Forward Accuracy Does Not Predict Parameter Bias

Approximate forward models, whether neural surrogates or coarse discretizations, are qualified by forward accuracy against a converged reference. We show this accuracy does not predict the parameter bias they cause inside an inverse problem, and can invert model rankings. On a field-dependent conductivity inverse problem, a surrogate whose forward error is smaller than the reference solver's ($0.66\%$ vs $0.69\%$ rms) incurs $10\times$ the bias in the threshold-field parameter, propagating to a $4.6$--$8.8\%$ error in predicted ablation volume against under $1\%$ for the solver it matched. The failure is not specific to neural surrogates: refining the reference's own mesh improves its forward error $2.6\times$ and leaves its bias no better. It also appears within one training recipe, where three models differing only in random training seed span $17\times$ in error alignment, the most accurate carrying the largest bias. The cause is geometric: in the linearized problem, the fit residual is exactly the projection of the model error onto $\mathrm{range}(J)^{\perp}$, blind to the bias by construction, while the bias is one projection onto a direction $q$ available from one adjoint solve before any data are fit. The same measurement also reduces the damage: dividing out the first-order bias cuts the median $\Ec$ error from $3.62\%$ to $0.58\%$ over six envelope cases, at the cost of one reference sweep per geometry. We therefore conclude that, to control this bias, the alignment must be measured on the checkpoint that will be deployed, not the recipe that produced it.