Physical Evidence for a Non-Local Gravitational No-Collapse Regularization of Navier-Stokes: 0D and 1D Models with FFT Methodology for 3D

A Regularized Navier-Stokes Model with a Non-Local Gravitational Non-Collapse Principle. Jorge Clavijo Hernández - ORCID: 0009-0001-3855-9593 - A Coruña, Spain - July 2026 We present a definitive version of a regularized incompressible Navier-Stokes system with a non-local dissipative term of gravitational type on T³. The system is ∂_t u + P[(u·∇)u + ε u Φ_eff] = ν Δu, Φ_eff = |u|² * K_δ, K_δ = 1/(|x|+δ), where P is the Leray projector. We correct the previous energy identity: ∫ P(uΦ_eff)·u = G_δ = ∫∫ |u(x)|²|u(y)|²/(|x-y|+δ) ≥0, yielding d/dt E + ν||∇u||² + ε G_δ =0. For fixed ε≥0, δ>0 we prove global existence of Leray-Hopf weak solutions via Galerkin and Aubin-Lions. The family is uniformly bounded in L∞L² ∩ L²H¹ and converges as ε→0 to a classical Leray-Hopf solution, with ε∫G_δ→0. Critical scaling is intentionally broken. We provide an integral no-collapse bound ε∫_0^T G_δ ≤ E(0) and discuss the open pointwise no-collapse problem. 3D pseudo-spectral DNS (32³, 64³, 128³) of Taylor-Green vortex shows gradient saturation (10-12% reduction of max|∇u| at 32³, mean 1.5% reduction of ||∇u||² at 64³) and faster energy decay, consistent with G_δ as an extra sink. This work does NOT claim a solution to the classical Millennium problem; it is a regularized model in the spirit of Leray-α.[u] Author's Note:This work does not arise from a desire to challenge mathematical complexity out of mere intellectual pride, but from a deep contemplation of the harmony within universal design. The persistence and stability of vortices in nature, from the microscopic scale to galactic structures, remind us that the cosmos operates under a perfect order that rejects the chaos of infinite collapse. The limitations of our current mathematical models do not reflect an error in creation, but rather the limit of our own human understanding, as expressed in the Book of Job: "Where were you when I laid the earth's foundation? Tell me, if you have understanding. Who marked off its dimensions? Surely you know!" (Job 38:4-5). This regularized model is a humble attempt to sketch a mathematical map that acknowledges the physical limits inscribed in that original wisdom.