Exact quantum factoring over a fixed finite gate set
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Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices. (OpenAI’s Lean note)
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Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
There is a polynomial-time uniform family of polynomial-size quantum circuits that outputs the complete prime factorization of every input integer N\ge2 with probability one. The number of gates and qubits is bounded by a fixed polynomial in the bit length of N. One fixed finite gate set suffices: NOT, CNOT, Toffoli, Hadamard, \operatorname{diag}(1,i), their inverses, and their singly controlled versions.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
We give a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer N ≥ 2 with probability one. A fixed finite set of bounded-arity gates suffices, and both the gate count and the number of qubits have polynomial worst-case bounds in the input length.
One Lean statement in this family
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For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
There is a polynomial-time uniform family of polynomial-size quantum circuits that outputs the complete prime factorization of every input integer N\ge2 with probability one. The number of gates and qubits is bounded by a fixed polynomial in the bit length of N. One fixed finite gate set suffices: NOT, CNOT, Toffoli, Hadamard, \operatorname{diag}(1,i), their inverses, and their singly controlled versions.
The Lean source’s comment reads: /-- Source Theorem 1.1 (thm:main): exact factoring over the fixed gate set,
with a single finite-alphabet polynomial-time length-uniform generator and
polynomial worst-case gate and qubit counts. -/
The Lean statement whose source cites this theorem
The Lean source cites this theorem. No review of whether the statements mean the same thing is recorded.
ExactQuantumFactoring.lean, the release lists it as a supporting statement.
defMainTheorem : Prop := ∃ family : ℕ → Circuit, Uniform family ∧ PolynomialResources family ∧ ∀ (ℓ N : ℕ), 2 ≤ N → N.size = ℓ → ℓ + (paddedLength ℓ)^2 ≤ (family ℓ).qubits ∧ (family ℓ).correctProbability ℓ N = 1⋮theoremexact_quantum_factoring : MainTheorem := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note:exact_quantum_factoring : MainTheorem := ∃ family : ℕ → Circuit, Uniform family ∧ PolynomialResources family ∧ ∀ ℓ N, 2 ≤ N → N.size = ℓ → ℓ + (paddedLength ℓ)^2 ≤ (family ℓ).qubits ∧ (family ℓ).correctProbability ℓ N = 1 over a fixed 20-gate set (NOT/CNOT/Toffoli/Hadamard/phase with optional inverse and one control), where correctProbability is the exact Born probability of outputting the sorted prime factorization with multiplicities (CorrectEncoding) and Uniform uses a Mathlib TM2ComputableInPolyTime generator with finite alphabets. Matches the summary (exact, polynomial gates and qubits, fixed finite gate set).
Definitions the group asks readers to compare with the paper:Hand-built circuit model: Gate := (primitive, inverse, controlled), Primitive.matrix entries on List Bool, Instruction.matrix, Circuit.apply := foldl over instructions; the polynomial-time notion is Mathlib’s TM2ComputableInPolyTime applied to unary input replicate ℓ true with Finite (Γ k), which is standard; the quantum gate semantics are written by hand.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
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.
AI-drafted comparisons, with no recorded human review
Exact quantum factoring over a fixed finite gate setExactQuantumFactoring
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a polynomial-time uniform quantum circuit family over a fixed finite gate set that outputs the prime factorisation of every N ≥ 2 with probability exactly one.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Exact quantum factoring over a fixed finite gate setExactQuantumFactoring
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source
Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.
Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source
Chinese guided reading of this family that compares the manuscripts’ main theorems with their abstracts and flags boundary conditions and limit orders that change the physical reading; drafted with Codex and not a re-proof.
What these papers cite in the release, and what cites them
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Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
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Identify the theorem the catalogue describes
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Address a question raised in an independent assessment
The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state the headline as OpenAI wrote it.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.