`ComparatorChallenges/ExactQuantumFactoring.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

28 declarations, 1 contain sorry, 0 declared as axioms

inductive OAI.ExactQuantumFactoring.Primitive The five primitive gates exactly as in Section 2 line 12
Source documentation
The five primitive gates exactly as in Section 2.
L4DEL explanation
The five primitive gates exactly as in Section 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: OAI/Computability/QuantumFactoring/Basic.lean:12 (same declaration text).
```
inductive Primitive where
…
```
def OAI.ExactQuantumFactoring.Primitive.arity line 16
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:16 (same declaration text).
def OAI.ExactQuantumFactoring.Primitive.code line 21
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:21 (same declaration text).
structure OAI.ExactQuantumFactoring.Gate An inverse and/or one added control line 26
Source documentation
An inverse and/or one added control. This is a fixed set of 20 gate names, not an unbounded-control or arbitrary-angle gate model.
L4DEL explanation
An inverse and/or one added control. This is a fixed set of 20 gate names, not an unbounded-control or arbitrary-angle gate model.
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/Computability/QuantumFactoring/Basic.lean:26 (same declaration text).
```
structure Gate where
…
```
def OAI.ExactQuantumFactoring.Gate.arity line 32
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:32 (same declaration text).
def OAI.ExactQuantumFactoring.Primitive.matrix Matrix entries are indexed by lists in wire order (row first, column second) line 36
Source documentation
Matrix entries are indexed by lists in wire order (row first, column second). Entries on ill-sized lists are irrelevant and set to zero.
L4DEL explanation
Matrix entries are indexed by lists in wire order (row first, column second). Entries on ill-sized lists are irrelevant and set 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: OAI/Computability/QuantumFactoring/Basic.lean:36 (same declaration text).
```
noncomputable def Primitive.matrix : Primitive → List Bool → List Bool → ℂ
  | .not, [a], [b] => if a = !b then 1 else 0
  | .cnot, [a,b], [c,d] => if a = c ∧ b = Bool.xor c d then 1 else 0
  | .toffoli, [a,b,c], [d,e,f] =>
      if a = d ∧ b = e ∧ c = Bool.xor f (d && e) then 1 else 0
  | .hadamard, [a], [b] =>
      (if a && b then -1 else 1) / (Real.sqrt 2 : ℂ)
  | .phase, [a], [b] => if a = b then (if b then Complex.I else 1) else 0
  | _, _, _ => 0
```
Mentions: OAI.ExactQuantumFactoring.Primitive
def OAI.ExactQuantumFactoring.Gate.baseMatrix line 46
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:46 (same declaration text).
def OAI.ExactQuantumFactoring.Gate.matrix line 49
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:49 (same declaration text).
structure OAI.ExactQuantumFactoring.Instruction A primitive is placed on distinct actual wires, with no operation of unbounded arity hidden in a circuit instruction line 59
Source documentation
A primitive is placed on distinct actual wires, with no operation of unbounded arity hidden in a circuit instruction.
L4DEL explanation
A primitive is placed on distinct actual wires, with no operation of unbounded arity hidden in a circuit instruction.
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/Computability/QuantumFactoring/Basic.lean:59 (same declaration text).
```
structure Instruction (q : ℕ) where
…
```
abbrev OAI.ExactQuantumFactoring.Basis line 64
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:64 (same declaration text).
abbrev OAI.ExactQuantumFactoring.State line 65
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:65 (same declaration text).
def OAI.ExactQuantumFactoring.Instruction.matrix line 67
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:67 (same declaration text).
structure OAI.ExactQuantumFactoring.Circuit line 75
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:75 (same declaration text).
def OAI.ExactQuantumFactoring.Circuit.apply line 79
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:79 (same declaration text).
def OAI.ExactQuantumFactoring.inputBasis Input wires are first, least significant bit first; all other wires start zero line 83
Source documentation
Input wires are first, least significant bit first; all other wires start zero.
L4DEL explanation
Input wires are first, least significant bit first; all other wires start 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/Computability/QuantumFactoring/Basic.lean:83 (same declaration text).
```
def inputBasis (q ℓ N : ℕ) : Basis q :=
…
```
Mentions: OAI.ExactQuantumFactoring.Basis
def OAI.ExactQuantumFactoring.Circuit.outputState line 86
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:86 (same declaration text).
def OAI.ExactQuantumFactoring.paddedLength The cutoff and padding are the standing parameters in Section 2 line 91
Source documentation
The cutoff and padding are the standing parameters in Section 2.
L4DEL explanation
The cutoff and padding are the standing parameters in Section 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: OAI/Computability/QuantumFactoring/Basic.lean:91 (same declaration text).
```
def paddedLength (ℓ : ℕ) : ℕ := max 128 ℓ
```
def OAI.ExactQuantumFactoring.readBit line 93
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:93 (same declaration text).
def OAI.ExactQuantumFactoring.outputList n entries of n bits each, in the n² wires immediately after the input line 97
Source documentation
n entries of n bits each, in the n² wires immediately after the input.
