Algorithmic Motives: The Obstruction-Tower Diagram, Motivic Realizations, and the QFT Dictionary Across Fourteen Instantiations

We present the complete Algorithmic Motives (AM) framework, a unifying architecture that maps computational problems into super-Tannakian categories equipped with obstruction towers, motivic realizations, and structural correspondences inspired by quantum field theory. The framework is built on three layers: (1) Obstruction-Tower Diagrams (OTD): filtered towers with Cartan decomposition, involution, contraction structure, and realization filtration that separate continuous (Betti) and discrete (étale) information. (2) Bi-Obstruction-Tower Diagrams (Bi-OTD): paired towers with complementary negligible ideals, contraction mismatch, and bridge conditions governing transfer between realizations. (3) Motivic realizations: chains of fiber functors (Betti, étale, crystalline) whose computational accessibility determines problem complexity. A four-level hierarchy of computational difficulty is defined: Level 0: internal resolution within a single tower Level 1: unconditional bridge between towers Level 2: conditional bridge with realization gap (quantum advantage regime) Level 3: open bridge with interference (κ > 0) We classify fourteen canonical problems across this hierarchy, including lattice reduction, quantum error correction, integer factoring, the Principal Ideal Problem, modular representation theory, KAM dynamics, the Volume Conjecture, and the Collatz, abc, and Riemann hypotheses. The central structural result is the Realization-Switching Pipeline, which shows that Level 2 problems with κ = 0 admit optimal algorithms that dynamically route computation through the cheapest motivic realization. The Betti–Étale Duality Theorem establishes equivalence of total computational cost under dual realizations with inverted bottlenecks. The Complexity-Theoretic Period Conjecture proposes that classical polynomial-time solvability is governed by whether the obstruction lies in the image of the Betti realization. All results are explicitly classified as: Theorem Conditional theorem Structural correspondence Conjecture Heuristic The QFT dictionary is treated strictly as a structural correspondence, providing an organizational framework for identifying shared algebraic mechanisms across arithmetic, topology, and computation.