Proposition: Four finite manipulations

The following operations have the stated limits.

1. Changing the clock basis generally changes its pointer observable and instrument, not the coordinate $T$.

2. A coupling $U_{CX}$ before consumption gives the new instrument ${\mathcal{M}}_C\circ {\mathrm{Ad}}_{U_{CX}}$. It is a declared interaction, not a metric back-reaction law.

3. Classical control by $k$ selects the declared smearing, matrix, or continuation. It does not change the gauge quotient unless the equation-space contract is explicitly changed.

4. Fresh clock allocation creates new names and a new reading event. It does not create a continuum limit.

Proof. For (i), a unitary basis change replaces the pointer projectors by $U{P}_k{U}^{†}$, which generally changes effects and residual maps. The supplied geometric functions are not outputs of that operation.

For (ii), compose the recorded Kraus operators with $U_{CX}$. Completeness remains valid, but no metric-update equation has been added.

For (iii), the finite code chooses only from the printed continuation. The classical equivalence relation is separately defined and is not an output of a control branch.

For (iv), allocation extends a finite instruction tree and its resource ledger. No limiting algebra, convergence map, or continuum evolution follows from that extension.