The Rough Path to Complexity: P = Np via Critical Roughness and Path Signatures

This paper completes the ”Millennium Trinity” framework, unifying the Riemann Hypothesis (RH), Navier-Stokes Equations (NSE), and the P vs NP problem under a single geometric principle: Roughness. We present a geometric proof of P ̸= NP by mapping computational complexity classes into the framework of Rough Path Theory. We rigorouslydefine a Control Embedding Map Φ that transforms discrete Turing machine transitions into continuous control paths in a high-dimensional manifold. We establish a rigorous isomorphism between the time complexity T (n) and the Roughness Index α of the optimal trajectory. Crucially, we prove via topological constraints that NP-complete problems correspond to pathswith vanishing roughness α ∼ 1/n → 0, whereas P-class algorithms correspond to smooth paths(α ≈ 1). Using Terry Lyons’ Signature Uniqueness Theorem, we demonstrate that the signature of a path with α ≪ 1/2 cannot be approximated in polynomial time. Thus, P ̸= NPis a geometric consequence of the critical roughness threshold.