Proposition: Clock limits

The following hold:

1. if the labels ${\theta }_s$ are distinct, exact calibration to one actual readout event,

   $$p_s>0⟹{\theta }_s=T(e),$$

   forces one deterministic code, or admits no normalized input when $T(e)$ is not a label;

2. ${\mathcal{M}}_C$ has no universal left inverse;

3. finite matrices cannot satisfy $[T_C,H_C]=i{I}_C$;

4. the instrument's records do not uniquely determine the geometric dictionary.

Proof. Distinct labels allow at most one supported outcome and probabilities sum to one. For (ii), the clock states $(|{\tau }_0⟩\pm |{\tau }_1⟩)/\sqrt{2}$, with the same payload, have identical destructive outputs. For (iii), take the trace of the proposed commutator.

For (iv), keep all finite matrices unchanged and relabel history times or rotate and translate the local rod frame. Small regular laboratories for different prescribed deficits can also carry the same finite instrument. The record law does not change.