Continuation Anatomy of Periodic Navier–Stokes Breakdown

We develop a machine-checked continuation theory for smooth forced periodic three-dimensional Navier–Stokes data and use it to characterize the structure of any periodic classical breakdown datum. Whenever smooth physical periodic data admit no global smooth classical solution, the associated native H² evolution has finite maximal lifespan and its H² norm tends to infinity as the maximal time is approached. At the same time, kinetic energy remains uniformly finite.

For every finite Fourier cutoff, the H² mass above that cutoff tends to infinity, while for every fixed positive heat scale the regularized state remains uniformly H²-controlled. Finite maximality also excludes every uniform positive bounded-control continuation certificate near the endpoint. The theorem package identifies periodic strong breakdown as fine-scale regularity escape and collapse of strong continuation capacity inside a finite-energy flow.

The proof proceeds through an exact classical/native bridge. Smooth real periodic initial data descend to Fourier/Sobolev carriers, preserve incompressibility and reality, inhabit every finite Sobolev order, and return exactly to the original classical data after reconstruction. Smooth periodic forcing admits the corresponding exact round trip, including the 2π derivative normalization, viscosity conversion, and force rescaling. Consequently, a hypothetical infinite native trajectory would produce a global classical solution for the original datum itself.

A recently announced OpenAI construction supplies a first concrete application: OpenAI reports a formally verified proof of the forced breakdown alternatives (C) and (D) in the official Clay formulation. If accepted, either alternative resolves the mathematical problem as posed, while the historically central zero-force regularity question remains logically distinct. The continuation anatomy proved here stands independently: it states what any periodic smooth classical breakdown datum satisfying the declared interface must look like in the native H² theory.