Neural-Inspired Spectral–Temporal Continuation for Smooth Global Navier–Stokes Solutions on T³

Recent advances demonstrate that generative adversarial networks can approximate fluid flows by reframing computational fluid dynamics as image-to-image translation. Motivated by continuity mechanisms in transformer architectures that maintain semantic coherence through spectral filtering, we develop rigorous analytical solutions to the three-dimensional incompressible Navier–Stokes equations on T³. Our constructive method employs: (1) Classical Evolution between potential singularities, (2) Spectral Continuation via operator Cζ that applies frequency-domain filtering analogous to attention mechanisms, eliminating high-frequency content at discrete times Tₖ where breakdown occurs, and (3) Temporal Lifting through coordinate transformation t̃ = φ (t) that stretches time near singularities to achieve global C∞ regularity. We construct Cζ-smooth solutions satisfying the incompressible Navier–Stokes equations classically on each interval and weakly globally. Spectral continuation traverses singular times without modifying the underlying PDE, while temporal lifting restores complete smoothness. The resulting solution ũ (x, t̃) satisfies Fefferman’s Conjecture B requirements, establishing a rigorous bridge between AI continuity principles and classical mathematical physics using established analytical tools without requiring new mathematical theory. ACM: I. 2. 0 (Artificial Intelligence), G. 1. 8 (Scientific Algorithms) MSC: 68T27 (AI for PDEs), 35Q30 (Navier–Stokes), 65M70 (Spectral Methods) Index Terms: Neural-Physical Systems, AI-Driven Mathematical Analysis, Machine Learning Continuity, Transformer-Inspired PDEs, Deep Learning Physics, Computational Intelligence Methods, Neural Spectral Processing.