Spectral protection and the geometry of recognition polytopes

Abstract

This work develops the spectral geometry associated with the recognition principle established in the first paper of this series. From the normalized recognition function R(H,K) = m(H,K)/N, where m(H,K) denotes the maximum size of a compatible matching between uniform linear histories, we introduce the Recognition Polytope Pₖ as the convex set of symmetric k × k matrices with unit diagonal whose entries satisfy the subadditivity constraints induced by Recognition Geometry. We prove that every realizable recognition matrix belongs to Pₖ, making this polytope an outer approximation of the realizable set Rₖ. We then study the relationship between Pₖ and the cone Cₖ of positive semidefinite matrices. We prove that P₃ is contained in C₃ and identify a phenomenon of Spectral Protection in a symmetric cyclic four-point family, where the combinatorial boundary coincides with the spectral boundary. Finally, we construct explicitly an Anomalous Hexagon, a matrix belonging to P₆ but not to C₆, by exhibiting a negative eigenvalue. This result demonstrates that local recognition constraints are insufficient to characterize the global realizable structure. We therefore formulate the Recognition Obstruction Principle and identify open problems concerning the characterization of Rₖ, positive semidefiniteness of the recognition kernel, and the discovery of additional global constraints.