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Phys-23-A2 · revision r1 · on Phys-23

OpenAI claim summary for Phys-23

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 at unsolvedmath.com: partially solved. 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.

The theorem rules out the pair (d,k)=(6,7), but does not characterize all allowed pairs in non-prime-power dimensions. The theorem determines the existence pairs only at d=6, giving N(6)=3. The problem statement asks existence pairs across all dimensions, especially every non-prime-power dimension. This entry records partial support; complete coverage of the question is not established by this catalogue.

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

Manuscript

Contents
  1. Mutually Unbiased Bases in Non-Prime-Power Dimensions
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Result 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Mutually Unbiased Bases in Non-Prime-Power Dimensions

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Order-six Hadamard matrices outside the Tao class are claimed to satisfy the designated Fourier vanishing identities; exact certificates also exclude seven unbiased bases and imply an upper bound of five; The maximum MUB count in complex dimension six is claimed to be three, conditional on complete execution of the documented certified-arithmetic pipeline. Together they provide the following support within the scopes described below.

Mathematical statement of the claimed result

Result 1.
The assertion excludes seven pairwise mutually unbiased orthonormal bases of $\mathbb C^6$. More explicitly, there are no bases $B_r=(b_{r,0},\ldots,b_{r,5})$, $1\le r\le7$, with \[ |\langle b_{r,i},b_{s,j}\rangle|^2=1/6 \qquad(r\ne s,\ 0\le i,j\le5). \] It does not by itself determine the maximum number below seven.
Result 2.
For pairwise mutually unbiased bases, $|\langle b,c\rangle|^2=1/6$ whenever $b,c$ belong to different bases of $\mathbb C^6$. The source reports a computer-assisted exclusion of four such bases and concludes \[ N(6)=3, \] where $N(d)$ is their maximal number in dimension $d$. The result is explicitly conditional on the appendix's arithmetic/compiler requirements and the reported complete execution; those computational requirements are not omitted from the claim.

Outline of the argument

Phase-invariant characters are rewritten as injective moments. Exact algebraic identities separate the Tao class, and integer/rational certificates verify the required moment inequalities; the supplied verifier and the completion theorem yield the stated MUB consequence.

An exhaustive seed cover is refined into certified phase balls. Completion estimates within each ball exclude two additional bases, and finite recorded execution covers all arbitrary complex configurations under the stated binary64 and compiler conditions.

Formal verification

For family 266, the selected statements give Fourier cancellation outside the Tao class and a bound of five on the MUB family, with a further Hadamard-ratio cancellation lemma. They do not establish the claimed maximum of three or certify the binary64 four-basis exclusion. The listed Comparator files are HadamardCubeFiber.lean, MUBSix.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