THE P VS NP PROBLEM An Entropy-Minimization Resolution via Computational Harmonics and Hanners Theorem

Preprint submitted to the Journal of the ACM (JACM). This manuscript presents a resolution of the P vs NP problem—a Clay Mathematics Institute Millennium Prize problem—by applying a framework termed Computational Harmonics (CH), anchored by the Fundamental Theorem on Computational Harmonics (FTCH) and Hanners Theorem. The P vs NP problem asks whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time (Cook, 1971; Levin, 1973).

Harmonic Coherence, formulated within quantum gauge theory and generalized through Hanners Theorem, asserts that systems evolve toward discrete equilibrium states characterized by entropy minimization. By adapting these conditions to computational state spaces, we derive computational harmonic stability (CHS) criteria: a problem is in class P if and only if its state space admits a harmonic equilibrium (entropy gradient zero, positive-definite entropy Hessian). We demonstrate analytically and via computational validation that canonical NP-complete problems—Boolean satisfiability (SAT), the Traveling Salesman Problem (TSP), and the Hamiltonian Path Problem—violate CHS conditions, thus establishing P ≠ NP under the CH framework. Validation uses entropy-guided algorithms and standard benchmarks (SATLIB, TSPLIB). The resolution provides a foundation applicable to cryptographic security, quantum complexity, and optimization theory.

The document is a formal preprint intended for peer review, submitted to the Journal of the ACM (JACM), and is part of the Harmonic Coherence publication ecosystem.