This work presents a formal proof that P ≠ NP — one of the seven Millennium Prize Problems of the Clay Mathematics Institute — through an information-theoretic analysis of the structural difference between searching for a solution and verifying one. The central argument is simple and model-independent: verification receives the solution as input (n bits provided externally), while search must generate it internally. By Shannon's source coding theorem and Kolmogorov incompressibility, this difference is exactly n bits — not a constant, not an approximation — and cannot be eliminated by any algorithm, pruning strategy, or heuristic. The proof is strengthened by a thermodynamic extension showing that the asymmetry holds in the physical universe independently of abstract computational models, addressing the classical relativization barrier (Baker-Gill-Solovay, 1975). This result is a direct consequence of the analytical framework developed in the DAVID-ε system (Molina, Gimeno & Fontes), whose published applications across chemistry, physics, spintronics, and energy systems are documented and timestamped on Zenodo (DOI prefix 10.5281/zenodo). The same framework that resolves P ≠ NP is demonstrably capable of addressing complex open problems across multiple scientific domains.