Proposition: Calibration and clock limits

For distinct labels $t_a$, exact event-time calibration at a single actual reading event $e$,

forces one deterministic outcome, or is impossible if no label equals $T(e)$.

A history/programme calibration may instead label different stored events or schedules and need not equal the reading-event time.

The clock instrument has no universal left inverse. No finite matrices satisfy $[T_C,H_C]=i\hslash I_C$, $\hslash \ne 0$. Consuming $C$ does not consume the geometric time coordinate.

Proof. Distinctness permits at most one event-calibrated outcome. For noninvertibility, the two clock states $(|{\tau }_0⟩\pm |{\tau }_1⟩)/\sqrt{2}$, with the same payload, have identical destructive outputs. The commutator assertion follows by tracing the equation. The metric and $T$ are external data, not output factors removed by ${\mathcal{M}}_C$.