A Structural View of Query Misspecification in Causal Foundation Models

Causal Foundation Models (CFMs) pretrain amortized causal estimators on large collections of synthetic datasets sampled from structural causal model (SCM) priors. In optimal-capacity, they recover the corresponding interventional posterior predictive target for queries on the training query surface. We study the failure mode induced when an inference-time query includes a post-treatment covariate. Structurally, we decompose the resulting CATE bias into three components: loss of the natural indirect effect, an interaction penalty, and treatment-differenced selection bias. Distributionally, we prove that conditioning on post-treatment values yields strictly positive KL divergence from the marginal interventional law on a positive-measure subset, and we provide closed-form KL decompositions under linear-Gaussian SCMs. Empirically, removing post-treatment covariates from the query yields substantial reductions in PEHE across graph topologies and CFM models without retraining. We further introduce Treatment-Centric Local Discovery (TC-LD), a lightweight pre-inference filter that flags likely post-treatment variables and recovers most of this improvement on our synthetic benchmark.