`ComparatorChallenges/ContingencyTables.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

38 declarations, 3 contain sorry, 0 declared as axioms

abbrev OAI.MatchingFPRAS.Symbol line 7
Also declared: ComparatorChallenges/MatchingFPRAS.lean:25 (same declaration text), OAI/Combinatorics/MatchingCount/Model.lean:25 (same declaration text).
def OAI.MatchingFPRAS.encodeNat Binary numeral followed by a delimiter line 10
Source documentation
Binary numeral followed by a delimiter. Tape blank is 0; bits use 1 and 2.
L4DEL explanation
Binary numeral followed by a delimiter. Tape blank is 0; bits use 1 and 2.
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: ComparatorChallenges/MatchingFPRAS.lean:28 (same declaration text), OAI/Combinatorics/MatchingCount/Model.lean:28 (same declaration text).
```
def encodeNat (n : ℕ) : List Symbol :=
…
```
Mentions: OAI.MatchingFPRAS.Symbol
def OAI.MatchingFPRAS.encodeInt line 13
Also declared: ComparatorChallenges/MatchingFPRAS.lean:31 (same declaration text), OAI/Combinatorics/MatchingCount/Model.lean:31 (same declaration text).
def OAI.MatchingFPRAS.encodeRat A rational encoded by its reduced numerator and positive denominator line 17
Source documentation
A rational encoded by its reduced numerator and positive denominator.
L4DEL explanation
A rational encoded by its reduced numerator and positive denominator.
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: ComparatorChallenges/MatchingFPRAS.lean:35 (same declaration text), OAI/Combinatorics/MatchingCount/Model.lean:35 (same declaration text).
```
def encodeRat (q : ℚ) : List Symbol := encodeInt q.num ++ encodeNat q.den
```
Mentions: OAI.MatchingFPRAS.Symbol, OAI.MatchingFPRAS.encodeInt, OAI.MatchingFPRAS.encodeNat
structure OAI.MatchingFPRAS.RandomMachine A uniform randomized Post-Turing machine: a single finite transition table line 20
Source documentation
A uniform randomized Post-Turing machine: a single finite transition table.
L4DEL explanation
A uniform randomized Post-Turing machine: a single finite transition table.
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: ComparatorChallenges/MatchingFPRAS.lean:45 (same declaration text), OAI/Combinatorics/MatchingCount/Model.lean:45 (same declaration text).
```
structure RandomMachine where
…
```
structure OAI.MatchingFPRAS.Configuration line 25
Also declared: ComparatorChallenges/MatchingFPRAS.lean:50 (same declaration text), OAI/Combinatorics/MatchingCount/Model.lean:50 (same declaration text).
def OAI.MatchingFPRAS.tick A halted configuration is absorbing line 31
Source documentation
A halted configuration is absorbing. Each nonhalting tick is exactly one write or one move, or the detection of halt.
L4DEL explanation
A halted configuration is absorbing. Each nonhalting tick is exactly one write or one move, or the detection of halt.
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: ComparatorChallenges/MatchingFPRAS.lean:56 (same declaration text), OAI/Combinatorics/MatchingCount/Model.lean:56 (same declaration text).
```
def tick (A : RandomMachine) (c : Configuration A) (bit : Bool) : Configuration A :=
…
```
Mentions: OAI.MatchingFPRAS.RandomMachine, OAI.MatchingFPRAS.Configuration
def OAI.MatchingFPRAS.initial line 41
Also declared: ComparatorChallenges/MatchingFPRAS.lean:66 (same declaration text), OAI/Combinatorics/MatchingCount/Model.lean:66 (same declaration text).
def OAI.MatchingFPRAS.run Bounded execution on a finite prefix of the random tape line 45
Source documentation
Bounded execution on a finite prefix of the random tape.
L4DEL explanation
Bounded execution on a finite prefix of the random tape.
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: ComparatorChallenges/MatchingFPRAS.lean:70 (same declaration text), OAI/Combinatorics/MatchingCount/Model.lean:70 (same declaration text).
