Boundary Matching Multiplicity from Orthogonal Signings: Determinant Bounds, Stability, and Refined Cube Certificates

Let W be a Hermitian matrix supported on a graph G and satisfying W² = αI. Across a cut V(G) = S ⊔ T, the identity BB* = αI − A² gives a rank certificate. This manuscript develops local Hall-profile bounds via principal Gram matrices and quadratic effective rank, an elementary Cauchy–Binet determinant-to-matching transfer yielding explicit lower bounds on saturating boundary matchings, an equivalent Gaussian-entropy formulation, and perturbative stability for approximately flat Hermitian edge weightings. For Boolean cubes, it establishes a hierarchy of degree, local two-step, defect-matrix, and signed spectral certificates. The accompanying source archive includes the LaTeX manuscript, bibliography, README, revision notes, reproducible Python verification code, and exact recorded output for exhaustive verification through Q₄. The mathematical proofs do not depend on the computation. AI disclosure: OpenAI ChatGPT/Codex (GPT-5 family) assisted with exploratory proof development, literature searching, finite verification, consistency checks, English editing, and LaTeX preparation. The AI system is not an author; the human author assumes responsibility for the content.