Average Accuracy Does Not Certify Topology in Learned Hamiltonians

Learned electronic Hamiltonians are commonly evaluated by errors in matrix elements, band energies, or occupied subspaces, but when do these metrics certify topology? We show that no strictly positive finite-\(p\) integrated-error threshold universally certifies topology over unrestricted smooth gapped families: a topology-changing discrepancy can be confined to arbitrarily small support while retaining finite pointwise amplitude. Conversely, topologically distinct, well-defined rank-\(r\) occupied bundles must attain subspace infidelity at least \(1/r\) somewhere, motivating uniform, gap-aware control. Learned four-band models realize this mismatch: lower band MAE ranks a family with \(0/15\) correct topologies ahead of one with \(15/15\) correct. In realistic settings, a structured perturbation changes the Chern number of a \(32\)-orbital NaRuO${}_2$ Wannier Hamiltonian at \(8.074\) meV near-Fermi band MAE, while a released neural Hamiltonian undergoes gap collapse that invalidates a raw topological comparison. Equal-budget loop-structured sampling shows no stable advantage. In an exploratory \(10\)-meV band-matched study, all \(276\) evaluations valid at both endpoints preserve topology, and no target-valid plane becomes newly unsafe. Together, these results expose a certification gap between predictive accuracy and topological fidelity without implying that such failures are prevalent under generic small spectral perturbations.