Unified Triadic Phase Dynamics of the Navier–Stokes Equations: From Global Regularity to Kolmogorov Scaling and Constant Determination

We develop a unified dynamical framework for the three-dimensional incompressible Navier–Stokes equations in which global regularity and turbulent inertial-range structure emerge from a common underlying mechanism. Building on a recent result establishing global regularity via coherent-core reduction and phase non-persistence, we reformulate the nonlinear dynamics in terms of triadic interactions and their associated phase evolution. We show that nonlinear amplification is confined to a High–High interaction channel, which can be further localized to a coherent core characterized by low phase drift. The phase dynamics within this core exhibits a curvature-driven instability, implying that persistent phase coherence is dynamically impossible. As a consequence, nonlinear transfer is temporally localized, preventing cumulative growth and ensuring global regularity. Using this structure, we derive the inertial-range energy cascade directly from deterministic dynamics. The combination of time-localized interactions and scale-dependent triadic multiplicity yields a constant energy flux across scales without invoking statistical assumptions or closure models, leading to a first-principles derivation of the Kolmogorov −5/3 scaling law. Furthermore, we show that the Kolmogorov constant is not an empirical parameter but a dynamically determined quantity arising from phase-averaged triadic interactions. At the continuum level, the theory yields a structural formula together with a finite admissible interval. This remaining indeterminacy is resolved by extracting the coherent-phase quantities from a GOY shell model, used as a dynamically consistent reduced system that preserves local triadic interactions. The resulting value is thereby obtained without introducing phenomenological closure assumptions. These results establish that Navier–Stokes regularity, inertial-range cascade, and the determination of the Kolmogorov constant are not independent phenomena, but three manifestations of a single triadic phase dynamic. The mechanism that suppresses finite-time blow-up is identical to the mechanism that generates energy transfer across scales and fixes the Kolmogorov constant, providing a unified deterministic foundation for fluid dynamics.