The Structural Failure of Passive Observation and Recovery by Source-Independent Action: Minimal Results in a Finite, Elementary Setting

Using only elementary finite mathematics—the pigeonhole principle and the finite law of total probability, with no causal graph, no measure theory, and no axiom of infinity—we characterise when passive observation suffices to predict interventions and when it does not, obtaining three results. First, impossibility: every bounded representation f : X → Z with |X| > |Z| satisfying a sufficient-statistic condition admits a pair of environments that are observationally identical yet interventionally distinct, on which every passive strategy fails at every sample size. The obstruction performs no arithmetic on probabilities—distributions enter only through equality and its negation, so it holds verbatim with distributions replaced by abstract labels; arithmetic enters exactly once, a single inequality converting the failure into error floors uniform in the sample size. Second, sufficiency: under a source-invariance condition on the outcome mechanism, any non-adaptive policy that draws each action value from a source independent of the latent state recovers the interventional distribution by a single finite sum. Third, necessity: among fixed assignment sources, source independence is exactly the condition for recovery uniform over outcome mechanisms, and every dependent source manufactures the same two-point obstruction, with worst-case error equal to the dependence of the source. Together these give a tight two-sided characterisation: source-independent action is necessary and sufficient for recovery, a dependent source is as uninformative as passive observation, and source independence is what separates failure from recovery. The action operation is graph-free, coinciding with Pearl's do on acyclic structural causal models and extending to simultaneous-equilibrium environments on which do is undefined.