On the Regularity of Three-Dimensional Navier-Stokes Solutions: A Geometric Approach via the Biot–Savart Connection

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Abstract

We develop a geometric approach to the regularity problem for the three-dimensional incompressible Navier-Stokes equations, one of the seven Clay Millennium Problems. The central idea is to define a connection on the infinite-dimensional bundle of divergence-free velocity fields, constructed from the Biot–Savart kernel and the Leray projection at the natural regularity of Leray-Hopf weak solutions. By encoding the geometric content of the flow into the curvature of this connection, we remain entirely within the Leray-Hopf energy class throughout. The programme asks whether incompressibility, combined with the curvature constraints imposed by this connection, obstructs finite-time blowup. All definitions and theorem statements are formalised in Lean 4 with Mathlib, and key identities are independently verified with SymPy.