Velocity Field Divergence

The divergence-compression correlation on the Checkerboard benchmark is verified with Pearson r > 0.94 and p < 10⁻³⁰⁰ for all models tested, and DS-RectFlow achieves a further ~6% reduction in mean absolute divergence beyond vanilla reflow. The convergent component of divergence (∇·v < 0) drives trajectory crossings by compressing volume; expansion (∇·v > 0) separates particles and actually reduces crossing risk. The Helmholtz decomposition splits the velocity field into a divergence-free transport component that routes mass between distributions and an irrotational dipole component that carries all the field's compressibility. Demanding zero divergence everywhere is too restrictive because the continuity equation requires nonzero divergence for any flow transporting a Gaussian to a structured target. The Hutchinson trace estimator with Rademacher vectors approximates divergence at the cost of one vector-Jacobian product per sample, without requiring second-order computation. Trajectory bending is localized to the support of the irrotational dipole; where the dipole is negligible, neighbouring trajectories already travel in parallel under the transport component alone.