Nonorthogonal Nagaoka Equality Supports under Bayesian Weak Commutativity

This preprint studies equality conditions for the Bayesian Nagaoka–Hayashi bound in multiparameter quantum estimation. For an explicit two-parameter, finite-prior, full-rank model in Hilbert-space dimension four over Q(√2), it proves that Bayesian weak commutativity can coexist with two nonorthogonal real-axis Nagaoka equality-support lines of a global NH minimizer. The result follows from a mixed z–z̄ equality-support matrix, exact determinant elimination, root isolation, cofactor geometry, and a reverse-KKT construction, with an eight-test exact replay. A block KKT lift embeds the optimizer geometry in every dimension d ≥ 4. The certified pair cannot occur together in an equality POVM using at most four distinct one-dimensional support lines; the result does not exclude overcomplete POVMs or establish unrestricted NH nonattainability or a strict Bayesian risk gap.