Axioms and internal rigidity of spectral duality structures

Abstract

Spectral duality structures (SDS) explain why a distinguished family of initial distributions makes the absorption probabilities of resetting processes insensitive to the resetting mechanism. Here we study their axioms. We prove that the six defining conditions are mutually independent and determine what each one contributes. The key tool is a structure theorem valid for an arbitrary permutation and an arbitrary positive weight: every insensitive distribution is supported on a union of cycles balanced at a common level, and the level of a cycle is fixed by the geometric mean of the weight on it. From it, positivity selects the admissible root, the constant orbit product is equivalent to the existence of an insensitive distribution of full support, and the involution is equivalent to the response rank being equal to the number of free weight parameters. Without the duality relation, insensitivity of a single distribution is cheap and generically isolated; duality turns it into structural neutrality of the separatrix. The full-rank condition splits into two layers: a weaker condition is equivalent to maximal response rank, while full rank also excludes insensitivity at any other level. We then show that a single SDS is internally rigid: compatible semigroup structures force a constant weight, compatible abelian group structures obey a precise arithmetic condition, congruences form a simple lattice, and the automorphism group is computed explicitly. Finally, the axioms do not determine the orientation of the response; in the two-site case we give the exact condition under which the geometric and analytic sign criteria agree.