Machine-checkable certificates for mutually-unbiased-basis exclusion in dimension six: an independent reproduction of Brierley–Weigert (2009) for the Diţă class BH(6,8)

WHAT THIS IS. An independent reproduction, with a strengthened certificate, of a result first obtained by Brierley and Weigert in 2009. It is not a new theorem. We state that at the top because we did not know it when we ran the computation, and the discovery that we had reproduced rather than discovered is the more useful half of the work.

RESULT. For each of the three equivalence classes of Butson-type Hadamard matrices BH(6,8), no fourth mutually unbiased basis accompanies the class in C⁶. Companion vectors are permitted anywhere on the unit circle, not merely at roots of unity: we find 216 companions (120 + 48 + 48 by class), of which 18 form genuine unbiased bases (10 + 4 + 4), and none of the 57 resulting basis pairs (45 + 6 + 6) is mutually unbiased.

METHOD, WHICH IS THE ACTUAL CONTRIBUTION. The companion variety is encoded as a polynomial ideal over ℚ with the real and imaginary parts of ζ_q adjoined and pinned by the minimal polynomial of 2cos(2π/q), so no per-q coefficient table is needed. msolve (F4/FGLM) proves the variety zero-dimensional. Every candidate basis pair is then certified NOT mutually unbiased by exact rational interval arithmetic: for each pair we identify a witness pair of vectors u, v and compute a rational enclosure of |⟨u,v⟩|² that provably excludes 6. All 57 pairs are certified; 0 are unresolved. Interval arithmetic is sound only in this direction — an enclosure that misses 6 proves NOT mutually unbiased, while an enclosure containing 6 would prove nothing and would be reported as unresolved.

WHAT THE ATTACHED ARTIFACT CONTAINS, EXACTLY. The certificate published with this record lists, for all 57 pairs, the two bases as index sets, the witness pair as indices, and the computed enclosure of |⟨u,v⟩|² with the exclusion verdict. It does NOT yet contain the companion vectors those indices refer to, which means a reader cannot at present recompute the inner products from the file alone — they can check our arithmetic is internally consistent, not that it is right. A self-contained version carrying each witness vector's component-wise rational enclosures is being generated and will replace this file; only then will the certificate be independently re-checkable with no solver and no trust in our code. That property is the whole point of the work, so it is stated here as not-yet-delivered rather than assumed. (Caught in pre-publication audit, 2026-08-12.)

STATUS UPDATE 2026-08-15: the self-contained replacement described above has NOT been generated in the three days since. The promise stands as an intention, not as work in progress, and the certificate as published remains internally-checkable only. It should not be cited as a referee-checkable artifact until that file exists. Separately, note that the enclosures are serialised in the file as decimal strings (e.g. "[10.5298,10.5298]"), not as exact rationals; the interval arithmetic was carried out over the rationals, but the published serialisation does not preserve them, so the file cannot be replayed at full precision as-is.

CALIBRATION STATUS — ADDED 2026-08-15, AND IT CUTS BOTH WAYS. The pipeline that produced these numbers has a two-leg calibration. The 12th-root leg PASSES: 11 BH(6,12) classes recovered as expected, triplet-basis counts [4,1,1,0,...], 0 quadruples. The 24th-root leg DOES NOT: on 2026-08-12 it returned counts [0,0,0,0,0,0,0,0,0,0,0] against an expected [4,4,4,1,1,0,0,0,0,0,0], and that discrepancy is unexplained as of this writing. It was logged as a WARN by a job that then exited 0, which is why it did not block anything automatically — a defect in our harness, recorded rather than hidden. The 24th-root leg is a DIFFERENT parameter from the k=8 work reported here, so it does not directly bear on these numbers; but it is a failure of the instrument that produced them at a neighbouring setting, and a reader is entitled to know that before weighting our arithmetic. WHAT PARTIALLY OFFSETS IT: the k=8 figures reported above — 216 companions, 18 unbiased bases, 57 pairs (and their per-class splits 120/48/48, 10/4/4, 45/6/6) — reproduce Brierley & Weigert (2009) to the unit. That is external agreement with a refereed result, which is stronger evidence for these particular numbers than any self-calibration we could run. Read the result as: externally corroborated at k=8, on a pipeline with a known unexplained failure at q=24.

RELATION TO PRIOR WORK. Brierley & Weigert, Phys. Rev. A 79, 052316 (2009), §4.1, obtained the same counts by numerical Gröbner methods at ~20-digit precision, reporting for D(0) that sixty of the vectors form ten bases, none of them mutually unbiased. Their per-class numbers agree with ours to the unit. The three BH(6,8) classes are the Diţă matrices D₆(0), D₆(±1/8), studied under that name since roughly 2004. Our search missed this because we looked for the Butson designation of an object the literature indexes under its Diţă designation — a vocabulary gap rather than a knowledge gap, and one that a forward-citation sweep of twenty-two works did not close because the exclusion predates the 2020 Butson classification by eleven years.

WHAT IS NEW, STATED NARROWLY. Only the certificate, and only once it is self-contained. Brierley and Weigert did not present their result as mere numerics — they argued it rigorously with explicit error bounds, which is the same logical device as interval arithmetic. The narrow claim available to us is therefore about FORM, not rigour: an exclusion certificate a third party can re-check mechanically, without reproducing the Gröbner computation, is a useful artifact, and the 2026 review of MU bases in composite dimensions asks in print (its Problem 10.6) for rigorous proofs to replace numerical evidence of unextendibility. It is not a new mathematical fact.

SCOPE LIMIT. The method decides one matrix, or a finite set of them. It cannot decide a continuous family: K₆⁽³⁾ has three free parameters and no finite ideal describes all of its members at once. Any claim we make about a family is a claim about the specific parameter values we ran.

SEALED IN ADVANCE, AND A DISTINCTION THAT MATTERS. Two predictions were hashed before the computation began, and both were about the CLOSED-GRID problem — companion vectors restricted to eighth roots of unity. In that restricted setting the first was confirmed (zero mutually unbiased quadruples) and the second refuted: it predicted that at least one class would admit an unbiased basis, and at eighth roots the three classes yielded 8 companion vectors and zero bases.

The 216 companions and 18 bases reported above come from the UNRESTRICTED problem, where companions may sit anywhere on the unit circle. The two settings are not variants of one computation; they give wildly different objects, and every one of the 18 unrestricted bases has irrational phases that no root-of-unity search could see. Conflating them would make this abstract contradict itself, so: 18 bases exist unrestricted, zero exist on the closed grid, and neither fact is evidence about the other. Both seals are in the vault against their original hashes.