Critical Regularity Anatomy of Periodic Navier–Stokes Breakdown

A previous machine-checked continuation theory for smooth forced periodic three-dimensional Navier–Stokes evolution established that any smooth periodic classical breakdown datum gives a native strong trajectory with finite maximal H² lifespan, bounded kinetic energy, eventual H² divergence, high-frequency regularity escape, fixed positive heat-scale regularity, and collapse of uniform continuation reserve. The present paper resolves the next question: what additional analytic structure must every such finite-maximal trajectory exhibit?

We prove an exact spectral-transfer layer and a hierarchy of physical continuation failures on the same maximal trajectory. The quadratic Navier–Stokes term has zero total kinetic-energy contribution while admitting nontrivial derivative-weighted transfer across finite spectral regions. Smooth forcing has uniformly vanishing high-frequency Sobolev tails on compact time windows, and its derivative-weighted work is separated into viscous absorption plus a vanishing fine-scale remainder. Finite maximality then forces arbitrarily large positive integrated nonlinear H² work in finite annuli beyond every prescribed spectral scale, arbitrarily late in the trajectory.

The same trajectory necessarily leaves the gradient and vorticity continuation regimes and every nonendpoint Prodi–Serrin class formalized here. Its physical deformation rate and physical vorticity supremum are not time-integrable; its velocity fails the L2tL∞x continuation criterion; and for every finite q>3 the critical quantity ‖u(·)‖Lq2q/(q−3) is not integrable on the maximal interval.

The vorticity criterion is obtained through a periodic logarithmic div–curl estimate, physical heat-kernel analysis, and a forced H³ energy law. The Prodi–Serrin family is obtained from physical torus L^q norms, a periodic H¹→L⁶ realization, exact interpolation, and the critical Young exponent 2q/(q−3).

These statements are packaged in a theorem-owned ClassicalContinuationFailureMechanism. Finite periodic strong breakdown therefore has one simultaneous critical anatomy: kinetic energy remains finite, regularity escapes every fixed spectral scale, terminal nonlinear regularity work becomes unbounded, accumulated physical deformation and peak vorticity become nonintegrable, and every formalized nonendpoint Prodi–Serrin class fails. The endpoint L^∞_tL³_x criterion remains outside the present theorem and marks the next analytic frontier.