LRSC Odd-Ring Spectral Rank Theorem: Exact Channel-Space Rank for Near-Nyquist Odd Cyclic Systems — v1.2

This release presents an analytic family theorem for the single-step near-Nyquist LRSC operator on odd cyclic lattices. For odd M >= 9, if the binary mask is reflection-symmetric and has at least one inactive reflection orbit, the conjugate-balanced post-stencil channel matrix V has rank exactly equal to p, the number of active reflection orbits. The proof proceeds through four components: an exact closed form for the local shedding field; residue-class formulas proving non-vanishing of the required Fourier coefficients for M = 4L + 1 and M = 4L + 3; a Chebyshev/Vandermonde argument establishing pre-stencil independence; and injectivity of the circulant stencil on the masked channel span. A computational verification companion corroborates the algebra by checking 500,494 Fourier-coefficient comparisons over all odd M from 9 through 2001, together with exact cyclotomic non-vanishing checks for M = 99, M = 101, and M = 127. These finite checks are corroborative and do not replace the analytic proof. The result is a structural rank theorem. It does not claim a universal fixed-unit compression number K_tau, a microscopic physical law, or a multi-step dynamical theorem. Author: Prince Upadhyay Affiliation: Independent Research License: CC BY 4.0