Adaptive Cascade Capacity and First-Hitting Continuation for Three-Dimensional Navier--Stokes Flows

This preprint develops an adaptive, scale-local continuation criterion for the classical three-dimensional incompressible Navier--Stokes equations on R^3. Assuming a hypothetical finite singular endpoint, a first-hitting shell argument produces arbitrarily high shell events. On each event's natural viscous time window, we define an adaptive cascade capacity combining root-mean-square coarse strain, time-integrated activity over an adaptively chosen finite forward shell band, and an enstrophy remainder discounted by the square of the spectral gap. A moving-center backward-heat detector and fixed-support dyadic product routing yield a scale-uniform positive capacity lower bound at sufficiently high first hits. Vanishing of the capacity uniformly over terminal parabolic windows therefore implies smooth continuation. The theorem concerns the unmodified classical Navier--Stokes dynamics; it neither asserts unconditional global regularity nor assumes a positive microscopic length. A separate kinematic example shows that the capacity observable can vanish despite fixed-height critical dyadic bursts and loss of terminal critical Besov continuity; this example is not a Navier--Stokes solution and does not establish inclusion relations between solution-class criteria. The companion PAV/RPAV preprint is cited for the common first-hitting framework and normalization, while the present adaptive estimates are developed in this manuscript. This is a preprint and has not undergone journal peer review.