```
def run (A : RandomMachine) (input : List Symbol) {t : ℕ}
    (bits : Fin t → Bool) : Configuration A :=
…
```
Mentions: OAI.MatchingFPRAS.RandomMachine, OAI.MatchingFPRAS.Symbol, OAI.MatchingFPRAS.Configuration
def OAI.MatchingFPRAS.Outputs line 49
Also declared: ComparatorChallenges/MatchingFPRAS.lean:74 (same declaration text), OAI/Combinatorics/MatchingCount/Model.lean:74 (same declaration text).
def OAI.ContingencyTables.HasMargins line 61
Also declared: OAI/Combinatorics/ContingencyTables/Model.lean:23 (same declaration text).
def OAI.ContingencyTables.Table line 64
Also declared: OAI/Combinatorics/ContingencyTables/Model.lean:27 (same declaration text).
def OAI.ContingencyTables.BoundedTable line 66
Also declared: OAI/Combinatorics/ContingencyTables/Model.lean:30 (same declaration text).
theorem OAI.ContingencyTables.entry_le_row line 69
Also declared: OAI/Combinatorics/ContingencyTables/Model.lean:38 (same declaration text).
theorem OAI.ContingencyTables.entry_le_total line 74
Also declared: OAI/Combinatorics/ContingencyTables/Model.lean:43 (same declaration text).
instance OAI.ContingencyTables.table_finite line 78
Also declared: OAI/Combinatorics/ContingencyTables/Model.lean:48 (same declaration text).
instance OAI.ContingencyTables.table_fintype line 87
Also declared: OAI/Combinatorics/ContingencyTables/Model.lean:57 (same declaration text).
instance OAI.ContingencyTables.boundedTable_fintype line 90
Also declared: OAI/Combinatorics/ContingencyTables/Model.lean:60 (same declaration text).
abbrev OAI.ContingencyTables.Algorithms.Machine line 103
Also declared: OAI/Combinatorics/ContingencyTables/AlgorithmStatements.lean:20 (same declaration text).
abbrev OAI.ContingencyTables.Algorithms.Symbol line 104
Also declared: OAI/Combinatorics/ContingencyTables/AlgorithmStatements.lean:21 (same declaration text).
def OAI.ContingencyTables.Algorithms.encodeMatrix line 106
Also declared: OAI/Combinatorics/ContingencyTables/AlgorithmStatements.lean:23 (same declaration text).
def OAI.ContingencyTables.Algorithms.encodeMargins line 109
Also declared: OAI/Combinatorics/ContingencyTables/AlgorithmStatements.lean:26 (same declaration text).
def OAI.ContingencyTables.Algorithms.encodeSamplingInput line 114
Also declared: OAI/Combinatorics/ContingencyTables/AlgorithmStatements.lean:31 (same declaration text).
def OAI.ContingencyTables.Algorithms.encodeCountingInput line 117
Also declared: OAI/Combinatorics/ContingencyTables/AlgorithmStatements.lean:34 (same declaration text).
theorem OAI.ContingencyTables.Algorithms.alphabetSize_neZero The tape alphabet has positive cardinality line 122
Source documentation
The tape alphabet has positive cardinality.
L4DEL explanation
The tape alphabet has positive cardinality.
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/Combinatorics/ContingencyTables/AlgorithmStatements.lean:39 (same declaration text).
```
theorem alphabetSize_neZero : NeZero (8 : ℕ) := inferInstance
```
def OAI.ContingencyTables.Algorithms.OutputsTable A halted tape contains the row-major binary matrix encoding line 125
Source documentation
A halted tape contains the row-major binary matrix encoding.
L4DEL explanation
A halted tape contains the row-major binary matrix encoding.
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/Combinatorics/ContingencyTables/AlgorithmStatements.lean:42 (same declaration text).
