Phase Non-Persistence in Triadic Interactions: A Complete Resolution of the 3D Navier–Stokes Regularity Problem via Coherent Core Reduction

The present study offers a potential resolution to the 3D Navier–Stokes regularity problem, demonstrating that the global existence of strong solutions is sustained by the autonomous decorrelation of triadic phases. The proof is based on a structural reformulation of the nonlinear term, in which the Fourier-space triadic interactions are decomposed into perturbative channels and a single potentially dangerous High–High coherent core. All non-core interactions are shown to be perturbative and absorbable into viscous dissipation by means of paraproduct analysis and scale-localized estimates. The remaining High–High core is further reduced to a coherent set characterized by low phase drift and non-negligible amplitude. The continuation problem is thereby reduced to a single dynamical obstruction: the possibility of persistent phase coherence within this coherent core. The present analysis suggests that such persistence cannot occur. The key mechanism is a curvature-driven instability in the phase dynamics, expressed through a coercive lower bound on the curvature kernel associated with triadic interactions. This yields a quantitative phase non-persistence result, showing that the low-drift coherent set has vanishing measure at high frequencies. Consequently, the nonlinear energy transfer is compressed in time and cannot accumulate sufficiently to overcome dissipation. This leads to a shellwise absorption estimate for the High–High interactions, which closes the energy inequality in Sobolev spaces and precludes finite-time blow-up. The argument is non-circular and requires no external closure assumptions. Conceptually, the proof demonstrates that the Navier–Stokes regularity problem reduces to a single geometric–dynamical mechanism, and that this mechanism is intrinsically incompatible with sustained nonlinear amplification. The result also provides a rigorous link between deterministic PDE analysis, and the transient coherence observed in turbulent energy cascades.