SMT-Vol7 & STCT-Vol3: The Rough Path to Complexity: P = Np via Critical Roughness and Path Signatures

This paper completes the next phase of the Seonggil Rough Operator Algebra(ROA) framework, extending the unified geometric principles that recently resolved the Riemann Hypothesis (RH) and the 3D Navier-Stokes Equations (NSE). We present a rigorous geometric proof of P ̸ = NP by mapping discrete computational complexity classes into the continuous framework of Rough Path Theory, utilizing Seonggil Matrix Theory (SMT) and Seonggil Tensor Calculus Theory (STCT). We define a measurable Control Embedding MapΦ that translates discrete Turing machine transitions into continuous controlled rough paths within a high-dimensional Riemannian manifold. We establish a strict analytic isomorphism between the algorithmic time complexity T(n) and the Roughness Index α (related to pvariation, where α = 1/p) of the optimal computational trajectory. Crucially, we prove viaGromov-Hausdorff limits that mapping NP-complete problems requires navigating a fragmented, topologically constrained solution space, enforcing a macroscopic oscillation where the roughness index vanishes as α ∼ 1/n → 0. In contrast, P-class algorithms correspond to smooth gradient flows with bounded variation (α ≈ 1). Utilizing Terry Lyons’ Signature Theorem, we demonstrate that the infinite-dimensional signature tensor of a path with vanishing roughness cannot be computed or approximated by any polynomial-time deterministic process. Thus, P̸ = NP is established as an intrinsic geometric consequence of thecritical roughness threshold.