CFOP-03: Finite Forest Obstruction, Universal Budget Audit, Negative-Sobolev Forcing, Sparse/Dense Geometry, and Cycle-V Closure

Finite Forest Obstruction, Universal Budget Audit, Negative-Sobolev Forcing, Sparse/Dense Geometry, and Cycle-V Closure

CFOP-02 reduced the horizon-unbounded diffuse causal forest to five residual classes (R_F-ACT, R_F-CONG, R_F-CAP, R_F-TAIL, R_F-NSPARSE). This paper asks the decisive question: does standard Navier–Stokes theory supply a universal finite budget that closes all five? The answer is no. But the audit does rule out several residuals as primitive terms and identifies the actual missing estimate form. It proves an energy-class negative-Sobolev nonlinear forcing budget: the velocity nonlinear term lies in L_t^(4/3) H_x^(-1), vorticity forcing lies in L_t^(4/3) H_x^(-2), with upper bounds controlled by kinetic energy and dissipation. It recasts source-dominated forest cutsets as a dichotomy between "finite weak forcing" and "higher-order dual congestion," proving that infinitely many disjoint source-dominated cutsets force the quartic H² dual congestion itself to diverge, not the coercive action directly. It reclassifies weighted state-tail escape as adaptive spatial-capacity growth. Running a scale audit, it shows that the Leray enstrophy-time budget and the energy-class H^(-2) forcing budget are each summable per parabolic scale (O(R) and O(R^(4/3)) respectively) — so neither alone can exclude an infinite geometric cascade. Local regularity and geometric sparseness results are introduced only as external conditional safeguards/calibrations. Conclusion: the currently standard finite budgets cannot yield a complete finite forest obstruction; the missing theorem is a Forest Coercive Budget that is near-critical in scale and carries a universal Navier–Stokes finite upper bound — formally defining the "Forest Coercive Budget Problem" and handing off to NS-FCBP.