From Linear Domain Invariance to Deployment Robustness: A Construct-Validity Audit of First-Moment Erasure

Domain-invariant representations are commonly motivated by robustness under distribution shift, and commonly validated by showing that a probe can no longer recover the domain. We ask what success on that measurement justifies about deployment behaviour. Using closed-form first-moment erasure as a controlled intervention that can be driven to the criterion's ceiling, we find that across three benchmark families the linear criterion is fully satisfied while the false-positive rate of a detector at an unseen domain shows no detected improvement. Two properties of the measurement explain the separation. A per-domain first moment retains only the content-averaged nuisance response, and in a controlled nuisance study 68–86% of the measured tangent energy lies outside that component. The finite-sample version of the same summary adds a within-group covariance term that grows as groups shrink, so a refinement of the conditioning that carries no population information can appear to recover geometry. A matched-cardinality random partition exhibits exactly this artefact, and cross-fitting removes 96% of the apparent gain. The practical consequence for evaluation is that metric success, nuisance suppression, semantic preservation and deployment behaviour are separate claims and should be reported separately.