The Mandate of Pressure: A Structural Resolution of Navier–Stokes Regularity

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Abstract

We present a resolution of the global regularity problem for the three-dimensional incompressible Navier–Stokes equations by recasting it not as a problem of analytical control, but as one of logical coherence. We argue that the classical pursuit of bounding techniques is founded on a critical misreading: the conflation of representational limits with physical breakdown. Our framework is centred on the Mandate of Pressure: the principle that a fluid state is coherent if and only if its velocity field permits the existence of a well-defined pressure field capable of enforcing incompressibility across all scales. We demonstrate that this mandate is unbreakable in finite time for any finite-energy flow. Within this framework, a finite-time singularity is not an analytical divergence but a structural contradiction—a state into which the system is logically forbidden from evolving. We conclude that global regularity follows as a necessary consequence of the system's unwavering, scale-invariant internal logic.

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