```
def OutputsTable (A : Machine) (input : List Symbol) {t m n : ℕ}
    (bits : Fin t → Bool) (X : Fin m → Fin n → ℕ) : Prop :=
…
```
Mentions: OAI.ContingencyTables.Algorithms.Machine, OAI.ContingencyTables.Algorithms.Symbol
def OAI.ContingencyTables.Algorithms.tableMass line 132
Also declared: OAI/Combinatorics/ContingencyTables/AlgorithmStatements.lean:49 (same declaration text).
def OAI.ContingencyTables.Algorithms.haltMass line 138
Also declared: OAI/Combinatorics/ContingencyTables/AlgorithmStatements.lean:55 (same declaration text).
def OAI.ContingencyTables.Algorithms.samplingSize line 143
Also declared: OAI/Combinatorics/ContingencyTables/AlgorithmStatements.lean:60 (same declaration text).
def OAI.ContingencyTables.Algorithms.BoundedSamplingStatement Paper 200-01, Theorem main(i): bounded runtime on every random tape and total variation at most 2⁻ᵏ, with freely varying dimensions line 148
Source documentation
Paper 200-01, Theorem main(i): bounded runtime on every random tape and total variation at most 2⁻ᵏ, with freely varying dimensions.
L4DEL explanation
Paper 200-01, Theorem main(i): bounded runtime on every random tape and total variation at most 2⁻ᵏ, with freely varying dimensions.
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/Combinatorics/ContingencyTables/AlgorithmStatements.lean:65 (same declaration text).
```
def BoundedSamplingStatement : Prop :=
…
```
def OAI.ContingencyTables.Algorithms.ExactSamplingStatement Documented declaration: ExactSamplingStatement line 162
Source documentation
Paper 200-01, Theorem main(ii): every halted output is feasible, each table has limiting mass 1/|Ω|, termination has probability one, and the expected number of bit operations is bounded by a uniform polynomial.
L4DEL explanation
Paper 200-01, Theorem main(ii): every halted output is feasible, each table has limiting mass 1/|Ω|, termination has probability one, and the expected number of bit operations is bounded by a uniform polynomial.
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/Combinatorics/ContingencyTables/AlgorithmStatements.lean:79 (same declaration text).
```
def ExactSamplingStatement : Prop :=
…
```
def OAI.ContingencyTables.Algorithms.count line 176
Also declared: OAI/Combinatorics/ContingencyTables/AlgorithmStatements.lean:93 (same declaration text).
def OAI.ContingencyTables.Algorithms.countingTime line 179
Also declared: OAI/Combinatorics/ContingencyTables/AlgorithmStatements.lean:96 (same declaration text).
def OAI.ContingencyTables.Algorithms.goodCountingTapes line 184
Also declared: OAI/Combinatorics/ContingencyTables/AlgorithmStatements.lean:101 (same declaration text).
def OAI.ContingencyTables.Algorithms.CountingStatement Paper 200-02, Theorem main line 194
Source documentation
Paper 200-02, Theorem main. The input length includes the rational accuracy parameters, and infeasible inputs always produce exactly zero.
L4DEL explanation
Paper 200-02, Theorem main. The input length includes the rational accuracy parameters, and infeasible inputs always produce exactly zero.
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/Combinatorics/ContingencyTables/AlgorithmStatements.lean:111 (same declaration text).
```
def CountingStatement : Prop :=
…
```
theorem OAI.ContingencyTables.boundedSampling Challenge statement: Sampling and counting contingency tables with arbitrary margins sorry line 211
Source documentation
An almost-uniform sampler with polynomial cost on every execution.
