Representation Sufficiency for Closed-Loop Control: Decision Risk and the Occupancy Bridge

Learned representations for robot control are often evaluated through action prediction on demonstrated states, yet accurate prediction need not imply effective closed-loop control. We separate return loss into the information deficit imposed by a representation and the excess loss of a learned policy within the resulting policy class. On a fixed state distribution, action-value gaps bound the minimum decision risk in terms of Bayes optimal-action error. Relating this risk to attainable return requires accounting for the different state occupancies induced by the risk-minimizing decision rule and the best policy available under the representation. A localized correction further tightens the bounds. Under a fixed macro-action interface, removing direction information reduces the exact optimal discounted return from $0.990$ to $0.291$ ($\gamma =0.99$ per macro decision). The resulting information deficit is $0.699$, whereas decision risk under expert occupancy, scaled to the same return units, is only $0.0246$. Since both optimal returns are computed under the same interface, the deficit reflects the representation restriction rather than learned-policy excess. We further show that identical recorded demonstrations can be consistent with learned-policy excess ranging from zero to linear in the horizon, even when the representation permits optimal control. Finally, fixed-interface data aggregation improves state-based, frozen-latent, and visual policies without adding information at execution time, providing evidence that part of the learned-policy excess is recoverable within the same policy class.