Example: Carrier-only history dependence from a classical register

Let a persistent hidden classical bit $E$ be uniform. Initialize a visible qubit in $|0⟩$. Apply ${{\sigma }_x}^E$, measure in the $Z$ basis, and record $R_1$. Then reset the carrier to the same $|0⟩$, independently of $R_1$. Apply the same sector-controlled ${{\sigma }_x}^E$ and measure again.

Both first histories have probability $1/2$, and

The carrier input after the reset is identical in both histories. The retained history dependence is entirely classical, stored in $E$.