Closer Is Not Sharper: Where to Spend Simulator Budget Near a Tipping Point

Fitting a linear Ornstein–Uhlenbeck (OU) surrogate to simulator output and reading the distance to a critical threshold out of the fitted variance is standard practice. That surrogate is the imperfect scientific model here, and it promises that the information one observation carries about the control parameter $r$ diverges as $|r|^{-2}$, so budget spent nearest threshold should be the most valuable of all. Both halves are wrong, and differently. Conditioning on non-explosion — which every non-diverged simulator run has already undergone — removes the divergence: the snapshot Fisher information is bounded and non-monotone, overstated by a factor of two inside $|s|=0.61$ but \textit{under}stated by $14%$ outside $|s|=1.24$. That ceiling belongs to the simulator's stopped law, not to any estimator — and it is not what a practitioner should maximise. A run yields a snapshot only if it is still alive, so contraction per call from the snapshot is $J=S(|s|),\mathcal{I}s(s)$, whose optimum lies \textit{outside} the observability horizon at $90$–$93%$ survival, never at the peak of $\mathcal{I}s$; a neural posterior estimator measures its shape to within error. But an ensemble also sees \textit{how many} runs survived, and that channel dominates, moving the optimum back onto the peak and inside the horizon. The naive target is thus roughly right for a reason the surrogate cannot see — not that snapshots sharpen there, but that survival is informative there — and $J$ governs only a single record, which is the early-warning setting.