Rank versus signature at a fermionic defect: counting, control, and observables

We study the exact counting of zero modes bound to a codimension-one fermionic defect with matrix-valued mass. The number of modes interpolates between a Callias-type topological index, fixed solely by the asymptotic signatures, and a larger aligned value attained only on measure-zero strata in the space of profiles. We prove a structure theorem for the alignment map that sends each mass profile to its pair of Jost subspaces: the map is controllable by compactly supported deformations, every selection rule is a kernel statement, and the fragile matrix of degenerate perturbation theory is precisely the first two orders of its derivative. The same geometric object governs the energies and ordering of pair annihilation, the finite-size floor of protected modes, the spectral pump under twist, and the codimension law of the disorder-induced band foot. Rank and signature become independent observables, refined further by a measurable lattice of protected kernels. The counting theorem applies to the vacuum manifold of its own order parameter and, within a reduced-channel model, classifies the Kibble–Lazarides–Shafi walls of polar-distorted \(^3\)He-B as minimal signature walls.