发表机构
IUFFyM, Universidad de Salamanca（萨拉曼卡大学 IUFFyM）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立谱对偶结构（SDS）的代数与范畴基础，证明其构成分次幺半范畴，明确对象分类方式，关联Fisher-Rao几何，揭示其作为自主数学对象的性质。
AI 中文摘要
本文建立了谱对偶结构（SDS）的代数与范畴基础。我们证明所有SDS构成一个分次幺半范畴，其中次数K是标记分层的基本不变量。我们定义了结构的笛卡尔积与不交并，以及保持对合和权重的态射，并证明C* = 1/(1+√K)是在每个分层上为常值的函子。对象按同构由组合类型(k,f)、次数K及反演类的多重集[r]={r,1/r}分类。我们进一步证明响应秩恰好等于非平凡对k的数量，并将该范畴结构与姊妹论文中发展的Fisher-Rao几何相联系。SDS最初在随机重置问题中被识别，如今作为具有丰富代数结构与自然几何实现的自主数学对象出现。
英文摘要
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.
Comments26 pages
