Proposition: Calibration and clock limits

At one fixed actual reading event $e$, distinct calibrated labels $t_a$ satisfying

permit at most one positive-probability outcome. A nontrivial multi-outcome history or programme label therefore requires a separate interpretation, not that event-time equality.

The clock channel has no universal left inverse on its full input space. No finite matrices obey $[T_C,H_C]=i\hslash I$ for $\hslash \ne 0$. Clock consumption does not consume $T$, $t$, or a spacetime dimension.

Proof. Distinct labels prove the first assertion. The two clock states $(|{\tau }_0⟩\pm |{\tau }_1⟩)/\sqrt{2}$, with the same payload, give identical destructive outputs, excluding a universal inverse. Tracing the commutator equation gives a contradiction. The geometric coordinates remain fixed classical inputs.