Navier–Stokes Regularity via Scalar Potential Geometry: Codimension-Three Structure and Blow-Up Degeneracy

Abstract

We develop a geometric framework for the three-dimensional incompressible Navier–Stokes regularity problem based on a canonical triple of scalar potentials. Every zero-mean divergence-free velocity field is represented exactly as u = −ΔE with div E = 0, establishing a global correspondence between velocity fields and potential triples. The associated metric is, up to scale, the induced metric of a three-dimensional graph in R^6. This codimension-three formulation introduces intrinsic curvature, the second fundamental form, and normal-bundle curvature, all controlled by the potential Hessian. The energy identity further yields improved slope and geometric integrability bounds. On the doubly-flat locus, where intrinsic and normal-bundle curvature vanish, the shape operators diagonalize simultaneously into orthogonal bending channels. A multi-bending invariant identifies a rank-one regime in which the component Hessians share a common rank-one direction and incompressibility implies u · n = 0. Under an additional cylindrical-reduction hypothesis, the dynamics reduce to a transverse one-dimensional heat flow. We also introduce a scale-critical geometric degeneracy functional for blow-up analysis. Under the stated blow-up compactness hypotheses, its degeneration forces the limiting geometry toward the rank-one regime, although energy alone does not imply this degeneration. The paper does not claim a solution of the Clay Millennium Problem. It establishes a codimension-three potential geometry for the full incompressible Navier–Stokes system and isolates the remaining obstruction as either closing the 3/2-derivative analytic gap or proving asymptotic rank-one degeneration near a possible singularity.

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