Proposition: Record laws do not certify conditional evolution

There are recorded channels with the same clock outcome law on every input and different conditional system states.

Proof. Let $S$ contain a qubit, let $Z_S$ be its Pauli $Z$, and compare

Unitary conjugation preserves each block trace, so their record laws agree on every input. For input $|{\tau }_0⟩⟨{\tau }_0|\otimes |+⟩⟨+|$, the conditional qubit output is $|+⟩$ for $\mathcal{F}$ and $|-⟩$ for $\mathcal{G}$. The residual states are orthogonal.