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Mutually unbiased bases in dimension six

From OpenAI Aftermath

Mutually unbiased bases in dimension six concern the largest number N(6) of pairwise mutually unbiased orthonormal bases of the complex space of dimension six, the first dimension that is not a prime power. Zauner's conjecture asserts that N(6) = 3. A manuscript dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims N(6) = 3 by a computer-assisted proof, with a companion manuscript on order-six complex Hadamard matrices. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

  • As recounted in the introduction of the main manuscript:
    • Two orthonormal bases are mutually unbiased when every vector of one has squared overlap 1/d with every vector of the other. In every dimension d there are at most d + 1 such bases, and Wootters and Fields constructed complete sets in prime-power dimensions. In dimension six the tensor-product construction gives three bases.
    • Zauner's conjecture that three is the maximum in dimension six goes back to his 1991 diploma thesis and his doctoral thesis; Klappenecker and Rötteler stated N(6) = 3 as a conjecture. Grassl showed that the standard and Fourier bases cannot belong to four mutually unbiased bases, numerical searches by Butterley–Hall and Brierley–Weigert supported the conjecture, and Jaming, Matolcsi, Móra, Szöllősi and Weiner excluded extensions of the two-parameter Fourier family by a certified discretization, followed by a programme for arbitrary second bases.

The original results

  • Exactly three bases (24 September 2026). The main theorem, labelled computer-assisted, states that under the arithmetic and compiler conditions of an appendix, a complete execution documented in the manuscript certifies that no four pairwise mutually unbiased orthonormal bases exist in dimension six, so that N(6) = 3.
    • Method, as described: after normalization the second basis is a complex Hadamard matrix. Its phases are covered by finitely many seed balls, each ball is completed in blocks with certified displacement estimates, and two further bases are excluded uniformly on each ball by a finite search with rigorous error bounds in binary64 arithmetic.
  • Companion. A second manuscript of the same date, Exact Fourier certificates for complex Hadamard matrices of order six, claims a proof of the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture for order-six complex Hadamard matrices outside Tao's cubic equivalence class.
  • Sources. Commit 1581d4c of the openai/math repository, folders The maximum number of mutually unbiased bases in dimension six and Exact Fourier certificates for complex Hadamard matrices of order six.
    • The verification section of the main manuscript refers to an accompanying verification directory with source files, their digests and the records of the full run. The manuscript folder at commit 1581d4c contains the README, the PDF and the LaTeX sources only.
  • Formalization, as described by the repository. The scope note lean/docs/266.md states that the formalization proves a weaker family bound, that every family in its model has at most five members, together with a Fourier character-sum vanishing statement for order-six complex Hadamard matrices not equivalent to the Tao matrix and a supporting cube-fiber cancellation lemma. It states that the selected statement does not establish the upper bound of three or the computer-assisted exclusion of four bases. The Comparator statements are MUBSix and HadamardCubeFiber.
  • Kernel records. In the con-leche development journal's run over all 416 Comparator challenges (repository commit fd4aeeb, run of 10 October 2026; DESIGN.md, task #348), con-leche accepted MUBSix while Lean's official kernel reached its one-hour time limit.
  • Catalogue status. In Formal Conjectures, the open quantum problems entry on mutually unbiased bases (OpenQuantumProblems/13.lean) marks the bound of at most five bases in dimension six as solved, with a formal proof link to the OpenAI Lean file, and keeps the statement that the maximum is three open.

Public follow-up work

Date Work Relation Public claim Public status (checked 17 October 2026)
2026-10-08 H.-Y. Sun, arXiv:2610.12180 (v1), titled Mutually unbiased bases: common states, completion, and extension obstructions Related work Studies common unbiased states, their completion and further extension. In dimension six it proves strong unextendibility of mutually unbiased triples whose third-basis rays are invariant under a ternary clock. The paper cites the two OpenAI manuscripts of 24 September as context and states that neither is a premise of its independent proof. arXiv v1.
2026-10-16 JJardine919/openai-math-audit (GitHub; J. Jardine, Lattice24; archived at Zenodo 10.5281/zenodo.23422995), Finding 2 Critique (verifiability gap) Reports that the upper bound depends on one recorded execution whose cited code and run records are not in the public release, so it cannot be checked from what has been published. It describes this as a verifiability gap, not a demonstrated mathematical error, reports no error in the written mathematics of Sections 2 to 6, and states that releasing the verification directory and the run archive would close it. Part of a completed audit of the collection's manuscripts without Lean; one refutation pass kept the severity at major.
2026-10-16 JD Jones, JD-Jones-ASES/finite-quantum-designs-lean, registered in the Palomar registry as PALOMAR-2026-10-16-000014 Related formalization A Lean development of mutually unbiased bases and SIC-POVMs in the vocabulary of Formal Conjectures: at most d + 1 mutually unbiased bases in dimension d, complete sets in prime-power dimensions, three bases in every dimension at least 2, and in dimension six the bounds 3 ≤ N(6) ≤ 7 and N(6) ≠ 6. The registry record lists OpenAI's MUBSix statement of at most five bases as independent work and states that the bound of five is not proved there. Palomar registry, version 1; the record lists kernel checks by nanoda and con-ron.

Notes

  • The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
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