Guide, Not Bind: Why Defeasible Priors Fail in Augmented Lagrangian Causal Discovery

Differentiable causal discovery methods increasingly encode expert priors as forbidden-edge constraints enforced by an Augmented Lagrangian (ALM) penalty, on the assumption that a data-adaptive relaxation mechanism will discount and eventually override a rule the data consistently contradicts. We show this design, which we call guide, not bind, fails for two independent, precisely characterized reasons, and that directly repairing both restores it only partially. First, sequential penalty-ramping ALM suppresses a wrongly-forbidden true edge before any counterfactual check can detect it: we give three necessary conditions any adaptive relaxation must satisfy to avoid this (Proposition~refprop:conditions), prove that DADU---the natural relaxation rule this paper introduces as the object of study---violates all three (Corollary~refcor:dadufailure), and confirm the failure across 3,072 training runs spanning graphs from 4 to 32 nodes, where a single wrong prior suppresses a true edge in 87--97% of trials under DADU. Second, and independent of any fix to the mechanism, we prove in closed form that the standard correlation-matching objective ties a true edge and its reverse to an identical cost of exactly 2r2 (Lemma~reflem:tie), not because the underlying equal-variance model is unidentifiable, but because normalizing to correlation discards exactly the variance information that would make it identifiable; covariance matching instead separates the two directions by a provable margin of at least w04 (Lemma~reflem:separation).