Formally Verified Financial Contagion Thresholds: Counterparty Default Cascades as a Grade-2 Dynamical System

Financial contagion — the cascading failure of interconnected institutions through bilateral exposures — is the central mechanism behind systemic crises from Lehman Brothers (2008) to Silicon Valley Bank (2023). We present a formally verified analysis that identifies counterparty default cascades as a Grade-2 dynamical system with cascade rate \(h(D) = \beta D^2 - \alpha D\), where \(\alpha\) represents risk absorption (capital buffers, insurance, central bank interventions) and \(\beta\) represents counterparty contagion (bilateral default chains). The critical default fraction \(D_c = \alpha/\beta\) defines the systemic risk threshold.

We prove 18 theorems establishing: (1) threshold existence and stability analysis; (2) sub-threshold absorption guaranteeing system resilience; (3) super-threshold cascade dynamics with leverage amplification; (4) bailout sufficiency bounds with intervention guarantees; (5) pro-cyclicality — the formal proof that individually safe positions can combine to generate systemic risk; (6) too-big-to-fail characterization; (7) binary counting showing \(N \geq 4\) institutions needed for sustained cascade; and (8) delay penalty proving that late intervention always costs more.

All results are machine-verified via the proof kernel and exported to 317 lines of Lean 4. The algebraic identity \(h(D) = \beta D^2 - \alpha D \equiv f(\rho) = \beta\rho^2 - \alpha\rho\) (Kessler debris) \(\equiv g(S,I) = \beta SI - \gamma I\) (SIR epidemic) confirms that the same Grade-2 structure governs cascading failures across space engineering, epidemiology, and finance.