A Piecewise Smooth Weak Solution to the 3D Navier–Stokes Equations on the Torus via Spectral Continuation

We introduce a new mathematical framework for constructing global piecewise smooth weak solutions to the three-dimensional incompressible Navier–Stokes equations on the torus T3. The method is based on the Delta-Zeta Algorithm, a spectral continuation technique that extends classical solutions beyond potential singularities using a zeta-inspired continuation operator. At empirically determined vorticity thresholds where the L-infinity norm of omega exceeds a fixed maximum (‖ω (t) ‖∞ > ωₘax), the velocity field is continued using an exponential mode-dependent weight: gammadelta (n) = (1 + exp (a * |n|ᵖ / delta) ) ^ (-1), with a > 0 and p > 1. This continuation produces smooth post-threshold states while preserving divergence-free structure and energy stability. Although inspired by the decay behavior of the Riemann zeta function evaluated at complex argument (1/2 + i * |n| * lambda * nuᵍamma), the method avoids explicit zeta-function evaluation and thus circumvents complications related to the Riemann Hypothesis. We establish global vorticity boundedness, uniform energy control across continuation points, and weak solution compatibility under recursive spectral continuation. The resulting velocity field is smooth on open time intervals and globally weak on T3 × [0, ∞), providing a lawful analytic extension through potential singularities. This work presents the first constructively realized piecewise smooth weak solution to the 3D Navier–Stokes equations in the Delta-continuation class, with implications for high-Reynolds-number flows and machine-learned threshold control. MSC: 68T27 (AI for PDEs), 35Q30 (Navier–Stokes), 65M70 (Spectral Methods) ACM: I. 2. 0 (Artificial Intelligence), G. 1. 8 (Scientific Algorithms) Index Terms: Zeta-Modulated Regularization, Spectral Decay, Navier–Stokes Global Regularity, Divergence-Free Flow. (Revised 7-25-2025 from "Global Smooth" to "Piecewise Smooth")