L4DEL explanation
n entries of n bits each, in the n² wires immediately after the input.
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/Computability/QuantumFactoring/Basic.lean:97 (same declaration text).
```
def outputList {q : ℕ} (ℓ : ℕ) (x : Basis q) : List ℕ :=
…
```
Mentions: OAI.ExactQuantumFactoring.Basis
def OAI.ExactQuantumFactoring.CorrectEncoding Complete nondecreasing prime factorization, multiplicities included, padded by zeros, as in Section 6 line 104
Source documentation
Complete nondecreasing prime factorization, multiplicities included, padded by zeros, as in Section 6. There is no primality oracle in the circuit.
L4DEL explanation
Complete nondecreasing prime factorization, multiplicities included, padded by zeros, as in Section 6. There is no primality oracle in the circuit.
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/Computability/QuantumFactoring/Basic.lean:104 (same declaration text).
```
def CorrectEncoding (N n : ℕ) (xs : List ℕ) : Prop :=
…
```
def OAI.ExactQuantumFactoring.Circuit.correctProbability Unnormalized Born sum on correct outputs; success is exactly one, not positivity, postselection, or a limiting high probability line 110
Source documentation
Unnormalized Born sum on correct outputs; success is exactly one, not positivity, postselection, or a limiting high probability.
L4DEL explanation
Unnormalized Born sum on correct outputs; success is exactly one, not positivity, postselection, or a limiting high probability.
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/Computability/QuantumFactoring/Basic.lean:110 (same declaration text).
```
noncomputable def Circuit.correctProbability (c : Circuit) (ℓ N : ℕ) : ℝ := by
…
```
Mentions: OAI.ExactQuantumFactoring.Circuit
def OAI.ExactQuantumFactoring.natWord Unary natural words with delimiters make the circuit description explicit line 118
Source documentation
Unary natural words with delimiters make the circuit description explicit. Unary wire addresses cause at most polynomial overhead when width is polynomial.
L4DEL explanation
Unary natural words with delimiters make the circuit description explicit. Unary wire addresses cause at most polynomial overhead when width is 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/Computability/QuantumFactoring/Basic.lean:118 (same declaration text).
```
def natWord (k : ℕ) : List Bool := List.replicate k true ++ [false]
```
def OAI.ExactQuantumFactoring.Instruction.encode line 120
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:120 (same declaration text).
def OAI.ExactQuantumFactoring.Circuit.encode line 124
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:124 (same declaration text).
def OAI.ExactQuantumFactoring.Uniform A single finite-alphabet classical generator receives only the INPUT LENGTH line 131
Source documentation
A single finite-alphabet classical generator receives only the INPUT LENGTH. All stack alphabets, states and labels are finite: no large-integer unit-cost oracle is hidden in TM2 push/branch functions.
L4DEL explanation
A single finite-alphabet classical generator receives only the INPUT LENGTH. All stack alphabets, states and labels are finite: no large-integer unit-cost oracle is hidden in TM2 push/branch functions.
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/Computability/QuantumFactoring/Basic.lean:131 (same declaration text).
```
def Uniform (family : ℕ → Circuit) : Prop :=
…
```
Mentions: OAI.ExactQuantumFactoring.Circuit
def OAI.ExactQuantumFactoring.PolynomialResources line 136
Also declared: OAI/Computability/QuantumFactoring/Basic.lean:136 (same declaration text).
def OAI.ExactQuantumFactoring.MainTheorem Literal main endpoint: one length-uniform family, exact output for EVERY binary integer N≥2, and worst-case polynomial gates AND qubits line 143
Source documentation
Literal main endpoint: one length-uniform family, exact output for EVERY binary integer N≥2, and worst-case polynomial gates AND qubits.
L4DEL explanation
Literal main endpoint: one length-uniform family, exact output for EVERY binary integer N≥2, and worst-case polynomial gates AND qubits.
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/Computability/QuantumFactoring/Basic.lean:143 (same declaration text).
```
def MainTheorem : Prop :=
…
```
theorem OAI.ExactQuantumFactoring.exact_quantum_factoring Challenge statement: Exact quantum factoring over a fixed finite gate set sorry line 149
L4DEL explanation
The formalization constructs a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer $N\ge2$ with probability exactly one. The circuits use one fixed finite set of bounded-arity gates, and both the gate count and number of qubits have polynomial worst-case bounds in the bit length of $N$. The probability-one conclusion is exact, rather than an asymptotic or bounded-error guarantee.
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.Computability.QuantumFactoring.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Computability/QuantumFactoring/Main.lean:17 (same statement text; different proof or body). Qualify references as OAI.ExactQuantumFactoring.exact_quantum_factoring@path.
```
theorem exact_quantum_factoring : MainTheorem := by
…
```
Mentions: OAI.ExactQuantumFactoring.MainTheorem