Theorem: Neutral decoder failures preserve the classical bound

If the decoder-output law $p(O\mid x,j)$ factors through a $K\le 2$ cut as in (eq:classicalfactor), then

The decoder's neutral-output probability may depend on $m,j,\lambda$.

Proof. Replace every neutral decoder output by an independent fair binary answer. This replacement preserves the expected score and uses the same classical memory. Apply Theorem thm:classical.