Algorithmic Motives: Obstruction Towers, Super-Tannakian Reconstruction, and a Dictionary for Analytic Number Theory

This paper introduces Algorithmic Motives, a framework that extracts geometric and representation-theoretic structure from iterative computational processes. The central construction—the motivic extraction procedure—feeds obstruction towers (computational systems with tensor structure and involution) into Nori’s diagram category, detects a canonical Z/2Z-grading, and applies Deligne’s theorem to reconstruct a motivic Galois supergroup. The resulting category is super-Tannakian: C Repₐ (G, ), where the even sector encodes operations visible to the algorithm, while the odd sector encodes the intrinsic kernel of its observables (the negligible ideal), which is annihilated by all even-equivariant functionals. This structure is not imposed externally; it is forced by the failure of tensor-ideal closure under the involution. The main structural result is that any obstruction tower with tensor product and involution naturally generates a supergroup symmetry. This encoding is a theorem—the Translation Engine—providing canonical transport of structure into a superrepresentation category. The associated quantum field theory (QFT) terminology is interpretive; all structural results are formulated and proved in representation-theoretic terms. Applied to the Goldbach sieve tower, the framework produces unconditional structural results: The exact Cartan spectrum \0, 1, 4\ A categorical derivation of the Hardy–Littlewood singular series Identification of the parity barrier as confinement within the even Lie algebra sector Additional results include a Newton Contraction Theorem establishing doubly exponential suppression of the negligible ideal in Newton-type towers, and a classification of obstruction regimes across arithmetic, dynamical, and computational systems. The framework applies broadly to systems with iterative refinement, including sieve theory, quantum error correction, KAM theory, and optimization algorithms. In each case, the negligible ideal identifies the algorithm’s intrinsic blind spot, while odd derivations provide the symmetry-breaking mechanism required to bypass it.