bibkey: openai2026maximum authors: OpenAI year: 2026 title: The maximum number of mutually unbiased bases in dimension six doi: null url: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026/README.md claim: The fixed-seed completion section derives a positive compressed Gram defect, a uniform trace bound, and necessary flatness inequalities; these local statements do not by themselves establish the manuscript’s global computational conclusion. strata_touched: [] license: citation-only triage: anchor
The maximum number of mutually unbiased bases in dimension six
The manuscript is dated September 24, 2026 and credited to OpenAI in its supplied citation. No DOI is supplied there. The immutable source revision is adc7f1241b42e322a6451854ab7e4b4c146bf78a.
Local input used
The completion section fixes orthonormal complementary frames at a nine-coordinate dephased seed center. The center need not itself have an exact Hadamard completion. For every feasible completion its compressed block has a positive semidefinite defect 6I-Y*Y. The section obtains a seed-ball trace upper bound, converts it to a polar-factor radius, and uses that radius in norm and directional necessary conditions.
Section 40 of the existing MUB theory retains this model and derives a smaller uniform trace estimate from the rank-one coordinate supports forced by dephasing. The source’s polar conversion and its block-defect identity are literature inputs; the new coefficient estimate is a separate derivation, not a claim that the cited paper contains it.
Scope
This citation supports the stated local completion identities and necessary-condition construction. It does not certify all prescribed seed indices, all recursion branches, archived execution data, or the global claim about the maximum number of bases. The source explicitly requires successful processing with no failed seeds or unresolved cells before drawing its global coverage conclusion.