A Diagnostic Radius for Knowledge-Induced Causal Effect Estimation

Observational causal discovery usually identifies an equivalence class of causal graphs rather than a unique graph. In applied settings, background or expert knowledge can further constrain uncertain edge orientations, after which sound orientation rules propagate their logical consequences. We give a query-specific certificate: the fewest closure orientations that must be contradicted before a fixed adjustment set can fail, conditional on a correct equivalence class, computed exactly by retractions without enumerating the space of knowledge states. On synthetic graphs whose generator spreads instances across radius levels by targeting the treatment's separation from the adjustment set, this breakdown radius tracks survival to corrupted knowledge at a rank correlation of 0.75 to 0.81 under three- and four-tier corruption, against at most 0.14 for structural Hamming distance to the true graph. When the query is instead drawn without targeting separation, the radius ranks survival at 0.29 and the structural proxy separation at 0.34, both well above SHD, with the radius still leading SHD by about 0.3. On published networks, with knowledge scenarios constructed by large language model elicitation, it ranks survival within fixed knowledge states, where knowledge counts are constant. We release code and benchmarks so that a breakdown radius can be reported alongside any causal estimate that relies on an adjustment set.