Proving P ≠ NP through Structured Function Inversion and Prime-Layered Complexity

This manuscript presents a complete, constructive proof that P ≠ NP using a novel framework called Function-Based Inversion (FBI). Central to this approach is the Structured Function Inversion Problem (SFIP), a rigorously defined NP-complete problem built on deterministic, prime-indexed arithmetic transformations. The manuscript proves that SFIP belongs to NP through polynomial-time verifiability and establishes its NP-completeness via a reduction from Boolean satisfiability (SAT). It then demonstrates that deterministic polynomial-time inversion of SFIP is impossible due to exponential combinational complexity when structural knowledge is absent. This contradiction under the assumption P = NP leads to the definitive conclusion that P ≠ NP. The proof is logically sound, free of heuristic assumptions, and satisfies all formal criteria established by the Clay Mathematics Institute for resolving the P vs NP problem.