Energy Identities and a Conditional Enstrophy Estimate for a Hodge-Theoretic Viscous Flow Model on a Compact Cylinder

We introduce a four-dimensional scale-augmented covariant viscous flow model on 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 positive-definite geometry uniformly equivalent to the flat product metric. We formulate the flow equations using Hodge-de Rham operators and impose absolute boundary conditions to isolate the exact vorticity subspace from the absolute harmonic kernel. Under an explicitly assumed vortex-stretching absorption inequality and compatible impermeable boundary conditions, we derive a conditional Gr"onwall bound on the enstrophy. This note does not derive the nonlinear stretching estimate from first principles and does not resolve global regularity for the classical three-dimensional Navier-Stokes equations.