Conditional Global Regularity of Navier-Stokes Flow via Discrete Vacuum Geometry

Conditional Global Regularity of Navier-Stokes Flow via Discrete Vacuum Geometry

Abstract

The classical Navier-Stokes equations allow for the theoretical possibility of finite-time blow-up because continuum field theory permits arbitrarily fine localization of energy. We propose that this mathematical singularity is physically precluded by the granular structure of the vacuum. Modeling spacetime as a self-dual F₄ lattice (the 24-cell honeycomb), we demonstrate that the underlying microdynamics impose both a maximum energy density and a minimum localization scale as a consequence of finite local state space and causal propagation. We establish a theorem of conditional global regularity within a physically admissible class of flows, proving that the Navier-Stokes equations possess unique, globally smooth solutions provided the flow remains within the regime defined by the lattice geometry. This work resolves the paradox of singularity formation by identifying it as a breakdown of the effective continuum description rather than a divergence of the physical system.