Proposition: Four admitted manipulations

Within the declared interfaces:

1. a clock unitary $V$ before Enter replaces the measured projectors by $V^{\ast }{P}_{k}V$;

2. a pre-readout coupling $W$ on $C\otimes X$ produces branch maps $D_{k}W(\cdot )W^{\ast }{D}_k^{\ast }$;

3. record-controlled continuation by channels ${\Lambda }_k$ produces

   $$\sum _k|k⟩⟨k|\otimes {\Lambda }_k(D_{k}Z{D}_k^{\ast });$$

4. fresh clock allocation permits another readout, but does not recover or reuse a consumed quantum name.

These constructions are CPTP after retaining all branches and remain valid with untouched references.

Proof. The first two claims follow by substituting $VZ{V}^{\ast }$ and $WZ{W}^{\ast }$ into (eq:enter). For the third, the direct-sum channel ${⨁}_k{\Lambda }_k$ preserves the trace of each classical block and is completely positive. Fresh allocation is tensoring with a fixed new state; its name is distinct by definition.