L4DEL explanation
The formalization gives an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed row and column sums having equal totals. It terminates almost surely, every halted output is feasible, and each table has exactly the uniform limiting probability. Expected bit complexity is polynomial in the dimensions and the binary length of the margins. It also gives a sampler with a fixed polynomial time bound on every execution whose total-variation error is at most $2^{-k}$ for requested precision $k\ge1$. The same Comparator file includes the companion approximation scheme for counting tables with individual cell bounds. The formalization gives a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions may vary, the margins and bounds are binary encoded, and zero entry bounds are allowed. For rational $0<\varepsilon,\delta<1$, the algorithm returns a nonnegative estimate with relative error at most $\varepsilon$ with probability at least $1-\delta$, and returns zero on every execution when no table exists. The running time is polynomial in the encoded input size, $\varepsilon^{-1}$, and $\log(\delta^{-1})$ on every random tape. The same Comparator file also contains the companion sampling results for tables without cell bounds.
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.Combinatorics.ContingencyTables.UnconditionalMain as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean:19 (same statement text; different proof or body). Qualify references as OAI.ContingencyTables.boundedSampling@path.
```
theorem boundedSampling : Algorithms.BoundedSamplingStatement := by sorry
```
Mentions: OAI.ContingencyTables.Algorithms.BoundedSamplingStatement
theorem OAI.ContingencyTables.exactSampling Challenge statement: Sampling and counting contingency tables with arbitrary margins sorry line 214
Source documentation
Exact uniform sampling in polynomial expected bit time.
L4DEL explanation
The formalization gives an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed row and column sums having equal totals. It terminates almost surely, every halted output is feasible, and each table has exactly the uniform limiting probability. Expected bit complexity is polynomial in the dimensions and the binary length of the margins. It also gives a sampler with a fixed polynomial time bound on every execution whose total-variation error is at most $2^{-k}$ for requested precision $k\ge1$. The same Comparator file includes the companion approximation scheme for counting tables with individual cell bounds. The formalization gives a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions may vary, the margins and bounds are binary encoded, and zero entry bounds are allowed. For rational $0<\varepsilon,\delta<1$, the algorithm returns a nonnegative estimate with relative error at most $\varepsilon$ with probability at least $1-\delta$, and returns zero on every execution when no table exists. The running time is polynomial in the encoded input size, $\varepsilon^{-1}$, and $\log(\delta^{-1})$ on every random tape. The same Comparator file also contains the companion sampling results for tables without cell bounds.
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.Combinatorics.ContingencyTables.UnconditionalMain as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean:24 (same statement text; different proof or body). Qualify references as OAI.ContingencyTables.exactSampling@path.
```
theorem exactSampling : Algorithms.ExactSamplingStatement := by sorry
```
Mentions: OAI.ContingencyTables.Algorithms.ExactSamplingStatement
theorem OAI.ContingencyTables.counting Challenge statement: Sampling and counting contingency tables with arbitrary margins sorry line 217
Source documentation
A cell-bounded FPRAS, with zero output on every infeasible execution.
L4DEL explanation
The formalization gives an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed row and column sums having equal totals. It terminates almost surely, every halted output is feasible, and each table has exactly the uniform limiting probability. Expected bit complexity is polynomial in the dimensions and the binary length of the margins. It also gives a sampler with a fixed polynomial time bound on every execution whose total-variation error is at most $2^{-k}$ for requested precision $k\ge1$. The same Comparator file includes the companion approximation scheme for counting tables with individual cell bounds. The formalization gives a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions may vary, the margins and bounds are binary encoded, and zero entry bounds are allowed. For rational $0<\varepsilon,\delta<1$, the algorithm returns a nonnegative estimate with relative error at most $\varepsilon$ with probability at least $1-\delta$, and returns zero on every execution when no table exists. The running time is polynomial in the encoded input size, $\varepsilon^{-1}$, and $\log(\delta^{-1})$ on every random tape. The same Comparator file also contains the companion sampling results for tables without cell bounds.
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.Combinatorics.ContingencyTables.UnconditionalMain as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean:29 (same statement text; different proof or body). Qualify references as OAI.ContingencyTables.counting@path.
```
theorem counting : Algorithms.CountingStatement := by sorry
…
```
Mentions: OAI.ContingencyTables.Algorithms.CountingStatement