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Phys-84-A1 · revision r1 · on Phys-84

OpenAI claim summary for Phys-84

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.

Total Boolean functions are claimed to separate bounded-error randomized and quantum queries by nearly the fourth power, excluding every universal exponent below four. 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. The optimal randomized-versus-quantum query exponent for total Boolean functions
  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

The optimal randomized-versus-quantum query exponent for total Boolean functions

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Total Boolean functions are claimed to separate bounded-error randomized and quantum queries by nearly the fourth power, excluding every universal exponent below four.

Mathematical statement of the claimed result

Result 1.
For total Boolean functions, let $R(f)$ and $Q(f)$ be bounded-error randomized and quantum query complexities. The source claims the optimal universal power is four: \[ R(f)=O((1+Q(f))^4). \] For every fixed integer $k\ge2$, it supplies total functions $F_{k,m}$, for sufficiently large powers of four $m$, such that \[ Q(F_{k,m})\le C_k\sqrt m(\log m)^{b_k},\qquad R(F_{k,m})\ge c_km^{2-1/k}/(\log m)^2. \] Letting $k$ grow excludes any smaller universal exponent.

Outline of the argument

An addressed certificate with outward pointers totalizes a hard partial function. An exact quantum subroutine and certificate checking give the quantum upper bound, while randomized information and pointer constraints force the larger worst-case bit-query cost.

Formal verification

Family 284 has no Lean scope page in this release. Its selected result is not presented as formalized. 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