`ComparatorChallenges/RegularParity.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

16 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.QAC.Bit line 10
Also declared: ComparatorChallenges/QACParity.lean:10 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:10 (same declaration text).
abbrev OAI.QAC.Word line 11
Also declared: ComparatorChallenges/QACParity.lean:11 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:11 (same declaration text).
abbrev OAI.QAC.Operator line 12
Also declared: ComparatorChallenges/QACParity.lean:12 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:12 (same declaration text).
abbrev OAI.QAC.QubitOperator line 13
Also declared: ComparatorChallenges/QACParity.lean:13 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:13 (same declaration text).
def OAI.QAC.tensorMatrix Tensor product in the computational basis line 16
Source documentation
Tensor product in the computational basis.
L4DEL explanation
Tensor product in the computational basis.
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/QACParity.lean:16 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:16 (same declaration text).
```
def tensorMatrix {N : ℕ} (a : Fin N → QubitOperator) : Operator N :=
…
```
Mentions: OAI.QAC.QubitOperator, OAI.QAC.Operator
inductive OAI.QAC.Gate A one-qubit unitary or an unbounded-arity Toffoli gate line 20
Source documentation
A one-qubit unitary or an unbounded-arity Toffoli gate.
L4DEL explanation
A one-qubit unitary or an unbounded-arity Toffoli gate.
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/QACParity.lean:20 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:20 (same declaration text).
```
inductive Gate (N : ℕ) where
…
```
def OAI.QAC.Gate.support line 26
Also declared: ComparatorChallenges/QACParity.lean:26 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:26 (same declaration text).
def OAI.QAC.Gate.toffoliWord A Toffoli flips its target iff all controls are one line 31
Source documentation
A Toffoli flips its target iff all controls are one.
L4DEL explanation
A Toffoli flips its target iff all controls are one.
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/QACParity.lean:31 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:31 (same declaration text).
```
def toffoliWord {N : ℕ} (c : Finset (Fin N)) (t : Fin N) (x : Word N) : Word N :=
…
```
Mentions: OAI.QAC.Word
def OAI.QAC.Gate.matrix line 34
Also declared: ComparatorChallenges/QACParity.lean:34 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:34 (same declaration text).
structure OAI.QAC.PhysicalLayer line 40
Also declared: ComparatorChallenges/QACParity.lean:40 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:40 (same declaration text).
def OAI.QAC.physicalLayerMatrix line 44
Also declared: ComparatorChallenges/QACParity.lean:44 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:44 (same declaration text).
def OAI.QAC.physicalCircuitMatrix Chronological order: the rightmost matrix acts first line 48
Source documentation
Chronological order: the rightmost matrix acts first.
L4DEL explanation
Chronological order: the rightmost matrix 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: ComparatorChallenges/QACParity.lean:48 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:48 (same declaration text).
```
def physicalCircuitMatrix {N : ℕ} (layers : List (PhysicalLayer N)) : Operator N :=
…
```
Mentions: OAI.QAC.PhysicalLayer, OAI.QAC.Operator
def OAI.QAC.inputWord The computational-basis input with the remaining qubits initialized to zero line 52
Source documentation
The computational-basis input with the remaining qubits initialized to zero.
L4DEL explanation
The computational-basis input with the remaining qubits initialized to 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: ComparatorChallenges/QACParity.lean:52 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:52 (same declaration text).
```
def inputWord {n N : ℕ} (x : Word n) : Word N :=
…
```
Mentions: OAI.QAC.Word
def OAI.QAC.parity line 55
Also declared: ComparatorChallenges/QACParity.lean:55 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:55 (same declaration text).
def OAI.QAC.successProbability Born probability, summing all final garbage with no cleanliness assumption line 58
Source documentation
Born probability, summing all final garbage with no cleanliness assumption.
L4DEL explanation
Born probability, summing all final garbage with no cleanliness 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: ComparatorChallenges/QACParity.lean:58 (same declaration text), OAI/InformationTheory/QuantumCircuit/Model.lean:58 (same declaration text).
```
def successProbability {n N : ℕ} (W : Operator N) (out : Fin N) (x : Word n) : ℝ :=
…
```
Mentions: OAI.QAC.Operator, OAI.QAC.Word
theorem OAI.QAC.parity_lower_bound_polynomial_size Challenge statement: Polynomial-size parity specialization sorry line 63
Source documentation
Fixed-depth circuits with polynomially many total qubits cannot compute parity with worst-case success at least two thirds.
L4DEL explanation
Fixed-depth circuits with polynomially many total qubits cannot compute parity with worst-case success at least two thirds.
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.InformationTheory.QuantumCircuit.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/InformationTheory/QuantumCircuit/Main.lean:12 (same statement text; different proof or body). Qualify references as OAI.QAC.parity_lower_bound_polynomial_size@path.
```
theorem parity_lower_bound_polynomial_size (d k C : ℕ)
    (_hd : 0 < d) (_hk : 0 < k) (_hC : 0 < C) :
    ∃ n₀ : ℕ, ∀ n ≥ n₀, ∀ N : ℕ, n ≤ N → N ≤ C * (n + 1) ^ k →
      ∀ layers : List (PhysicalLayer N), layers.length ≤ d →
        ∀ out : Fin N, ∃ x : Word n,
          successProbability (physicalCircuitMatrix layers) out x < (2 : ℝ) / 3 := by
…
```
Mentions: OAI.QAC.PhysicalLayer, OAI.QAC.Word, OAI.QAC.successProbability, OAI.QAC.physicalCircuitMatrix