Intensity-Adaptive Vorticity-Direction Coherence at Viscous Scales in the 3D Navier–Stokes Equations

Geometric depletion arguments for the three-dimensional Navier–Stokes equations relate vortex stretching to the coherence of the vorticity direction near intense-vorticity regions. The individual ingredients are established, but their intensity-indexed composition is less explicit: how should a local directional coherence coefficient be compared when the observation radius itself follows the vorticity intensity? We formulate a finite-scale framework based on the response Ω ↦ r_c(Ω) ↦ K(Ω) ↦ χ(Ω), where r_c(Ω) = c·sqrt(ν/Ω) and χ(Ω) = r_c(Ω)^γ · K(Ω). Here K is a local γ-Hölder coefficient for the vorticity direction on an intensity-selected region. We prove that χ is exactly the unit-radius Hölder coefficient of the spatially rescaled direction field, characterize it through a multiscale angular profile, and isolate structural monotonicity generated by nested radii and nested absolute thresholds. We then derive the neutral scale-compensation benchmark p_c = γ/2 for a finite-range response model K(Ω) ~ Ω^p, with p_c = 1/4 when γ = 1/2, and show that a sub-neutral asymptotic bound forces projective directional coherence on the adaptive neighborhoods. A near-field trilinear estimate rewrites the classical angular-depletion mechanism directly in terms of χ. The framework removes the deterministic Hölder homogeneity associated with the chosen radius law, but it does not remove the structural change of the admissible pair set as the radius changes. No new regularity criterion or proof of depletion is claimed. The resulting question is finite-scale and testable: does the normalized directional response remain sub-neutral, neutral, or super-neutral under controlled changes of intensity, threshold, and radius convention?