FCBP-01: Critical Forest Coercivity, One-Derivative Gap, Dual Congestion Renormalization, and Structural Cancellation

Critical Forest Coercivity, One-Derivative Gap, Dual Congestion Renormalization, and Structural Cancellation

Continuing from the forest coercive budget problem left by CFOP Cycle V (finding a forest functional that is simultaneously globally finite, near-critical on the dangerous cut, and stable under branching/fragmentation), FCBP does not add a new forest taxonomy but asks: can the precise Navier–Stokes structure "lift" an already-known finite-scale weak budget into a critical or non-summable coercive budget? Proves a scaling theorem for the velocity and vorticity nonlinear forcing in Sobolev dual space, identifying that — at the same time exponent p=4/3 — the universal energy-class forcing upper bound and the scale-critical forcing topology differ by exactly one spatial derivative; proves a "weighted-to-unweighted critical lift" NO-GO that holds for every summable positive scale weight. Integrates three external Navier–Stokes-specific cancellation modules (Miller strain–vorticity orthogonality, diffusive absorption of filtered vortex stretching, pressure–flux weighted finite-chain telescoping); these cancellations exhibit genuine structural derivative/coercivity recovery, but leave explicit gaps in the model cone, far-field/commutator/localization, backscatter, observability, and weighted stacking. Defines the "critical lift problem": without assuming regularity, converting the universally finite scale-weighted cancellation ledger into a non-summable near-critical forest-cut budget — this round does not prove such a lift; the finite forest obstruction and regularity remain OPEN.