We study worst-case contagion losses in financial networks with interbank liabilities and common exposures to marketable assets. Existing linear-programming approaches provide exact stress-test values only within an insolvency-resilience margin, where all institutions remain solvent with respect to the external sector. We show that this margin is computed exactly by a finite family of linear programs, one per vertex of the uncertainty ball, whereas a single linear program based on the componentwise worst deterioration of external resources yields only a certified lower bound. We extend this framework beyond that margin in three directions. First, we derive an exact mixed-integer linear programming (MILP) formulation for the maximal clearing vector under arbitrary signed external net positions, providing a clearing oracle for robust stress testing when insolvency can occur. Second, we develop certified bounds on the worst-case loss beyond the margin, including insolvency-pattern enumeration, scenario lower bounds, fixed-pattern dual upper bounds, localized certificates, and a regime-generation procedure with monotone gap certificates. Third, we analyze the worst-case loss curve as a function of the perturbation radius. For $\ell_1$ and $\ell_\infty$ price-perturbation sets, the curve is convex, nondecreasing and piecewise linear up to the margin, with breakpoints generated by switches in worst-case shock directions, LP bases, and bank-level activity patterns. Beyond the margin, it remains nondecreasing and piecewise affine, but convexity may fail, the curve may jump upward, and the worst case need not be attained. The framework provides a certifiable MILP/LP methodology for parametric stress testing and interpretable diagnostics identifying the banks responsible for changes in systemic loss.

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