The exact fourth-vector feasibility wall of the canonical MUB triple in dimension six

For the canonical product MUB triple

{Z₂ ⊗ Z₃, X₂ ⊗ Q₀, Y₂ ⊗ Q₁}

in dimension six, known to admit no vector unbiased to all three bases, we prove an exact two-sided theorem for the projector-space feasibility functional f, where f(v) = 1 would characterize a candidate fourth unbiased vector.

We show that the global maximum of f is

W = (88 + 3√6) / 100 ≈ 0.9534846923.

The bound is attained by an explicit algebraic unit vector, verified entirely in exact arithmetic over the field Q(ζ₂₄, √5). Its symmetry orbit contains 72 distinct maximizing rays.

The matching global upper bound f ≤ W is established via an exact positive-semidefinite certificate at the level-2 Veronese lift. The certificate consists of a symmetric matrix over Q(√2, √3), whose polynomial identity and positivity are machine-verified in exact arithmetic and closed deterministically under the triple's 432-element symmetry group. No randomized computation or floating-point arithmetic appears anywhere in the proof chain.

We further prove, via exact separating functionals, that degree-4 and degree-6 sum-of-squares certificates do not exist. In this precise sense, the lift level employed is minimal and necessary.

Companion results include:

• an exact product-vector ceiling for aligned product triples;

• the exact wall value for the second known triple;

• a complete Weyl-board census in dimension six, together with a Lean 4 kernel-verified combinatorial layer.

These results completely characterize the pinned canonical triple. They do not resolve the MUB(6) problem, which remains open.

The deposit contains two files:

(1) the manuscript (PDF);

(2) a complete verification bundle (ZIP; 91 files) containing all computational modules, tests, evidence receipts, execution logs, a SHA-256 manifest, and a minimal independent verifier (verify_global_certificate.py).

The verifier is implemented in standard-library-only Python, shares no code with the certificate-construction pipeline, loads the stored exact certificate matrix, and independently re-establishes both directions of the global bound in approximately 150 seconds. Reproduction is available via:

make verify-global

Code is released under the Apache License 2.0 (with SPDX headers throughout the bundle). The manuscript text is released under CC BY 4.0.

This research made extensive use of AI systems for conjecture generation, computation, proof drafting, and adversarial review. However, no mathematical claim depends on the correctness of any AI system. Every reported result is verified by exact rational arithmetic, exact interval arithmetic, or the Lean 4 kernel, and all verification artifacts are included in the deposit.