Theorem: Amplitude visibility or erasure

For ${\iota }_1$ and $P\in {\mathcal{P}}_4$, the following are equivalent:

1. The complete record is invariant on (eq:orbit).

2. $B_y=C_y=0$ for every reported output $y$.

3. The complete record is independent of $a$ on $\mathbb{R}$.

For the classical law alone, the corresponding condition is ${\beta }_y={\gamma }_y=0$ for all $y$.

Proof. Differentiate (eq:selective):

where $s,t$ are evaluated at $\lambda c_{\epsilon }$. Distinct classical blocks cannot cancel. At $\lambda =0$, invariance forces $B_y=0$. At any nonzero $\lambda$, $st\ne 0$, so it then forces $C_y=0$. The converse follows from (eq:selective). Taking traces proves the law statement.