Endpoint Rigidity in Periodic Navier–Stokes Flow

Let u:[0,T_max)→H²(T³) be the canonical smooth forced periodic three-dimensional Navier–Stokes trajectory. Papers I and II establish the finite-maximal continuation anatomy below the velocity endpoint, including bounded kinetic energy, eventual H² divergence, escape of H² mass beyond every fixed Fourier cutoff, terminal nonlinear derivative-weighted work, collapse of uniform continuation reserve, and failure of every formalized nonendpoint Prodi–Serrin control. The remaining critical boundary is L^∞_tL³_x.

The paper proves that bounded L^∞_tL³_x norm forces global continuation. Equivalently, finite maximal time forces infinite terminal L³ limsup. The statement is deliberately limsup-scoped: eventual divergence of the L³ norm is not asserted. The classical endpoint principle is realized here on the exact forced periodic maximal trajectory, retaining forcing at finite scale and allowing it to disappear only under blow-up magnification.

Assuming finite maximal time and bounded terminal L³, parabolic magnification at an endpoint singular point produces a nontrivial unforced suitable ancient state in L^∞_tL³_x. Canonical exterior vorticity vanishes on each negative terminal window. Every negative interval contains an intrinsic global H¹∩L³ restart slice, and weak–strong identification recovers the same ancient state from the mild restart.

The literal Oseen history and its analytic representative are then identified with that authoritative state. Exterior curl zero propagates across each connected spatial slice, harmonic L³ rigidity forces zero slices on every negative interval, and weak critical-time continuity gives the zero ancient state, contradicting the retained terminal nontriviality.

Several failed recovery geometries and invalid shortcut inferences remain explicit obstruction results. A separate companion theorem quantifies critical-root contraction under pre-existing smallness and is not used in the endpoint contradiction.