Theorem: Exact fixed-table channel distance

Keep the same pinching, history labels, payload interface, and report words, and change only $v_{ki}$ to ${\tilde{v}}_{ki}\in [-1,1]$. Then

Proof. Include an arbitrary reference and apply both maps to an arbitrary operator $X$. The output $(k,i,y)$-flags are orthogonal. For a fixed input block, the sum of the absolute differences of the two sign probabilities is $|v_{ki}-{\tilde{v}}_{ki}|$. Hence

The sum is the trace norm of the pinched input and is at most ${\Vert X\Vert }_1$. Pinching is an average of unitary conjugations, so this inequality applies to arbitrary operators and references. This proves the diamond upper bound. A basis input supported on a maximizing $(k,i)$, with any normalized payload, attains it.