Paper 100: Holosphere Resolution of the Navier–Stokes Regularity Problem A Physically Motivated Derivation of Global Regularity from Angular Coherence Constraints

We present the Holosphere–Navier–Stokes (HNS) system, a coherence-governed upgrade layer for incompressible Navier–Stokes derived from a discrete SU(2) angular-strain lattice. The model is controlled by a fixed triad: a hard angular cap smax⁡=πs_{\max}=\pismax=π, a finite healing rate γ\gammaγ with coherence time τcoh=γ−1\tau_{\mathrm{coh}}=\gamma^{-1}τcoh=γ−1, and a dropout threshold θbreak\theta_{\mathrm{break}}θbreak for loss of long-range transport. These inputs define a strain-rate governance scale Ssat=smax⁡/τcohS_{\mathrm{sat}}=s_{\max}/\tau_{\mathrm{coh}}Ssat=smax/τcoh and motivate two continuum controls: bounded stress saturation (amplitude control) and a strictly dissipative fourth-order term (curvature control). On the periodic torus we prove global well-posedness using a monotone-operator/Galerkin compactness framework and derive an exact dissipation identity with classical viscosity, saturation dissipation, and bi-Laplacian hyperviscosity. We establish O(ε)O(\varepsilon)O(ε) convergence to classical Navier–Stokes on any interval where the classical solution remains smooth and within the low-strain envelope, yielding a coherence-conditional regularity criterion based on bounds relative to SsatS_{\mathrm{sat}}Ssat. Pseudospectral benchmarks (2D shear layer and 3D Taylor–Green vortex) exhibit bounded strain, monotone energy decay, and negligible dropout over the simulated windows. This work does not resolve the Clay Millennium problem for unmodified 3D Navier–Stokes; it shows that global regularity follows for the coherence-constrained HNS closure while recovering Navier–Stokes in the smooth, low-strain regime.