Recovery Equivalence Theory: The structure of non-identification, and decision over the equivalence class

Reconstructing the network of dependence between economic units from how they move together rarely singles out one structure. This note builds the theory of that ambiguity. It characterises the recovery equivalence class, the set of structures consistent with a given observation, and computes the size of the indeterminacy exactly. The quantities the data fix are precisely the invariants that survive it, which turns a three-way reading of identification, firmly identified, identified only up to a calibration constant, or beyond what the data can fix, from a convention into a theorem. Non-identification is given a closed-form measure, the class varies continuously with the observation except where the leading spectral gap closes, and a confidence-weighted decision rule nests outright refusal and point-identified action as its two limits, with worst-case regret that vanishes as the data resolve more of the structure.