Multi-Scale Coherence of Represented Flows

Abstract

Many problems in nonlinear and statistical physics are formulated through represented flows, including physical-space vector fields, phase-space drift fields, and truncated renormalization-group beta functions. We introduce a complementary representation-dependent diagnostic for testing whether finite-separation flow geometry is stable across observational resolution. For two separated points, states, or theories, the method compares the direction of the corresponding vector-field increment after the field has been smoothed at two resolutions. Averaging this normalized comparison over sampled separations gives a coherence matrix tied to the chosen variables, coarse graining, metric, and sampling protocol; it is a consistency test, not a coordinate-invariant quantity. We demonstrate the diagnostic in three settings. Synthetic divergence-free fields with identical Fourier amplitudes, spectra, and scalar two-point correlations nevertheless produce distinct coherence matrices, showing that second-order statistics do not determine cross-resolution increment geometry. Lorenz phase-space tests show that a smooth coordinate wrinkling changes represented drift geometry without changing the underlying dynamics, and that a weak model perturbation lowers finite-separation coherence even when local stretching proxies remain closely matched. Finally, for functional renormalization-group flows of the three-dimensional \(O(1)\) scalar theory, projected \(M=4,5,6\) LPA beta fields remain internally coherent, while cross-truncation coherence decreases as higher-order coupling directions are activated. The diagnostic provides a practical field-level check of how representations, models, and truncations preserve finite-separation flow geometry, complementing rather than replacing standard local, spectral, or fixed-point diagnostics.