arXiv2026
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 symmetric monoidal category, where the degree $K$ is the fundamental invariant indexing the strata. We define operations of Cartesian product and disjoint union, morphisms preserving involution and weights, and prove that $C^\ast=1/(1+\sqrt K)$ is a functor constant on each stratum. We classify objects up to isomorphism by the combinatorial type $(k,f)$, the degree $K$, and a multiset of inversion classes $[r]=\{r,r^{-1}\}$. In each degree, skeletons are exactly the finite $C_2$-sets over the additive group of the reals with $C_2$ acting by $t\mapsto -t$, which explains the monoidal structure and identifies the classification with the decomposition into orbits. Furthermore, we prove that the response rank is the rank of an explicit matrix built from the amplitudes and the weights, and hence exactly the number of non-trivial pairs $k$ under the full-rank condition, and we connect the categorical structure with the Fisher--Rao geometry of Paper V. The SDS, originally discovered in stochastic resetting problems, is thus an autonomous mathematical object with a rich algebraic structure and a natural geometric realisation.