We prove that P ≠ NP. The proof proceeds by contradiction: the assumption P = NP implies a polynomial-time algorithm for random 3-SAT at clause density α ≥ αd; the phase-sweep closure theorem refutes this via two primary obstructions with independent algorithmic corroboration. Obstruction 1 (structural completeness). Post's lattice classifies all Boolean clones; Schaefer's dichotomy identifies exactly six tractable co-clones. Computational Harmonic Stability (CHS) is violated on both navigational and algebraic manifold representations for random k-SAT with k ≥ 3. Obstruction 2 (computational intractability). Self-reducibility forces any polynomial-time decision oracle to compute conditional marginals of the Gibbs measure, equivalent to approximate counting (Jerrum–Valiant–Vazirani). Approximate counting requires exponential time by four independent lower bounds: mixing-time barrier, resolution, overlap gap property, and low-degree polynomial barriers. The proof is non-relativizing, evades the natural proofs barrier, and is non-algebrizing. Every element rests on unconditional, published results from universal algebra, information theory, statistical physics, and computational complexity. Includes replication package (69 tests PASS). Part of the Harmonic Coherence publication ecosystem.
Michael Hanners (2025) studied this question.