Daniel Schoch

In general relativity, the strong equivalence principle is underpinned by a geometrical account of fields on spacetime, by which all fields and bodies probe the same geometry. This geometric account implies that the parallel transport of all spacetime tensors and spinors is dictated by a single affine connection. No similar account of gauge theory is put forward by standard textbooks, which use principal bundles to coordinate the parallel transport of different, interacting particles. Nonetheless, here I argue that gauge theory does afford such a geometric account, obviating the need for principal bundles.

Recent advances demonstrate that generative adversarial networks can approximate fluid flows by reframing computational fluid dynamics as image-to-image translation, and 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: Classical Evolution between potential singularities, 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 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, where spectral continuation traverses singular times without modifying the underlying PDE while temporal lifting restores complete smoothness, and 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 Meth- ods, Neural Spectral Processing.