Detector, Not Predictor: Why the Residual Hessian Cannot Predict Capacity-Limited Catastrophes Without External Physics

The residual Hessian has been proposed as a diagnostic tool for inverse problems. A recent critique warned that the Hessian is a "detector", not a "predictor": it can identify geometric fragility, but cannot foresee physical catastrophes without external constraints. In this technical note, we test this distinction numerically on a 1D network flow problem (Darcy flow with a bottleneck). We demonstrate two results. First, the condition number of the regularized Hessian is blind to the bottleneck when the observation operator is the identity. The correct geometric detector is the condition number of the graph Laplacian. Second, even with the correct detector, the posterior variance (derived from the Hessian) anti-correlates with the proximity to a physical catastrophe (capacity saturation). We conclude that the Hessian is a microscope for ill-posedness, not a crystal ball for physical events. A predictor requires coupling geometric diagnostics with external physical loads.