Proposition: A simple finite-cut non-determination witness

A finite list of channel values at externally specified cuts does not in general determine the intermediate continuous-time family.

Proof. Take one co-located qubit and finitely many sampling times $s_0<\cdots <s_m$. The identity family agrees at those times with

where $\vartheta$ is a nonzero smooth function supported strictly between two successive cuts and chosen not always to be an integer multiple of $\pi$. At every sampled time $U(s_j)=I$, but ${\mathrm{Ad}}_{U(s)}$ is nontrivial at some intermediate time.