A Cogenetic Proof of Global Regularity for the Three-Dimensional Incompressible Navier–Stokes Equations

Abstract

This manuscript presents a Cogenetic proof of global existence, smoothness, and uniqueness for the three-dimensional incompressible Navier-Stokes equations on R3 with positive viscosity and arbitrary smooth divergence-free rapidly decaying initial velocity. The proof identifies the recursively closed CNS-N1/CNS-R1 recovery relation as the invariant mechanism. Each finite recovery retains complete flow ancestry, projected field content, redistributed content, residual, and the transformed next differential. The exact amplitude-intrinsic-area ledger shows that attained magnitude changes projected coordinates without changing the recursively returned output type. Classical continuation failure is translated into a complete tail-stable concentration carrier and rejected when terminal noncontinuation requires deletion of the residual and continuing differential returned at every finite closure. The foundational finite-to-global causal extension is isolated explicitly, with classical continuation, uniqueness, pressure recovery, and energy behaviour recovered downstream. Supporting material provides claim-labelled symbolic, carrier, causal-recovery, metadata, and build verification.