Beyond State Sufficiency: The Dimension of Gain-Bounded Predictive Realization

Once a representation fully specifies the state, can regular prediction still require additional dimensions? We establish a quantitative separation between state identification and regular predictive realization. Transition structure can impose an additional dimensional requirement, even when no state information is missing. We define predictive realization dimension and derive verifiable certificates for exact prediction under a shared regularity bound. These certificates rule out all state-preserving encodings below certified thresholds and permit non-invertible transitions. Grid experiments preserve every state identity after dimension reduction, yet the walled systems require more dimensions even when small expansions are allowed. Our results identify a limit of information-based dimensionality assessment: coordinates can be redundant for state identification but necessary for predictive geometry.