Example: A carrier-only causal-break witness with classical memory

Let a hidden classical bit $E$ be uniform. Allocate a carrier in $|0⟩$, apply ${{\sigma }_x}^E$, and destructively read it in the $Z$ basis, obtaining $R_1$.

After that readout allocate a fresh carrier in the same $|0⟩$, irrespective of $R_1$. Apply ${{\sigma }_x}^E$ again and read $R_2$.

Then

but

The post-break carrier preparation is identical in both histories. The surviving information is the classical register $E$. All carrier states can remain diagonal.