Long-range primitives: exactly which guarantee fails

A non-product channel need not be transmissive: a constant Bell-pair preparation has non-product output but no input dependence.

The extension has an exact cone for the shortened metric

This is the shortest-path metric obtained by adding an edge of length $r$. A shortest path need use that edge at most once; finite $d_W$-balls are contained in a finite union of finite $d$-balls.