Renormalized Lambert-W Cascade and Finite-Time Amplification and Blowup for reconstructed b Dynamics on T3 for the 3D Navier Stokes Equations

This article extracts and consolidates the renormalized Lambert-W branch-point cascade, its distinguished phase reduction, the exact characteristic invariant and finite-time amplification mechanism, and the extended reconstructed b i equation on T 3 . Repeated historical derivations are removed while the principal proofs and terminal reconstruction estimates are retained. The presentation separates exact finite-depth statements from coupled-depth asymptotics and records the hypotheses required for the extended PDE reconstruction. This paper further supports a recent paper \cite {moschandreou2026exploration} published by the corresponding author which claims that the Navier Stokes equations lose smoothness in finite time from initial smooth data.