Intensity-Adaptive Vorticity-Direction Coherence at Viscous Scales: Framework and a Size-Calibrated Measurement in Isotropic Turbulence

Geometric depletion arguments for the three-dimensional Navier–Stokes equations relate vortex stretching to the coherence of the vorticity direction near intense-vorticity regions. At an intense event, however, three quantities vary simultaneously: vorticity magnitude, the spatial scale associated with that magnitude, and the directional organization observed within that scale. Comparing a local Hölder coefficient at two different radii can therefore mix a true geometric response with the deterministic change induced by measuring the same type of quotient in different physical coordinates. This paper first formalizes that composition and then measures it. Selecting the observation radius from the inverse-square-root viscous scale r_c(Ω) = c·√(ν/Ω) and evaluating a local γ-Hölder coefficient K(Ω) of the vorticity direction there, we prove that the normalized quantity χ = r_c^γ·K is exactly the unit-radius Hölder coefficient of the spatially rescaled direction field, and characterize it through a multiscale angular profile. The response model K ~ Ω^p then carries the neutral scale-compensation benchmark p_c = γ/2, equal to 1/4 at the one-half exponent of the directional-coherence criteria: growth at exactly that rate is what the shrinking ball produces by itself. We isolate the structural monotonicities generated by nested radii and nested absolute thresholds, and rewrite a classical near-field angular-depletion estimate directly in terms of χ. We then report a measurement of the finite-range response exponent on two independently selected populations of intense-vorticity events (|ω|/ω_rms ∈ [2.0, 8.6]) from a single snapshot of the JHTDB isotropic8192 dataset (Re_λ = 613), with c = 8: a 15-event population with densified sampling and a 78-event prospective population acquired under a frozen replication protocol. The inferential machinery is itself calibrated: simulating from the fitted variance components under the exact block structures, we measure the finite-sample size of every interval method used, and find the originally pre-registered block-percentile bootstrap anticonservative on the small population (false-exclusion rate 12% at nominal 5%), while the classical t-interval attains nominal size in both. Under size-correct inference the small population is uninformative and earlier benchmark-exclusion readings of it are withdrawn as interval artifacts; the prospective population measures a positive response, p̂_eff = +0.19 with CI (+0.07, +0.32), excluding p = 0 (p-value 0.003) and containing p_c = 1/4. The raw coefficient therefore grows with intensity, while the normalized coefficient χ ∝ Ω^(p−1/4) is consistent with constancy or mild decay — and exact constancy is also what a saturated estimator returns, since the angular numerator is bounded by one. The design cannot yet separate genuine geometric growth from that ceiling, which the available resolution cannot diagnose directly; we state it as the principal open systematic. The results are finite-range, single-snapshot statements at one radius prefactor; they establish no depletion, regularity, universality, or asymptotic law. What they do establish is a finite-scale, falsifiable question with its benchmark fixed by exact algebra and its inference calibrated against known truth.