Gauss–Newton is Blind to Embedded Saddles: Value-Trained Surrogates as a Derivative Oracle for Amortized Inversion

Amortized inversion—one nonlinear least-squares solve per target, across many targets sharing one forward model—commonly initialises the full model from the fit of a reduced one. By the embedding principle of nested models, such a fit, when interior and stationary, is an exact critical point of the richer model's landscape; we characterise it in closed form as a strict saddle whenever a sign test on the residual holds, and for coalesced Gaussian components generically. The saddle is invisible to the Gauss–Newton matrix $J^{\top }J$: it carries exactly zero curvature in the escape subspace, so Gauss–Newton, Levenberg–Marquardt and secant-corrected methods report convergence at the non-minimiser. The escape lives in the residual-weighted term $C={\sum }_i{r}_i{\nabla }^2{f}_i$ that Gauss–Newton discards; taking it from the true model costs $O(n^2)$ evaluations per finite-difference probe or an interpolation model, restarts multiply the cost, and under evaluation noise a single-sample stencil faces a bias–variance trade-off. We show the term can be learned: a neural surrogate of the single-component shape, trained on function values alone with input dimension independent of $n$, supplies the Jacobian and the curvature at zero true-evaluation cost, and the fidelity that matters is low leakage into the well-determined subspace, not value accuracy. On Voigt-profile families with $n=8$–$32$, Surrogate-Guided LM matches the strongest classical escapes to within a few points (twelve behind at transmission $n=8$) at a cost flat in $n$, $12$–$36\times$ below them at $n=32$; an ablation attributes the cost to the free Jacobian and the first-attempt escape to the curvature (a single split perturbation reaches $63$–$81\%$, the curvature step from it $79$–$96\%$). With $0.1\%$ evaluation noise it meets the accuracy criterion on $73$–$88\%$ of targets against at most $52\%$ for single-sample alternatives. On an external stellar-synthesis code at $2$ s per call, initialised at the single-star fit of a synthetic binary, it meets the success criterion on all $52$ non-trivial targets at $3$–$9\times$ fewer syntheses than the tested classical escapes—a saving the free Jacobian supplies, since a perturbed split without the curvature step does as well—the surrogate's cost repaid after about seven solves.