FCBP-05: Long-Age Observability, Sharp Half-Exponent Window Threshold, Combined Anti-Phantom Detection, and Paid-Side Recurrence

Long-Age Observability, Sharp Half-Exponent Window Threshold, Combined Anti-Phantom Detection, and Paid-Side Recurrence

FCBP-04 splits the horizon time problem into two branches — "long-age observability" or "thin-window amplification" — and shows the filter endpoint problem is no longer the main obstruction. This round addresses: can dangerous information stay observable across many local parabolic ages? On horizon-aligned thin windows, how fast can the observability constant degrade while still keeping a non-summable exhaustion schedule? Can fresh vorticity input be forced into a single signed pressure–flux work channel? Can paid-side backscatter/leakage be controlled by endpoint energy alone? Proves that a direct universal bridge from fresh vorticity sources to a single signed kinetic-work channel is impossible (the Navier–Stokes nonlinear term is globally energy-orthogonal even when its curl is nonzero) — and replaces the single-channel target with combined pressure/flux/positive-energy/adjoint-trace/model-cone observability. Proves a sharp sequencing theorem for finite-time horizon scheduling: the normalized horizon-thickness exponent a=1/2 is the sharp threshold, with an explicit logarithmic schedule that makes the series diverge — yielding the sharp time threshold for exhaustion effectiveness, γq=1/2. Integrates Tao's quantitative L³ backward propagation as an external strong-critical long-age state propagation module.