The Physical Blueprint of P \neq NP: Topological Information Erasure and the Geometric Resolution of Circuit Complexity

This manuscript presents a deterministic, physical framework aimed at resolving the conceptual core of the P versus NP problem. By transitioning Boolean circuit complexity from purely abstract combinatorics into the realm of topological physics and thermodynamic information erasure (Landauer's principle), I demonstrate that the polynomial-time inversion of One-Way Functions (OWFs) is physically and geometrically prohibited. Utilizing the phase-locking mechanisms of the \Phi-ND model—previously applied in my derivations of Navier-Stokes regularity and macromolecular formation—I establish that deterministic logic gates inherently destroy information, manifesting a strict topological mass gap. Consequently, attempting to reverse this computational process requires an exponential explosion in circuit size to mathematically reconstruct eroded informational states. This manuscript provides the extended logical and geometric derivation of this phenomenon, including the matrix mechanics of inverse graphs and the decoupling of circuit depth from entropy. While I leave the final algebraic and combinatorial translation to the broader mathematical community, this framework formulates the rigorous physical lower bound necessary to bypass the Razborov-Rudich barrier, concluding from a topological perspective that P \neq NP.