Spectral collapse and algebraic exclusion of the μ5 branch in the d = 6 mutually unbiased bases problem

This manuscript provides a complete algebraic proof excluding the μ_5 branch—the most challenging generic branch in the longstanding open problem of constructing a third mutually unbiased basis (MUB) in dimension d=6 (related to Zauner's conjecture). The proof introduces a novel algebraic framework based on the Schur-closure identity of the Gram matrix, yielding a fundamental factorized master equality. Through rigorous phase-alignment reduction and subresultant elimination, we demonstrate that the regular physical regime inevitably forces a "spectral collapse". This mechanism mathematically annihilates the Vandermonde determinant, strictly driving the physical phase vector to the trivial zero vector and yielding an absolute contradiction. Furthermore, we provide explicit Cramer coefficients, boundary-lifting lemmas, and a self-contained numerical witness certificate to ensure full mathematical traceability. Our results definitively show that the μ_5 branch carries no non-degenerate physical solution, establishing a major milestone toward the full algebraic resolution of the d=6 MUB problem.