`ComparatorChallenges/CoordinateSweeps.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

64 declarations, 2 contain sorry, 0 declared as axioms

Conditional moments for one ordered coordinate sweep on power-of-two rectangular grids. The line law mixes uniform permutations with a binary coordinate sweep. The trace is unnormalized, and the trajectory cost uses falling factorials.

structure OAI.CoordinateSweeps.FiniteLaw line 19
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:28 (same declaration text).
instance OAI.CoordinateSweeps.FiniteLaw.(instance at line 28) line 28
theorem OAI.CoordinateSweeps.FiniteLaw.ext line 30
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:39 (same declaration text).
theorem OAI.CoordinateSweeps.FiniteLaw.sum_mass line 38
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:47 (same declaration text).
def OAI.CoordinateSweeps.FiniteLaw.uniform line 41
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:51 (same declaration text).
def OAI.CoordinateSweeps.FiniteLaw.map line 49
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:60 (same declaration text).
abbrev OAI.CoordinateSweeps.Cube line 60
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:177 (same declaration text).
def OAI.CoordinateSweeps.pairSwitch line 62
Distinct definitions of this name: OAI/Probability/CoordinateSweeps/Grid.lean:180 (same statement text; different proof or body). Qualify references as OAI.CoordinateSweeps.pairSwitch@path.
def OAI.CoordinateSweeps.binaryLayer line 74
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:193 (same declaration text).
def OAI.CoordinateSweeps.orderedProduct Coordinate zero acts first line 79
Source documentation
Coordinate zero acts first.
L4DEL explanation
Coordinate zero acts first.
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/Probability/CoordinateSweeps/Grid.lean:198 (same declaration text).
```
def orderedProduct {G : Type*} [Monoid G] {d : ℕ} (g : Fin d → G) : G :=
…
```
abbrev OAI.CoordinateSweeps.BinaryChoices line 82
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:202 (same declaration text).
instance OAI.CoordinateSweeps.(instance at line 84) line 84
def OAI.CoordinateSweeps.binarySweep line 86
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:207 (same declaration text).
def OAI.CoordinateSweeps.binaryLaw line 89
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:211 (same declaration text).
structure OAI.CoordinateSweeps.Grid line 92
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:221 (same declaration text).
def OAI.CoordinateSweeps.Grid.size line 99
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:228 (same declaration text).
abbrev OAI.CoordinateSweeps.Grid.Slot line 101
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:230 (same declaration text).
abbrev OAI.CoordinateSweeps.Grid.Line line 103
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:232 (same declaration text).
abbrev OAI.CoordinateSweeps.Grid.Choices line 105
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:234 (same declaration text).
def OAI.CoordinateSweeps.Grid.fiberPerm line 107
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:237 (same declaration text).
def OAI.CoordinateSweeps.Grid.stage line 113
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:243 (same declaration text).
def OAI.CoordinateSweeps.Grid.boundary line 117
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:248 (same declaration text).
def OAI.CoordinateSweeps.Grid.sweep line 120
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:251 (same declaration text).
def OAI.CoordinateSweeps.Grid.Allowed The side lengths are in [R, R²], where the single global scale is R = 2^r line 123
Source documentation
The side lengths are in [R, R²], where the single global scale is R = 2^r.
L4DEL explanation
The side lengths are in [R, R²], where the single global scale is R = 2^r.
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/Probability/CoordinateSweeps/Grid.lean:263 (same declaration text).
```
def Allowed (G : Grid) (r : ℕ) : Prop := ∀ j, r ≤ G.bits j ∧ G.bits j ≤ 2 * r
```
Mentions: OAI.CoordinateSweeps.Grid
structure OAI.CoordinateSweeps.Grid.Holes Full disjoint paths, including every boundary of the ordered sweep line 126
Source documentation
Full disjoint paths, including every boundary of the ordered sweep.
L4DEL explanation
Full disjoint paths, including every boundary of the ordered sweep.
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/Probability/CoordinateSweeps/Grid.lean:266 (same declaration text).
```
structure Holes (G : Grid) (h : ℕ) where
…
```
Mentions: OAI.CoordinateSweeps.Grid
def OAI.CoordinateSweeps.Grid.Holes.Compatible line 136
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:276 (same declaration text).
def OAI.CoordinateSweeps.Grid.Holes.Feasible line 139
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:279 (same declaration text).
def OAI.CoordinateSweeps.Grid.Holes.lineCount line 141
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:282 (same declaration text).
def OAI.CoordinateSweeps.Grid.Holes.cost line 145
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:287 (same declaration text).
def OAI.CoordinateSweeps.Grid.Holes.stabilizer The symmetric group on the unmarked input sites line 151
Source documentation
The symmetric group on the unmarked input sites.
