An Exact Bravais-Shell Identity for Coordination-Normalized Spectral Anisotropy

This preprint proves an exact closed-form identity for the coordination-normalized spectral anisotropy coefficient of any full-rank Bravais lattice in arbitrary spatial dimension, evaluated on its distinct directed nearest-neighbor shell. The main result shows that the Brillouin-zone averaged spectral anisotropy coefficient equals a dimension-coordination baseline plus an exact correction determined by the second-moment isotropy defect of the nearest-neighbor direction shell. Consequently, the entire deviation from the dimension-coordination floor is exactly the second-moment defect. Equality holds precisely when the shell has isotropic second moment. The simple cubic, body-centered cubic, and face-centered cubic nearest-neighbor classes follow as direct corollaries with coefficients 1/3, 1/4, and 1/6 respectively. The corresponding two-dimensional coefficients are 1/4 for the square shell and 1/6 for the six-direction hexagonal/triangular shell. The ordering result of the earlier two-dimensional comparison is retained, while the absolute coefficient normalization is corrected by the trace-square identity proved here. The proof is self-contained and proceeds through reciprocal-torus orthogonality, a cosine-product pair average, and two trace-average identities. The manuscript also proves exactness of midpoint reciprocal-torus quadrature from a finite Fourier-support bound. A separate appendix analyzes finite direction shells and spherical 4-design saturation, while explicitly separating those finite-shell diagnostics from Brillouin-zone coefficients. Basis-indexed supports such as HCP-like layered structures are excluded from the single-symbol Bravais formalism pending a basis-aware extension. The paper is intended as a standalone result in periodic discrete operator geometry and as a formal operator-geometric component of the UGM_02 programme. It does not assign an energetic interpretation, construct a Hamiltonian, or promote the coefficient to a dynamical functional.