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 bi 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 published by the corresponding author which claims that the Navier Stokes equations lose smoothness in finite time from initial smooth data.