Decision-weighted prediction redirects what a capacity-limited encoder retains

Capacity-limited predictive encoders are standard in model-based control: each compresses a controller's belief into a code too small to hold it, and no guarantee says the code keeps the directions the control cost depends on. The unweighted objective ranks belief directions by predictive energy, which the control cost does not enter. We prove, using standard spectral arguments, that one reweighting of the target redirects the same capacity onto the directions the cost weights most. The weight is a fixed matrix, the factor that makes prediction error measure control cost. That statement is read at an exact optimum with no tie in the cost's ranking, so our finitely trained runs illustrate it rather than test it. Such a factor exists exactly when the predictive map discards no decision-relevant direction; a counterexample shows the reweighting cannot be dropped. Two companion results drop encoder linearity, show decoder linearity cannot be dropped, and make the condition checkable in advance; every theorem is machine-checked in Lean. On synthetic linear-Gaussian systems the weighted target lowers decision-relevant error wherever the capacity constraint binds.