Sharp Phase Transitions in Financial Contagion: A Formal Theory of Systemic Risk Cascades

We develop a formal theory of systemic risk cascades and prove that financial contagion exhibits a sharp phase transition at a critical default fraction \(D_c = \alpha/\beta\), where \(\alpha\) is the absorption rate (capital buffers) and \(\beta\) is the contagion rate (bilateral exposure). The cascade dynamics are governed by a Grade-2 equation \(h(D) = \beta D^2 - \alpha D\), which admits an exact factorization \(h(D) = D(\beta D - \alpha)\). We establish twelve formal theorems including: (i) sub-threshold stability (\(D < D_c \Rightarrow h(D) < 0\)), (ii) super-threshold blowup (\(D > D_c \Rightarrow h(D) > 0\)), (iii) pro-cyclicality (individually subcritical exposures can be collectively supercritical), and (iv) isomorphism between financial cascades and SIR epidemics. The network topology layer proves that contagion intensity \(\beta = \bar{k} \cdot e\) factors into average connectivity times exposure per link, and that central counterparties raise the threshold by reducing \(\beta\). All theorems are formalized in the Platonic proof language with 28 verified declarations and 0 errors. The bailout sufficiency and delay penalty theorems provide formal foundations for intervention timing.

Keywords: systemic risk, financial contagion, phase transition, network contagion, central counterparty, Grade-2 dynamics

JEL codes: G01, G21, G28, C61