Radius-dependent looseness of the far-field Calderón–Zygmund bound in turbulent vortex stretching

Author's accepted manuscript of an article accepted for publication in Physical Review Fluids (accepted 21 August 2026), doi:10.1103/std4-f34j. Please cite the version of record. © 2026 American Physical Society. The far-field contribution to vortex stretching at a high-vorticity point can be bounded by an unsigned Calderón–Zygmund (CZ) singular-integral estimate. We separate this bound into the quantity it bounds and the quantity it uses to bound it, and measure both by direct summation of the contracted Biot–Savart stretching integral on direct numerical simulation (DNS) data, as functions of the outer integration radius R. Across forced isotropic turbulence at Taylor-scale Reynolds numbers Re_λ ≈ 433, 611, and 1300, the unsigned capacity—the magnitude the estimate permits—grows approximately logarithmically in R, whereas the realized signed magnitude—what the same integral actually delivers—grows with only 10–19% of the capacity slope. The empirical tightness C_far = |σ_far|/σ_far^abs therefore decreases by 21–39% over R/η = 25–60, with block-bootstrap confidence intervals excluding zero at both median and extreme-vorticity targets. This decline is robust: it is unchanged when target classes are defined by a threshold pooled across all sampled cutouts rather than locally, it survives a sixth-order finite-difference recomputation of the full pipeline, and it is insensitive to the inner cutoff. The looseness of the bound is thus a radius-dependent margin, not a fixed factor. Expressed as the dimensionless tightness rather than the dimensional capacity, the curves nearly collapse across the threefold range in Re_λ. The realized far field is dynamically substantial: the magnitude ratio |σ_far|/|σ_total| reaches 0.50 at median targets and 0.44–0.48 in the extreme-vorticity tail at R/η = 60, so by the triangle inequality the contribution complementary to the measured far-field band has magnitude at least approximately half of the total at the median and at least 52–56% at the most intense events, with the far field a near-equal partner at the largest measured radius. Cancellation is systematically weaker at extreme-vorticity targets than at the median: the tail-to-median tightness ratio is 1.2–1.7 and grows with radius, so the sign organization that suppresses the far field weakens precisely at the events most relevant to amplification. A per-shell cancellation diagnostic shows that the looseness is already present within individual radial shells and strengthens with radius in both intensity classes.