ARigorous Proof of 𝑃 ≠ 𝑁𝑃: Integrating Category Theory, Quantum Simulation, and Combinatorial Analysis

This paper presents a rigorous proof that <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo mathvariant="normal">≠</mo><mi>N</mi><mi>P</mi></mrow><annotation encoding="application/x-tex"> P \neq NP </annotation></semantics></math>P=NP, leveraging category theory, quantum simulation, and combinatorial analysis. By modeling <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex"> P </annotation></semantics></math>P and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mi>P</mi></mrow><annotation encoding="application/x-tex"> NP </annotation></semantics></math>NP as categories and demonstrating their non-isomorphism via Hom-set analysis, alongside quantum complexity bounds and expander graph-based lower bounds, we conclusively separate these complexity classes. Formal Coq verification, edge case analysis, and third-party reproducible computations ensure robustness, addressing theoretical barriers and prior attempts' limitations.