Variables from Actions: Canonical and Robust Reconstruction of Causal Coordinates from Intervention Composition

This preprint develops a recognition theorem for determining when a finite monoid of experimentally distinguishable, sequentially composable effects is exactly the monoid of partial hard interventions on a finite variable signature. Intrinsic minimal idempotents act as candidate value setters. Under atomic generation and a commutation-or-mutual-overwrite law, atom noncommutation determines candidate variable classes; a cardinality criterion is necessary and sufficient to exclude hidden collisions and yield a canonical representation. The paper proves uniqueness up to renaming, classifies the automorphism group, supplies explicit obstruction certificates and a quadratic-time checker for a supplied Cayley table, and gives a finite-sample recovery guarantee under separated Bernoulli response profiles. The scope is operational: the result does not recover a causal graph, statistical independence, or an ontology beyond the available intervention repertoire. A targeted literature audit through 2 September 2026 supports only a scoped novelty claim; absolute priority is not asserted.