A Formal and Spectral view of P ≠ NP via Decision Tree Lower Bounds and Coq Verification

This work provides a rigorous and formally verified proof that P ≠ NP by combining three independent methods: decision tree complexity, cyclic XOR-SAT hardness, and spectral analysis. We show that any algorithm solving SAT must explore an exponential number of paths, rendering polynomial-time solutions impossible. A new class of XOR-SAT problems is introduced that systematically resists heuristic and probabilistic solvers. Spectral graph theory is used to demonstrate that the Laplacian eigenvalues of NP problems diverge exponentially, unlike those of P problems. The proof is implemented and formally verified using the Coq proof assistant, ensuring mathematical rigor and logical completeness. Accompanying simulations confirm the expected gap in complexity between NP and P instances. This result resolves one of the most fundamental open problems in theoretical computer science. Included are two documents:– A human-readable exposition of the argument– A machine-verifiable Coq formalization of the complete proof