Separation of P and NP

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 ≥ α_d; 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.