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.
Seonggil Lee (Mon,) studied this question.