slack, one uniform constant, and a correct program at every size.
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/LinearAlgebra/MatrixMultiplication/Model.lean:88 (same declaration text).
```
def AdmissibleExponent (F : Type*) [Field F] (τ : ℝ) : Prop :=
…
```
def OAI.MatrixMultiplication.Arithmetic.omega line 95
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:93 (same declaration text).
def OAI.MatrixMultiplication.Arithmetic.innerSize The inner dimension of an n × n^k by n^k × n multiplication line 98
Source documentation
The inner dimension of an n × n^k by n^k × n multiplication.
L4DEL explanation
The inner dimension of an n × n^k by n^k × n multiplication.
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/LinearAlgebra/MatrixMultiplication/Model.lean:96 (same declaration text).
```
def innerSize (n : ℕ) (k : ℝ) : ℕ := ⌈(n : ℝ) ^ k⌉₊
```
def OAI.MatrixMultiplication.Arithmetic.RectangularAdmissibleExponent line 100
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:98 (same declaration text).
def OAI.MatrixMultiplication.Arithmetic.rectangularOmega line 105
Also declared: OAI/LinearAlgebra/MatrixMultiplication/Model.lean:103 (same declaration text).
def OAI.MatrixMultiplication.Arithmetic.complexAlpha The dual exponent is a supremum of exact exponent-two rectangular shapes line 109
Source documentation
The dual exponent is a supremum of exact exponent-two rectangular shapes.
L4DEL explanation
The dual exponent is a supremum of exact exponent-two rectangular shapes.
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/LinearAlgebra/MatrixMultiplication/Model.lean:107 (same declaration text).
```
def complexAlpha : ℝ :=
…
```
theorem OAI.MatrixMultiplication.complex_omega_le_nine_quarters Complex matrix multiplication has exponent at most 9/4 sorry line 118
L4DEL explanation
We prove that the exponent of matrix multiplication over the complex numbers is at most 9/4.
— OpenAI 9/4 manuscript (abstract)
We’re releasing a broad range of new mathematical results produced by an internal frontier model.
— OpenAI mathematics release announcement
natural, obvious and centuries-old methods for solving natural, important computational problems may be far from the fastest.
— Avi Wigderson and Binyi Zu (2023), page 3
This is an asymptotic arithmetic-complexity statement: for every positive slack ε, sufficiently large complex n×n matrices can be multiplied using O(n^(9/4+ε)) additions, subtractions, and multiplications in a division-free program. It does not claim that the construction is fastest at practical matrix sizes or in bit complexity.
1. The manuscript first bounds the asymptotic tensor rank of matrix multiplication by analysing restrictions and asymptotic rates.
2. Those tensor identities yield algorithms at every exponent strictly above 9/4; the formal definition of ω then turns the family of bounds into ω(ℂ) ≤ 9/4.
3. The Lean target is the exact square-exponent conclusion. Its comparator configuration points to the released solution module, while the pinned manuscript supplies the mathematical exposition.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI 9/4 manuscript (abstract), Pinned OpenAI formalization note 107, OpenAI mathematics release announcement, Avi Wigderson and Binyi Zu (2023), page 3
Distinct definitions of this name: OAI/LinearAlgebra/MatrixMultiplication/Main.lean:14 (same statement text; different proof or body). Qualify references as OAI.MatrixMultiplication.complex_omega_le_nine_quarters@path.
```
theorem complex_omega_le_nine_quarters :
    Arithmetic.omega ℂ ≤ (9 : ℝ) / 4 := by
…
```
Mentions: OAI.MatrixMultiplication.Arithmetic.omega
theorem OAI.MatrixMultiplication.complex_alpha_gt_93_div_200 The complex dual matrix-multiplication exponent is greater than 0.465 sorry line 122
L4DEL explanation
the dual exponent satisfies α > 0.465
— OpenAI rectangular matrix-multiplication manuscript (abstract)
The dual exponent α is the supremum of aspect ratios k for which multiplying an n×n^k matrix by an n^k×n matrix can be done with exponent 2, up to arbitrary positive slack. This theorem proves α > 93/200 = 0.465; it is a strict asymptotic bound, not a benchmark at a fixed dimension.
1. The accompanying manuscript develops rectangular matrix-multiplication bounds over characteristic-zero fields.
2. A concrete construction attains exponent 2 for an aspect ratio strictly larger than 0.465.
3. The Lean target states that strict lower bound directly, with the comparator selecting the released matrix-multiplication solution module.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI rectangular matrix-multiplication manuscript (abstract), Pinned OpenAI formalization note 107, OpenAI mathematics release announcement
Distinct definitions of this name: OAI/LinearAlgebra/MatrixMultiplication/Main.lean:19 (same statement text; different proof or body). Qualify references as OAI.MatrixMultiplication.complex_alpha_gt_93_div_200@path.
```
theorem complex_alpha_gt_93_div_200 :
    (93 : ℝ) / 200 < Arithmetic.complexAlpha := by
…
```
Mentions: OAI.MatrixMultiplication.Arithmetic.complexAlpha
theorem OAI.MatrixMultiplication.complex_rectangular_omega_lt_523_div_250 Rectangular complex matrix multiplication at aspect ratio 0.709 has exponent below 2.092 sorry line 126
L4DEL explanation
ω(1, 0.709, 1) < 2.092
— OpenAI rectangular matrix-multiplication manuscript (abstract)
For multiplication of an n×n^0.709 matrix by an n^0.709×n matrix over ℂ, the arithmetic exponent is strictly less than 523/250 = 2.092. The strict inequality means some uniform positive exponent improvement exists; it does not specify a practical crossover size.
1. The manuscript turns tensor constructions into explicit upper bounds for rectangular matrix multiplication in characteristic zero.
2. Substituting aspect ratio 709/1000 yields a bound strictly below 523/250.
3. The Lean target isolates that numerical consequence, while the pinned formalization note records its intended arithmetic model and scope.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI rectangular matrix-multiplication manuscript (abstract), Pinned OpenAI formalization note 107, OpenAI mathematics release announcement
Distinct definitions of this name: OAI/LinearAlgebra/MatrixMultiplication/Main.lean:25 (same statement text; different proof or body). Qualify references as OAI.MatrixMultiplication.complex_rectangular_omega_lt_523_div_250@path.
```
theorem complex_rectangular_omega_lt_523_div_250 :
    Arithmetic.rectangularOmega ℂ ((709 : ℝ) / 1000) < (523 : ℝ) / 250 := by
…
```
Mentions: OAI.MatrixMultiplication.Arithmetic.rectangularOmega