L4DEL explanation
The symmetric group on the unmarked input sites.
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/Probability/CoordinateSweeps/Grid.lean:337 (same declaration text).
```
def stabilizer (H : G.Holes h) : Subgroup (Equiv.Perm G.Slot) where
…
```
def OAI.CoordinateSweeps.Grid.Holes.reference line 160
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:348 (same declaration text).
theorem OAI.CoordinateSweeps.Grid.Holes.reference_compatible line 162
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:350 (same declaration text).
def OAI.CoordinateSweeps.Grid.Holes.residual A reference completion identifies the free input and output sites line 166
Source documentation
A reference completion identifies the free input and output sites.
L4DEL explanation
A reference completion identifies the free input and output sites.
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/Probability/CoordinateSweeps/Grid.lean:354 (same declaration text).
```
def residual (H : G.Holes h) (hf : H.Feasible) (ω : {ω : G.Choices // H.Compatible ω}) :
    H.stabilizer :=
…
```
def OAI.CoordinateSweeps.Grid.choiceWeight Independent line permutations with law (1-z) U + z B line 177
Source documentation
Independent line permutations with law (1-z) U + z B.
L4DEL explanation
Independent line permutations with law (1-z) U + z B.
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/Probability/CoordinateSweeps/Grid.lean:365 (same declaration text).
```
def choiceWeight (G : Grid) (z : ℝ) (ω : G.Choices) : ℝ :=
…
```
Mentions: OAI.CoordinateSweeps.Grid
def OAI.CoordinateSweeps.Grid.Holes.probability line 186
Also declared: OAI/Probability/CoordinateSweeps/Grid.lean:673 (same declaration text).
def OAI.CoordinateSweeps.Grid.eventDistance line 191
Also declared: OAI/Probability/CoordinateSweeps/SparseBound.lean:562 (same declaration text).
structure OAI.CoordinateSweeps.UnitaryIrrep line 197
Also declared: OAI/RepresentationTheory/Unitary/Basic.lean:10 (same declaration text).
def OAI.CoordinateSweeps.schattenMoment Unnormalized Schatten 2q moment, without division by the dimension line 206
Source documentation
Unnormalized Schatten 2q moment, without division by the dimension.
L4DEL explanation
Unnormalized Schatten 2q moment, without division by the dimension.
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/Matrix/TensorMoment.lean:614 (same declaration text).
```
def schattenMoment {D : ℕ} (q : ℕ) (K : Matrix (Fin D) (Fin D) ℂ) : ℝ :=
…
```
def OAI.CoordinateSweeps.Grid.Holes.conditionalAverage line 213
Also declared: OAI/Probability/CoordinateSweeps/Conditioning.lean:980 (same declaration text).
def OAI.CoordinateSweeps.a line 221
Also declared: OAI/Probability/CoordinateSweeps/Conditioning.lean:1035 (same declaration text).
def OAI.CoordinateSweeps.c0 line 222
Also declared: OAI/Probability/CoordinateSweeps/Conditioning.lean:1036 (same declaration text).
def OAI.CoordinateSweeps.e0 line 223
Also declared: OAI/Probability/CoordinateSweeps/Conditioning.lean:1037 (same declaration text).
def OAI.CoordinateSweeps.c line 224
Also declared: OAI/Probability/CoordinateSweeps/Conditioning.lean:1038 (same declaration text).
def OAI.CoordinateSweeps.e line 225
Also declared: OAI/Probability/CoordinateSweeps/Conditioning.lean:1039 (same declaration text).
def OAI.CoordinateSweeps.logMoment The logarithm of a zero moment is minus infinity line 228
Source documentation
The logarithm of a zero moment is minus infinity.
L4DEL explanation
The logarithm of a zero moment is minus infinity.
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/Probability/CoordinateSweeps/Conditioning.lean:1042 (same declaration text).
