FCBP-04: Moving-Filter Telescoping, Continuous Filter Drift, Horizon Alignment, Time-Thickness Barrier, and Borderline Critical Lift

Moving-Filter Telescoping, Continuous Filter Drift, Horizon Alignment, Time-Thickness Barrier, and Borderline Critical Lift

FCBP-03 produces the first non-summable pressure–flux scale schedule (∑r_k²<∞ but ∑r_k/r_0=∞), proving a genuine "schedule lift" — but a schedule lift is not yet a "horizon critical lift." This round handles two compatibility problems: the filter must move with the physical scale, and the window slices must genuinely approach the singular horizon with the correct parabolic geometry. The first problem can largely be closed; the second reveals a new time-thickness barrier. Proves an L² filtering-scale derivative estimate for smooth compactly-supported mollifiers, deriving the exact coarse-grained Navier–Stokes equation for a continuous time-dependent filter; under slow schedules, the filter-drift work of a linearly moving relative filter is universally controlled by the Leray energy/dissipation budget — so a discrete filter-switch stack can be bypassed by continuous filter drift. However, proves a parabolic-alignment summability theorem: if adjacent full-thickness parabolic window slices accumulate at T* while staying within O(1) local parabolic age of the horizon, then ∑r_k must be finite — a non-summable slow schedule must therefore lose fixed parabolic horizon alignment. The second route uses horizon-aligned thin window slices, where a co-moving heat-semigroup filter can exactly absorb the filter drift, but the universal time-thickness theorem proved here shows that even when ∑w_k diverges, the bounded normalized power can still be summable.