Geometric Regularization and Conditional Regularity for a Scale-Augmented Covariant Hodge-Navier-Stokes Model

We introduce a four-dimensional scale-augmented covariant fluid model embedded in a compact cylinder M̄ = T³ × [0,1]. By applying a regular conformal modification using the Gamma function to the background product metric, we establish a strictly positive-definite, non-degenerate geometry with uniformly bounded curvature. We utilize a Hodge-de Rham formulation for the viscous dissipation and enforce an energy-compatible absolute boundary realization to derive exact, closed energy identities for both the velocity 1-form and the exact vorticity 2-form. Finally, by verifying the trivial intersection of the exact vorticity subspace with the absolute harmonic kernel, we establish a topological spectral gap. This yields a formal conditional regularity criterion: subject to a solution-uniform stretching bound, the fluid's enstrophy cannot blow up in finite time, preserving global regularity.