Finite compression and implementation bounds

For compatible finite channel sequences, the same one-factor-at-a-time expansion gives

Keeping history registers inside every comparison makes the bound apply to the entire history instrument.

Conditional normalization is more delicate. If the complete history channels are within $ϵ$ in half-diamond norm, a fixed unnormalized branch differs in trace norm by at most $2ϵ$. For positive branch probabilities at least $p_{\min }$, the normalized conditional trace distance is at most $2ϵ/p_{\min }$, by adding and subtracting one unnormalized block divided by the other probability. There is no uniform conditional estimate across arbitrarily rare or zero-probability branches.

Numerical compression does not replace an exact event guard. The reversible event rule continues to use the exact finite parameter word $j=0$, not an approximate length report.