The Pump Cycle Bridge: Structural Isomorphism Between Gravitational Singularities and Debris Cascades

We prove a structural isomorphism between gravitational noncollision singularity (NCS) thresholds and orbital debris cascade thresholds. Both phenomena require a "pump cycle" mechanism—the sustained transfer of energy or momentum between subsystems—which in turn requires at least two independent binary subsystems. Since N bodies yield at most floor(N/2) binaries, the threshold N = 4 is universal: N ≤ 3 cannot sustain pump cycles, N ≥ 4 can. We formalize this connection in dependent type theory with 29 verified theorems, 8 classical axioms, and a capstone Painlevé-Kessler correspondence theorem establishing the structural equivalence. The proof architecture shows that the discrete threshold (N = 4) corresponds to a continuous bifurcation in Grade-2 ODEs (subcritical vs supercritical regimes), unifying discrete and continuous threshold phenomena.

Keywords: pump cycle, gravitational singularity, Painlevé, Kessler syndrome, debris cascade, noncollision singularity, dependent type theory, formal verification

MSC 2020: 70F10 (N-body problems), 70F15 (Celestial mechanics), 37N05 (Dynamical systems in classical mechanics)