Separation of the P and NP Complexity Classes via Quiver Cohomology

Title: Proof of the Separation of the P and NP Complexity Classes via Quiver Cohomology Author: Charles EDOU NZE (Independent Researcher, [email protected]) Abstract This memoir presents an unconditional proof of the separation of the computational complexity classes P and NP, establishing that P ≠ NP. By transposing 3SAT propositional formulas into the geometric framework of subspace arrangements and the algebraic setting of quiver representations, we prove the existence of an unavoidable topological obstruction. To each formula Φ, we associate a quotient quiver algebra ΛΦ whose second Hochschild cohomology HH2(ΛΦ) governs the infinitesimal deformations of stable representations. We prove that the unsatisfiability of Φ is equivalent to the non-vanishing of a global obstruction class [θΦ] ∈ HH2(ΛΦ). Through the study of the wild representation type of the algebra and the singularities at the boundary of the moduli space of semi-stable representations, we show that vanishing or detecting this obstruction class requires a super-polynomial number of algebraic operations. The proof bypasses the three classical barriers (relativization, naturalness, algebrization) by exploiting the intrinsically non-natural and global character of the homological invariants of subspace intersections. Machine-Checked Formal Verification The core exponential circuit lower bound and quiver cohomological entropy barriers have been formally verified in the Lean 4 interactive theorem prover (via Mathlib) with zero axioms, zero linter warnings, and zero sorry placeholders (module: PvsNPQuiverEntropy.lean). Interactive Web Showcase & Research Repository: https://maths-proofs.edounze.com | GitHub: millennium- prize-problems