Parabolic Atomic Vanishing and Residual-Stress Continuation for Three-Dimensional Navier–Stokes Flows

Description We introduce a scale-critical dual continuation criterion for the classical three-dimensional incompressible Navier–Stokes equations on $\mathbb{R}^3$, based on dyadic Littlewood–Paley shells and normalized backward-heat test packets. Using the dissipation-wavenumber continuation framework, a hypothetical finite-time singular endpoint produces arbitrarily high first-hitting shell events. Each sufficiently high first hit forces a strictly positive dimensionless pairing between the full quadratic stress and a backward-heat atom. The same witness yields a scale-critical recent-window enstrophy lower bound. This leads to Parabolic Atomic Vanishing (PAV): scale-wise vanishing of the maximal positive atomic response of the quadratic stress. PAV excludes finite-time singularity without requiring cross-scale summability. For a fixed positive resolution $\ell_* > 0$, let $S_{\ell_*}$ denote convolution with a finite-resolution kernel whose Fourier multiplier decays like $(1+\vert{}\eta\vert{})^{-\beta}$. We prove, for $\beta > 1/2$, that the resolved stress $S_{\ell_*}(u \otimes u)$ is automatically PAV under the energy bound. The key step is an exact atomic–caloric defect identity, which identifies atomic stress response with the critical caloric defect of the corresponding shell velocity. Consequently, PAV for the full quadratic stress is equivalent to PAV of the unresolved residual $$R_{\ell_*}^{\mathrm{sub}} = (I - S_{\ell_*})(u \otimes u),$$ a condition termed Residual-PAV (RPAV). RPAV therefore provides a continuation criterion for the classical equation. We also establish a kinematic obstruction showing that a positive resolution scale and finite kinetic energy alone do not imply RPAV: smooth divergence-free spacetime fields with uniformly bounded kinetic energy can retain order-one unresolved atomic responses at arbitrarily high scales. Thus finite-resolution decomposition isolates the nonlinear residual property required for continuation but does not by itself supply that property. The Navier–Stokes equation is not filtered or modified in the argument. No unconditional conclusion is claimed for the zero-resolution limit $\ell_* \downarrow 0$, and the classical three-dimensional global-regularity problem remains open. Reserved DOI: 10.5281/zenodo.23049603 Creator Lee Byoungwoo Independent Researcher, Daejeon, Republic of Korea Keywords Navier–Stokes equations; three-dimensional incompressible flow; regularity criteria; continuation criteria; Parabolic Atomic Vanishing; PAV; Residual-PAV; RPAV; first-hitting shell events; Littlewood–Paley decomposition; backward heat kernels; backward-heat packets; dissipation wavenumber; atomic–caloric defect; caloric defect; dyadic frequency localization; residual stress; subgrid stress; coarse graining; finite-resolution filtering; enstrophy; nonlinear stress; high-frequency regularity