Algebra of spectral duality structures

Abstract

This paper establishes the algebraic and categorical foundations of spectral duality structures (SDS). We show that the class of all SDS admits the structure of a graded monoidal category, in which the degree K is the fundamental invariant indexing the strata. We define the Cartesian product and the disjoint union of structures, together with morphisms preserving the involution and the weights, and we prove that C* = 1/(1+sqrt(K)) is a functor constant on each stratum. Objects are classified up to isomorphism by the combinatorial type (k,f), the degree K, and a multiset of inversion classes [r] = {r, 1/r}. We further prove that the response rank equals exactly the number of non-trivial pairs k, and we connect the categorical structure with the Fisher-Rao geometry developed in a companion paper. The SDS, originally identified in problems of stochastic resetting, thus emerges as an autonomous mathematical object with a rich algebraic structure and a natural geometric realisation.