Theorem: Hybrid-controller support bound

Every output observable supported on $Y$ pulls back to an input observable supported on $S_0$. Equal input reductions on $S_0$ therefore give equal output reductions on $Y$.

If $X$ has a metric and $\mathrm{diam}{R}_t\le \Delta$, then

Proof. If $R_t$ is disjoint from the current support, the observable is the identity on that entire envelope. The channel adjoint is unital, so the envelope contributes no support. If they meet, support can enlarge only to their union. Induction proves (eq:local-backwards).

A new point lies within distance $\Delta$ of a point in the previous backwards set. The triangle inequality gives the metric bound. Finally, equality of input reductions gives equality of every pulled-back output expectation.