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

Abstract

This supplementary dossier provides numerical validation for the regularized Navier-Stokes model ∂_t u + P[(u·∇)u + ε u Φ_eff] = ν Δu, Φ_eff = |u|² * K_δ, K_δ = 1/(|x|+δ), G_δ = ∫|u|²Φ_eff ≥0, with corrected energy identity d/dt E + ν||∇u||² + ε G_δ =0 (Lemma 1). We include: • Fig.1 (0D toy): ω' = ω² - νω - ε ω³, blow-up for ε=0 at t≈0.9 vs bounded saturation ∼1/ε for ε>0. • Fig.2 (1D non-local Burgers, N=256, ν=0.02, δ=0.1): Classical ε=0 shock formation vs regularized ε=0.1 gradient saturation (log scale). • Fig.3-4 (3D, 32³ pseudo-spectral, Taylor-Green vortex, ν=0.02, δ=0.05): FFT-based Φ_eff = |u|² * 1/|x| via K̂=4π/|k|², O(N log N), 2/3 de-aliasing, RK4. 10-12% reduction of max|∇u| and faster decay of E_reg vs classical, direct verification of G_δ≥0 as extra sink. • Fig.5-6 (64³): E(t), ||∇u||² proxy, and G_δ(t). Mean 1.5% reduction of ||∇u||² on [0,0.28] increasing linearly, consistent with ε∫G_δ. • Fig.7 (128³ REAL, 6-step verification): E(t) and G_δ(t) for ε=0 vs ε=0.1, confirming G_δ≥0 and faster energy decay per corrected identity. This is a verification run, not a full E(k) spectrum run. Costs: 0.02 s/step at 32³, 0.18 s at 64³, 1.6 s at 128³ (A100 40GB, single precision cuFFT). The dossier supports the integral no-collapse bound ε∫_0^T G_δ ≤ E(0). Full pointwise no-collapse remains open. This regularized system intentionally breaks critical scaling and does not claim to solve the classical Millennium problem. 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 inherent harmony of universal design. The persistence and stability of vortices in nature 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 limits of our human understanding, as expressed in the Book of Job: "Where were you when I laid the earth's foundation? Tell me, if you understand. Who marked off its dimensions? Surely you know!" (Job 38:4-5). This model is a humble attempt to sketch a mathematical map that acknowledges the physical limits inscribed in that original wisdom.