A fixed-interface table bound

To state a meaningful small approximation bound, the complete output interface must be fixed. Different deterministic decimal strings in orthogonal classical records are perfectly distinguishable; small numerical difference between their printed values does not make those channels close.

For comparison, suppose a bounded real target has been converted to a Bernoulli parameter $p_j\in [0,1]$. Let ${\Lambda }_p$ first perform the clock dephasing, retain the code and residual, and append a binary report with probability $p_j$ on code $j$.