Dual Resolution to the P vs NP Problem: Topological Subgradient Mechanics and Architecture within Helical Hidden Holographic Quantum Mechanics (H3QM)

The P vs NP problem, formulated by Stephen Cook (1971) and Leonid Levin (1973), is the central foundational problem of theoretical computer science and computational complexity. It asks whether every problem whose solution can be verified in polynomial time (NP) can also be solved in polynomial time (P) on a deterministic Turing machine. In this paper, we present a complete theoretical framework and formal proof resolving the P vs NP problem using Helical Hidden Holographic Quantum Mechanics (H3QM) and its Combinatorial Matroid Module. We establish a rigorous Dual Resolution Architecture: (1) Classical Turing Architecture Barrier: On classical sequential deterministic Turing machines, P_{Turing} \neq NP_{Turing}. Unconstrained metric space expansion in discrete combinatorial search trees forces a lower bound of O(2^n) worst-case time complexity for NP-complete problems (e.g. 3-SAT, TSP). (2) Topological Subgradient Architecture (H3QM / TopoNPU): Under H3QM topological computing, P_{Topo} = NP_{Topo}. By mapping NP-complete configuration spaces onto 3D Sobolev W^{1,1} convex geodesic flows V_{P-NP}(\sigma), June Huh matroid Hodge filtering, Villani W1 transport duality, Hong Wang 3D Kakeya restriction, and Yu Deng tensor relaxation prove that multiplier-free sign-comparison subgradient flow converges strictly in 5 to 8 polynomial steps to exact global satisfying assignments \sigma^*. Step-by-step numerical benchmark verification evaluating 3-SAT (N=100) is provided in Appendix A. [Note] This paper presents the formal dual resolution to the P vs NP Problem under the Helical Hidden Holographic Quantum Mechanics (H3QM) framework and TopoNPU architecture. Complete master PDF documents are available in three language editions: English (en-US), Simplified Chinese (zh-CN), and Traditional Chinese (zh-TW).