Physical Regularization of Navier--Stokes Blow-Up via Nonlinear Vacuum Response

Physical Regularization of Navier–Stokes Blow-Up via Nonlinear Vacuum ResponseVersion 3.14 (September 2026). Supersedes version 1 (December 2025) and withdraws its separate technical appendix. The paper considers the three-dimensional incompressible Navier–Stokes equations augmented by a nonlinear damping term −λeff|u|αu, α ≥ 2, motivated as the response of the physical vacuum, treated as a medium, to extreme fluid motion. The coupling is built from fundamental constants, λ = G/c³, rather than tuned. The paper states precisely what is known about the global regularity of the modified system, corrects the dimensional analysis of the coupling, and validates the term numerically with an open-source package released alongside. What is claimed A physical motivation for the damping term and for building its coefficient from G and c. The identification of the modified system with the convective Brinkman–Forchheimer (damped Navier–Stokes) equations of the PDE literature, exponent r = α + 1. Known results: global strong solutions exist for α > 2 and every λeff > 0 (Cai & Jiu 2008; Zhou 2012), and for the critical case α = 2 under the coefficient condition 4νλeff ≥ 1 (Hajduk & Robinson 2017). Under the Navier–Stokes scaling the α = 2 term scales exactly like the viscous term: it is critical, and a small coefficient does not change the scaling class of the equation. A corrected dimensional analysis: λ = G/c³ has dimension s·kg⁻¹; the identity λ = 1/(8πΞc) of version 1 is false (that quantity is G/c⁵); Ξ = c⁴/(8πG) is a force, not a pressure; and λρ = Gρ/c³ has dimension s·m⁻³ whereas the equation requires s·m⁻², so the SI form of the coupling is one power of length short. This missing length is stated as an open problem. Numerical validation with moss-reg v0.2.1: an exact operator-splitting substep for the damping ODE, a symmetric particle integrator, a dealiased 2D spectral solver verified against the exact Taylor–Green decay (relative error 5×10⁻¹⁵) and the energy identity (residual 4×10⁻⁷), a self-validation command with 25 PASS/FAIL checks, and a resolution × timestep × kernel × softening × speed-of-light sweep of a 1D gravitational-collapse benchmark. The sweep shows that the cubic term acts as a speed limiter with terminal velocity vterm = c·(g/Gρ)1/3, delays shell crossing by about one particle spacing divided by c, and leaves the collapse unchanged whenever the free-fall speed is below c. What is not claimed (erratum to version 1) No proof of global regularity for the physically motivated case α = 2 with small λeff. The proof of version 1 used a non-scale-invariant interpolation inequality and is withdrawn; the case is open, and for the physical coefficient the condition 4νλeff ≥ 1 fails by roughly thirty-six orders of magnitude. No statement about the Clay Millennium Problem, which concerns the unmodified equations. No arrest of shell crossing at physical parameters. The arrest shown in version 1 was a finite-resolution effect at a speed of light of 0.005 code units (a collapse superluminal in the model's own units), compounded by a double-damping bug in release 0.1.0 of the software. No parameter-free stabilizer for engineering CFD: with the physical coefficient the term is inert in every regime where classical fluid dynamics is valid (damping rate ≈ 10⁻²⁷ s⁻¹), and a coefficient large enough to act is a tuned regularization. Numerical blow-up of a discretization is not the mathematical blow-up of the equations. The Moss Soliton is kept as a conjecture, and the connection to the Theory of Temporal Spheres is kept as a clearly marked speculative appendix, with the identification Ξ = EP/ℓP³ of version 1 corrected (EP/ℓP³ = c⁷/(ħG²) is a different quantity). Open physical questions listed in the paper: the missing length ℓ in λeff = Gρℓ/c³, the rest frame relative to which the damping acts, and the origin of the exponent α. Files Physical_Regularization_of_Navier-Stokes_Blow-Up_via_Nonlinear_Vacuum_Response_v3.14.pdf — the paper (20 pages). Supplementary_Numerical_Material_v3.14.pdf — generated by moss-reg supplement from the machine-readable report.json of the software release: provenance (commit, environment, command lines), the full validation table, every parameter and metric of every run, the complete sweep grid and the sensitivity tables. No number in it is typed by hand. Software moss-reg v0.2.1 (MIT), github.com/psldm/moss-regularization. One-command reproducibility: moss-reg validate (25 checks: dimensional formulas, exact damping solution, integrator regression, convergence orders, kernels, Jacobian monitor, Taylor–Green decay, divergence, energy identity, dealiasing), moss-reg compare --type {particles,fluid} (classical vs. moss, figure plus report.json with commit hash, environment, parameters, metrics and checks), moss-reg sweep, moss-reg run-all, moss-reg supplement. The paper's numbers were produced at commit 33851d4 (assets regenerated at 8692f44). Changes relative to version 1 Title kept; subtitle "critical-exponent regularization, numerical validation and open problems" added. Section "Resolution of the Millennium Problem" removed; the Planck analogy kept as a statement of intent in the Discussion. Global-regularity theorem replaced by the literature results with sources and by an explicit erratum. Dimensional table corrected (Ξ in N, λ in s·kg⁻¹, λeff with the missing length marked open). Shell-crossing figure replaced by the full sweep and by a physical-regime comparison (compactness 0.1). CFD section rewritten around what the term can and cannot do. The separate technical appendix of version 1 is withdrawn entirely and replaced by the generated supplementary material. ---Ξυα Mσςς[email protected]