When Regularization Prefers the Wrong Answer: Dam Seepage Inversion as a Negative Result

We report a computational experiment applying the regularization framework to a 2D synthetic dam seepage problem. The forward model uses the steady-state Darcy equation on a layered domain with two Gaussian sources: a main leak and a hidden second leak of half the amplitude. The inverse problem is posed on a 4800-cell grid with up to 20 training piezometers. The central finding is a negative result: neither L1, L2, nor Elastic Net recovers the correct depth of the main source. Instead, the L1 prior actively prefers a shallow artifact. The residual Hessian explains this via the "Hessian-Murphy" phenomenon: the shallow artifact has a higher diagonal observability than the true source (13.4 vs 6.2 normalized units). Because L1 minimizes the norm, it selects the more observable, but incorrect, cell. We prove that exact-cell localization is structurally impossible at this operator without a sensor within approximately 2 meters of the target. Furthermore, we record a formal refinement to the Manifesto: at D = Identity, the condition number decreases with alpha, contradicting the theoretical prediction and requiring a D = Gradient (Total Variation) refinement in future experiments. This work demonstrates that regularization cannot substitute for observation geometry, and that the residual Hessian can serve as a diagnostic tool to predict exactly where and why the algorithm will fail.