Critical Regularity Anatomy of Periodic Navier–Stokes Breakdown

This preprint develops the second stage of a machine-checked continuation program for smooth forced periodic three-dimensional Navier–Stokes evolution. Building on the finite-maximal H2 continuation anatomy established in the first paper, the present work proves an exact spectral-transfer layer and a unified family of classical continuation failures on the same maximal strong trajectory. Finite maximality forces arbitrarily large positive integrated nonlinear H2-weighted work in finite Fourier annuli beyond every prescribed spectral scale, arbitrarily late in the evolution, even though the quadratic nonlinearity contributes zero net kinetic energy. The same trajectory necessarily leaves the physical gradient and vorticity continuation regimes and every nonendpoint Prodi–Serrin class formalized here. In particular, ∥∇u(⋅)∥L∞∉L1(0,Tmax⁡),∥curl⁡u(⋅)∥L∞∉L1(0,Tmax⁡),∥u(⋅)∥L∞2∉L1(0,Tmax⁡), and, for every finite q>3, ∥u(⋅)∥Lq 2q/(q−3)∉L1(0,Tmax⁡). The development includes explicit modewise kinetic-energy balances, finite-annulus H2 budgets, smooth-forcing high-frequency remainder control, physical gradient continuation, a periodic logarithmic div–curl estimate and forced H3energy law for vorticity continuation, and physical-torus Prodi–Serrin estimates with the exact critical exponent 2q/(q−3). These results are assembled in a machine-checked \\texttt{ClassicalContinuationFailureMechanism}. The endpoint Lt∞Lx3criterion is not proved here and remains the next analytic frontier. The principal results are formalized in Lean 4.