```
def logMoment (t : ℝ) : EReal := if t = 0 then ⊥ else (Real.log t : EReal)
```
def OAI.CoordinateSweeps.ConditionalMain line 230
Also declared: OAI/Probability/CoordinateSweeps/Conditioning.lean:1046 (same declaration text).
def OAI.CoordinateSweeps.MomentBound line 237
Also declared: OAI/Probability/CoordinateSweeps/Main.lean:100 (same declaration text).
def OAI.CoordinateSweeps.ScaleLarge line 242
Also declared: OAI/Probability/CoordinateSweeps/BaseCase.lean:346 (same declaration text).
def OAI.CoordinateSweeps.baseWidth line 245
Also declared: OAI/Probability/CoordinateSweeps/BaseCase.lean:458 (same declaration text).
def OAI.CoordinateSweeps.baseZeta line 246
Also declared: OAI/Probability/CoordinateSweeps/BaseCase.lean:459 (same declaration text).
structure OAI.CoordinateSweeps.ScalarInduction.MainSizeReady line 250
Also declared: OAI/Probability/CoordinateSweeps/Scales.lean:244 (same declaration text).
def OAI.CoordinateSweeps.SparseScalar.theta line 260
Also declared: OAI/Probability/CoordinateSweeps/SparseBound.lean:34 (same declaration text).
def OAI.CoordinateSweeps.SparseScalar.K0 line 261
Also declared: OAI/Probability/CoordinateSweeps/SparseBound.lean:35 (same declaration text).
def OAI.CoordinateSweeps.SparseScalar.B0 line 262
Also declared: OAI/Probability/CoordinateSweeps/SparseBound.lean:36 (same declaration text).
def OAI.CoordinateSweeps.SparseScalar.xi line 263
Also declared: OAI/Probability/CoordinateSweeps/SparseBound.lean:37 (same declaration text).
def OAI.CoordinateSweeps.SparseScalar.rho line 264
Also declared: OAI/Probability/CoordinateSweeps/SparseBound.lean:38 (same declaration text).
def OAI.CoordinateSweeps.SparseScalar.q line 265
Also declared: OAI/Probability/CoordinateSweeps/SparseBound.lean:39 (same declaration text).
def OAI.CoordinateSweeps.SparseScalar.uniformRate line 267
Also declared: OAI/Probability/CoordinateSweeps/SparseBound.lean:384 (same declaration text).
structure OAI.CoordinateSweeps.SparseScalar.SizeReady line 272
Also declared: OAI/Probability/CoordinateSweeps/Perturbation.lean:63 (same declaration text).
structure OAI.CoordinateSweeps.SparseScalar.ScaleReady line 283
Also declared: OAI/Probability/CoordinateSweeps/Perturbation.lean:81 (same declaration text).
def OAI.CoordinateSweeps.SparseScalar.slack line 289
Also declared: OAI/Probability/CoordinateSweeps/Perturbation.lean:100 (same declaration text).
def OAI.CoordinateSweeps.SparsePerturb line 294
Also declared: OAI/Probability/CoordinateSweeps/Perturbation.lean:316 (same declaration text).
theorem OAI.CoordinateSweeps.all_grid_moment Challenge statement: Conditional coordinate-sweep moment estimate sorry line 300
Source documentation
The quantitative all-grid estimate under the explicit scale and perturbation bounds.
L4DEL explanation
The quantitative all-grid estimate under the explicit scale and perturbation bounds.
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 13 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
3. The repository's comparator configuration selects OAI.Probability.CoordinateSweeps.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Probability/CoordinateSweeps/Main.lean:412 (same statement text; different proof or body). Qualify references as OAI.CoordinateSweeps.all_grid_moment@path.
```
theorem all_grid_moment {r : ℕ} (hr : ScaleLarge r) (hrS : SparseScalar.ScaleReady r)
    (hS : ∀ G : Grid, G.Allowed r → ScalarInduction.MainSizeReady SparseScalar.q G.size)
    {zStar : ℝ} (hζ : SparsePerturb r zStar) {z : ℝ} (hz : z∈Set.Icc 0 zStar) :
    ∀ G : Grid, G.Allowed r → MomentBound G SparseScalar.q z := by
…
```
Mentions: OAI.CoordinateSweeps.ScaleLarge, OAI.CoordinateSweeps.SparseScalar.ScaleReady, OAI.CoordinateSweeps.Grid, OAI.CoordinateSweeps.ScalarInduction.MainSizeReady, OAI.CoordinateSweeps.SparseScalar.q, OAI.CoordinateSweeps.SparsePerturb, OAI.CoordinateSweeps.MomentBound
theorem OAI.CoordinateSweeps.conditional_main Challenge statement: Conditional coordinate-sweep moment estimate sorry line 307
Source documentation
One global choice of scale, moment order, and positive perturbation interval.
L4DEL explanation
One global choice of scale, moment order, and positive perturbation interval.
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 13 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
3. The repository's comparator configuration selects OAI.Probability.CoordinateSweeps.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Probability/CoordinateSweeps/Main.lean:454 (same statement text; different proof or body). Qualify references as OAI.CoordinateSweeps.conditional_main@path.
```
theorem conditional_main : ConditionalMain := by
…
```
Mentions: OAI.CoordinateSweeps.ConditionalMain