Intervention Geometry in Polynomial Causal Models: Exposure, Power, and Experiment Design

Closed-loop scientific agents must decide not only *what* to perturb, but *how strongly*. We study intervention-level selection for nonlinear polynomial structural causal models (SCMs). If a candidate mechanism omits one genuine parent $r$, hard interventions $do(X_r=u)$ trace a vector-polynomial path in its reduced coefficients. We prove that $d_r+1$ distinct levels deterministically expose an omitted degree-$d_r$ parent, that this bound is worst-case sharp, and that two independently sampled continuous levels expose it almost surely. Finite-sample detectability is ranked by a Wald information score $\lambda_n(u,v)=n\delta(u,v)^\top\Omega(u,v)^{-1}\delta(u,v)$, explaining why perturbation distance can be misleading. Across 324 frozen synthetic conditions, $\lambda_n$ ranks empirical HC3-Wald power with Spearman $\rho=0.9918$. On sci-Plex3 A549 perturbations, a training-replicate information score predicts held-out dose-pair separation better than log-dose distance ($\rho=0.494$ vs. $0.142$). In strict offline next-dose replay, a cross-compound acquisition model reaches $94.18\%$ of held-out oracle utility versus $86.68\%$ for uniform random choice.