Proposition: Clock-conditioned source promises

If a source command acts after clock readout, its token promise must hold for the conditional source reduction in every positive-probability branch that receives the source interpretation. A promise on the unconditional source marginal alone is insufficient.

Proof. The actual branch input is ${\sigma }_a=D_a\rho {D_a}^{†}/p_a$. Its source expectation is evaluated in ${\sigma }_a$, not in their average. For example, a clock classically correlated with source states $|0⟩,|1⟩$ can have unconditional source expectation zero but conditional expectations $1,-1$. The channel identity remains valid without any promise; only the source interpretation requires the branchwise condition.