The Double Gate Theorem: Cascade Instabilities Require Two Independent Conditions

We prove that gravitational cascade instabilities (such as the N-body problem's runaway collisions) and debris cascade instabilities (such as the Kessler syndrome in orbital mechanics) share a common mathematical structure: both require the simultaneous satisfaction of two independent threshold conditions. Gate 1 (Discrete): the system must contain at least four bodies (\(N \geq 4\)) to support the binary pump cycle mechanism. Gate 2 (Continuous): the density must exceed a critical threshold (\(\rho > \rho_c\)) for supercritical growth dynamics. Neither condition alone suffices—a five-body dilute system remains stable, as does a three-body system at arbitrarily high density. Only when both gates are open does cascade become inevitable. We formalize this as the Double Gate Theorem and prove 22 theorems establishing the four-quadrant stability/instability classification, with fully verified proofs in the Platonic kernel.