Structural Reduction Framework and Residence-Time Compression of Coherent Same-Scale Triadic Interactions in the 3D Navier–Stokes Equations

We develop a structural framework for the three-dimensional incompressible Navier–Stokes equations in which the nonlinear dynamics are reorganized in terms of triadic interactions, dyadic shells, and helical modes. Within this formulation, all interactions are classified into Low–Low, Low–High, and High–High channels, and it is shown that the Low–Low and Low–High contributions are perturbatively controlled through scale-localized estimates without introducing external assumptions. Consequently, potentially non-perturbative contributions are confined, within the present framework, to a class of same-scale High–High interactions. This class is further reduced, through geometric and dynamical constraints, to a coherent core characterized by amplitude activity and low phase drift. The resulting reduced dynamics is expressed in terms of family-level phase variables and associated curvature quantities. The main result establishes a quantitative residence-time compression principle for this coherent regime. Specifically, it is shown that intervals on which both amplitude activity and low phase drift persist must have small total measures, due to an absolute-value coercivity property of the curvature combined with bounded-variation control of the phase dynamics. This implies that coherent same-scale interactions cannot occupy a macroscopic portion of any bounded time interval, even though local re-entry into low-drift configurations is not excluded. Consequently, the nonlinear transfer associated with coherent triads becomes temporally localized and admits a shellwise compressed representation. These results provide a structurally reduced description of a candidate mechanism for cumulative same-scale amplification within the present dyadic–triadic framework. They do not claim a framework-level structural exclusion of the global regularity problem. Rather, they identify and analyze, within an explicit structural setting, a minimal mechanism for non-perturbative amplification, and establish a quantitative constraint on its temporal persistence.