Holonomy-First Navier–Stokes Regularity: A Geometric Proof via the Davis Field Equations

This repository contains the full manuscript, figures, and supporting materials for Holonomy-First Navier–Stokes Regularity, a geometric proof of global regularity for the three-dimensional incompressible Navier–Stokes equations formulated within the Davis Field-Equation framework. The work introduces a new governing relation—the Davis Law—which constrains the rate at which a dynamical system may traverse configuration space by coupling a tolerance budget to a geometric curvature barrier. In this framework, capacity represents the maximum admissible rate of semantic progress, while curvature represents the geometric cost of changing structural features of a fluid state. Rather than analyzing Navier–Stokes solutions pointwise, the framework organizes the space of velocity fields into semantic basins defined by an augmented cache consisting of: a finite holonomy (circulation) signature, and a vorticity-scale coordinate derived from the maximum vorticity magnitude. At a fixed tolerance, these quantities partition state space into basin interiors with guaranteed separation margins, yielding a coarse but geometrically meaningful discretization of fluid behavior. The proof architecture proceeds in four steps: Holonomy-first discretizationA fixed circulation skeleton induces a finite-dimensional holonomy signature. Together with the vorticity scale, this defines semantic classes of fluid states at finite resolution. Curvature barrier for basin transitionsAny transition between distinct basin interiors requires accumulation of a strictly positive geometric barrier. This result relies on two explicit bridge postulates that define the compatible Riemannian geometry on state space:(i) vorticity magnitude controls circulation deformation along state-space paths;(ii) any positive change in vorticity scale incurs positive integrated barrier cost. Blowup forces infinite refinementBy the Beale–Kato–Majda criterion, finite-time singularity implies divergence of the vorticity scale. Continuity of the augmented cache then forces the trajectory to pass through infinitely many basin interiors before the putative blowup time. Capacity–time contradictionThe Davis Law imposes a minimum physical time cost for each basin transition. Infinitely many transitions therefore require infinite time, contradicting finite-time blowup. The resulting conclusion is global regularity of Navier–Stokes solutions under the Davis Field-Equation axioms. Importantly, this work does not attempt to derive the Davis Law or the bridge postulates from classical Navier–Stokes analysis. They are stated explicitly as additional governing structure, analogous to adopting Einstein’s equations in general relativity or the Yang–Mills action in gauge theory. All classical inputs (local well-posedness and the Beale–Kato–Majda criterion) are isolated in a minimal analytic interface. To ensure scientific accountability, the repository includes: an explicit falsifiability section describing concrete conditions under which the framework would be invalidated; a gap ledger enumerating all axioms, classical results, and bridge postulates; computational evidence demonstrating consistency with established Navier–Stokes phenomenology, including BKM behavior, Kolmogorov scaling, and the Arnold curvature–enstrophy correspondence. This work is intended as a foundations-level contribution proposing a new geometric law governing admissible fluid evolution. Evaluation of the result therefore hinges on acceptance or rejection of the Davis Field-Equation framework itself, not on hidden analytic assumptions. Keywords: Navier–Stokes, global regularity, geometric analysis, holonomy, vorticity, curvature, Beale–Kato–Majda, fluid topology, axiomatic physics, field equations, nonlinear PDE