Foundations of the Algorithmic Motives Framework: The Universal Functor, the κ-Classification, the Spectral Operator, and the Comparison Cost

This paper develops four foundational constructions underlying the Algorithmic Motives (AM) framework. First, it constructs a canonical functor Φ : C^_K → Rep(G_Kᵐᵒᵗ, ε) from a rigorously defined category of cyclotomic diagrams—projective systems of finite subsets of class groups with norm-compatible transition maps—to the super-Tannakian category of representations of the motivic Galois group. The construction proceeds via a linearization functor, the Kummer embedding into Nori’s diagram category, and Tannaka reconstruction. Second, it formalizes the κ-obstruction as an invariant of the underlying Nori motive rather than of a specific algorithm. It proves invariance under motivic equivalence, monotonicity under polynomial-time reduction, and a model-dependent quantum query lower bound in the Boolean oracle model. It also analyzes the role of κ in the classical post-processing stage of known quantum algorithms for arithmetic problems. Third, it realizes the OTD Hamiltonian as a closed, densely defined operator on a Gel’fand triple and shows that its regularized trace recovers a form of the Weil explicit formula. This gives the spectral side of the framework a rigorous operator-theoretic interpretation. Fourth, it introduces a quantitative comparison-cost theory for switching between Betti and étale realizations, including explicit bounds and a composition theorem for sequential realization switches. The paper is intended as a foundations paper for the AM program: it makes explicit several constructions used implicitly in companion papers and clearly distinguishes between proved results, model-dependent statements, and conjectural extensions.