First-Hit Energy and Logarithmic Viscous Memory: A Continuation Criterion for the Three-Dimensional Navier–Stokes Equations

Version v1.0 This work develops first-hit continuation criteria for the classical three-dimensional incompressible Navier–Stokes equations near a hypothetical first singular time. The construction begins at the first time a high dyadic shell reaches the scale-invariant dissipation threshold. Exact shell capture then yields a quantitative high-frequency energy floor at the terminal event. FirstHit_LogMemory_NavierStokes… The first main result is a global complementary-filter theorem. For a normalized low/high filter pair satisfying \(G_\ell^2+H_\ell^2=I\), the unresolved high-pass energy obeys an exact global energy ledger whose nonlinear term is the standard signed subgrid-scale (SGS) stress–strain work, with no boundary, moving-frame, or cutoff-commutator remainder. Combining this identity with viscous coercivity and a logarithmic memory interval gives an explicit singularity-production obstruction. In the endpoint case \(\vartheta=0\), the continuation hypothesis can be written purely in terms of the positive SGS work \([\Phi_q^G]_+\) relative to unresolved energy; the viscous dissipation appears only in the proof through coercive damping. FirstHit_LogMemory_NavierStokes… The second main result is a spatially resolved refinement. A shrinking mesoscopic observation region is transported by a coarse velocity to remove rigid sweeping from the local energy budget. The resulting moving high-pass energy is compared with an adaptive two-scale viscous capacity, and its nonlinear transfer is connected exactly to standard coarse-grained subfilter work up to explicit boundary-transport and filter/localization commutator terms. A logarithmic-memory argument then yields localized continuation criteria together with standard-flux, stress–strain, and fully expanded sufficient formulations. FirstHit_LogMemory_NavierStokes… The results are continuation criteria for the unmodified three-dimensional Navier–Stokes equations. They do not assert unconditional global regularity. FirstHit_LogMemory_NavierStokes…