Theorem: Sharp continuum speed

Assume C1–C5. Define $c_{\mathrm{cont}}$ as the supremum of

over positive-duration signalling pairs of limit events, with sender modifications supported arbitrarily close to $A$. Then $c_{\mathrm{cont}}=c$.

Proof. If the ratio exceeds $c$, a sufficiently small neighborhood of $A$ is at positive distance from $J_{\nu }^{-}(B)$. Theorem thm:contDOD excludes signalling, so $c_{\mathrm{cont}}\le c$.

C5 gives pairs on the boundary. Total-variation convergence and the triangle inequality preserve a gap at least $\gamma$ in their limit laws. Thus $c_{\mathrm{cont}}\ge c$.