Butson BH(6,8) — no fourth mutually unbiased basis

Unsealed Prediction

{ "P1": { "claim": "ZERO quadruples exist among the 3 BH(6,8) classes \u2014 i.e. no 4th mutually unbiased basis of 8th-root type accompanies any of them", "confidence": "~85% \u2014 every closed-grid census in this family since 2007 has returned zero quadruples, and Zauner's conjecture (max 3 MUBs in C^6) predicts it", "falsification": "ANY quadruple found refutes P1 and would be a major result \u2014 a 4th MUB in dimension 6 of Butson type" }, "P2": { "claim": "at least one BH(6,8) class admits >=1 unbiased BASIS (a triplet), by analogy with the 12th-root case where 3 of 11 classes did", "falsification": "all three classes yielding zero bases refutes P2" }, "context": { "calibration_passed": "2026-08-12: reproduced Bengtsson et al. 2007 census EXACTLY at 12th roots [4,1,1,0x8] with 0 quadruples, and at 24th roots [4,4,1,1,0x7] with 0 quadruples, using exact arithmetic in Z[zeta_12]/Z[zeta_24]. A lift bug (base-12 exponents read as base-24) was caught BY this gate and fixed.", "why_open": "Chen-Yu exclusion chain invalidated April 2025 (McNulty-Weigert, J.Phys.A 58 168001); these cells reverted to numerical evidence only. Complete BH(6,k<=17) classification exists since 2020 (Lampio-Ostergard-Szollosi) with no exclusion work run against it." }, "engine": "euler_shadow/butson_mub.py, exact cyclotomic arithmetic, calibration-gated", "scope_honesty": "This is the CLOSED-GRID version: companion bases restricted to 8th roots. The companion-UNRESTRICTED theorem is the publishable target and requires Grobner/exact methods beyond this census. Do not conflate them when reporting.", "sealed": "2026-08-12T02:27:07-0500", "target": "Butson-restricted MUB exclusion in dimension 6 at k=8: can any of the 3 BH(6,8) equivalence classes appear in a set of 4 mutually unbiased bases in C^6, with companion bases restricted to 8th roots of unity?" }

Verification Notes

SPLIT — one claim confirmed with a machine-checkable certificate, one refuted. P1 (zero quadruples) CONFIRMED and certified: all 57 basis pairs across the three BH(6,8) classes are certified NOT mutually unbiased by exact rational interval arithmetic, 0 unresolved. See the accompanying certificate. P2 (at least one class admits an unbiased basis) REFUTED: predicted by analogy with the 12th-root case, where 3 of 11 classes did. In the closed-grid run the eighth-root classes yielded 8 companion vectors and zero bases. The honest headline is elsewhere. This computation reproduces Brierley & Weigert, Phys. Rev. A 79, 052316 (2009): the three BH(6,8) classes ARE the Diţă matrices, studied under that name since ~2004, and their per-matrix counts (120+48+48 companions, 10+4+4 bases, 45+6+6 pairs) equal ours exactly. We searched for the Butson name of an object the literature indexes under its Diţă name. A vocabulary gap, not a knowledge gap. What survives is the certificate, not the result. CAVEAT ADDED 2026-08-15: the pipeline's 24th-root calibration leg fails (counts all zero vs expected [4,4,4,1,1,0,...], unexplained since 2026-08-12); the 12th-root leg passes. The k=8 counts independently match Brierley-Weigert (2009) to the unit, which is the stronger check, but the calibration regression is material and is disclosed on the publication record.