Operational Memory: Predictive Spectra and Realization

A state representation of a dynamical system is typically judged by how accurately it predicts future observations. In systems with memory, however, two histories with the same observed present can respond differently to the same intervention. A model may therefore predict trajectories accurately while discarding historical information that remains predictive of intervention responses. We call this information Operational Memory. We formalize this distinction and show that accurate trajectory prediction need not imply accurate prediction of response differences under intervention. To characterize Operational Memory, we introduce the Operational Predictive Memory Spectrum (OPMS), which decomposes the dependence of future intervention responses on history into predictive directions; a context-resolved form prevents cancellation when history–response relationships vary across matched contexts. We further develop Operation-Complete Predictive Realization (OCPR), which constructs a finite predictive state whose time and intervention updates can be identified and reused recursively. In two spin–boson environments, updates identified using responses with at most two interventions compose to predict responses after a third intervention, with Causal Skill 0.966/0.971. On superconducting qubit data, context resolution reveals dependence missed by the pooled spectrum; an independent quantum jump dataset supports the same principle.