Theorem: Conditional Page-history transfer

Suppose one physical model and one receiver supply complete access-preserving comparisons to a fixed sampled finite proxy $U$, at every declared cut $k$. Let the common accessible comparison dimension be $L_k$, including any necessary padding, and suppose

For every fixed compared input,

If the physical model itself has an ensemble coupled to the Haar proxy with these allowances, then

For coupled single histories, the all-prefix deviation obeys

Hence the normalized leading Page law transfers whenever

and the other hypotheses of Theorem thm:uniform hold.

Proof. The channel comparison bounds output trace distance by ${\eta }_k$. Apply the sharp finite-dimensional entropy continuity estimate Audenaert. Maximizing its right-hand side over trace distances at most ${\eta }_k$ gives $c_{L_k}({\eta }_k)$. Take expectations or the maximum and use the triangle inequality.