When the Simulator Is Wrong, Do Better Derivatives Improve Decisions?

Scientific surrogates are often trained on values but consumed through their derivatives: molecular-dynamics integrators use forces rather than energies, and force matching is one form of derivative supervision. Derivative labels are the costly part of that data, and which to buy is decided on thin evidence: per-derivative error needs a reference few simulators supply. Ours supplies one: on a two-state stochastic-volatility PDE, after a predeclared mask, up to four numerical legs agree on values to $10^{-3}$ relative and give a measured uncertainty envelope per derivative, so which orderings resolve is measured. At two regimes excised from training and validation, a label set pays for a derivative it does not contain: the rung adding \emph{only} curvature labels improves an unlabelled second-state derivative by $40$ and $26$ percent, resolved at both. Against two unsupervised baselines it cuts held-out second-derivative and second-state RMSE by $82$ to $93\%$; but at the next rung only the newly labelled derivative improves resolvably at both anchors, so the benchmark does not establish monotonic improvement in label count. Under model-class departure the downstream gains are not derivative-specific: at zero cost a value-only arm captures at least $96\%$ of each of rung 3's eight wins over the residual-only baseline, and at the registered $1\%$ cost rung 3 is worse in all ten departed cells. The pre-registered confirmatory endpoint, a coefficient-shift cell, also fails. The ladder is nested, so we claim no general ordering; the results motivate testing whether Hessian supervision improves force accuracy in coarse-grained force fields.