Geometric vorticity coupling: A geometric diagnostic of anisotropic nonlinear structure formation

Geometric Vorticity Coupling (GVC) is investigated as a local scalar diagnostic that combines the amplitudes and relative orientation of a density gradient and a rotational velocity field. We evaluate the diagnostic in three-dimensional periodic particle calculations using mass-weighted Eulerian velocity reconstruction, complemented by controlled rotational benchmarks, coarse-graining tests, spectral robustness analyses, density-Hessian environment classification, and amplitude-conditioned angular null models. Decomposing GVC into an amplitude factor and an angular factor shows that its statistical variation is predominantly amplitude-associated in the reconstructed flow. The relation between GVC and its amplitude control has an exponent of 1.0017 ± 0.0003, while coarse graining strongly attenuates both quantities and produces only modest angular reweighting. Their reference spectral-slope difference is 0.019 ± 0.020 and varies from approximately −0.029 to 0.032 across numerical robustness tests, providing no configuration-independent spectral enhancement. At one-cell coarse graining, the filament-to-void mean ratios are 1.386 ± 0.094 for GVC and 1.398 ± 0.099 for its amplitude control, whereas the angular-factor ratio is 0.9881 ± 0.0020; the GVC contrast reverses sign without smoothing, demonstrating scale dependence. Finally, angular enrichment of high-GVC regions relative to unrestricted shuffling is reproduced by an amplitude-conditioned null model, leaving a residual strong-angle fraction of −0.001 794 ± 0.000 474. These results establish GVC as a scale-dependent amplitude–orientation diagnostic: orientation provides local geometric weighting, but the present calculations resolve neither a positive angular enrichment beyond amplitude-conditioned selection nor a robust independent spectral enhancement.