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Phys-56-A1 · revision r1 · on Phys-56

OpenAI claim summary for Phys-56

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Uniform polynomial-size circuits over one fixed finite bounded-arity gate set are claimed to output every integer factorization with probability one. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Exact quantum factoring over one fixed finite gate set
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Exact quantum factoring over one fixed finite gate set

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Uniform polynomial-size circuits over one fixed finite bounded-arity gate set are claimed to output every integer factorization with probability one.

Mathematical statement of the claimed result

Result 1.
There is a polynomial-time uniform quantum circuit family with $\mathbb P(\text{output is the prime factorization of }N)=1$ for every $N\ge2$. Resources grow polynomially in $\lceil\log_2N\rceil$, with separate bounds for gates and qubits. One fixed finite gate set suffices: \[ \mathrm{NOT},\ \mathrm{CNOT},\ \mathrm{Toffoli},\ H,\ \operatorname{diag}(1,i), \] together with inverses and singly controlled versions. This is exact success, rather than merely success approaching one.

Outline of the argument

Order trials are arranged to have prescribed success probabilities while preserving exact classical records. Circuit phase functions and fraction-label recovery replace arbitrary-angle approximations, and exact amplification plus recursive arithmetic gives a worst-case polynomial factorization family.

Formal verification

For family 279, the selected theorem gives exact probability-one factorization by a polynomially uniform circuit family over a fixed finite bounded-arity gate set. The listed Comparator files are ExactQuantumFactoring.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources