Continuation Anatomy for Periodic Navier Stokes Breakdown

This preprint develops a machine-checked continuation theory for smooth forced periodic three-dimensional Navier–Stokes evolution in a native H2 framework. The central result characterizes the structure of any smooth periodic classical breakdown datum satisfying the stated interface. If no global smooth classical solution exists for the given data, then the associated maximal H2 trajectory has finite lifespan and its H2 norm diverges at the endpoint, while kinetic energy remains uniformly finite. The divergent regularity escapes above every fixed Fourier cutoff, whereas every fixed positive heat-regularization scale remains uniformly H2-controlled. Finite maximality also forces the collapse of every uniform positive bounded-control continuation certificate near the endpoint. The development includes an exact bridge between smooth classical periodic data and the native Fourier/Sobolev evolution, including incompressibility, reality, Sobolev regularity, the 2π normalization, forcing transport, pressure recovery, maximal-history construction, and reconstruction of the original classical equation. The paper also discusses the recently announced forced Navier–Stokes breakdown result as an application of this general continuation theorem package, while keeping the independent continuation mathematics logically separate from that external result. The principal analytic and bridge results are formalized in Lean 4.