Corollary: Sharp isotropic continuum maximal speed

Define continuum event signalling by matched sender modifications localized in arbitrarily small neighborhoods of an event $A$, a common finite readout at $B$, and a nonzero unconditioned readout-law gap. For positive-duration pairs,

Every Euclidean direction attains the boundary speed in this limiting sense.

Proof. If the ratio exceeds $c$, a sufficiently small neighborhood of $A$ is at positive distance from $J_c^{-}(B)$. Theorem thm:DOD excludes a gap. Thus $c_{\mathrm{op}}\le c$.

Theorem thm:relay supplies a matched boundary pair in every direction with gap one. Its two limiting readout laws are still ${\delta }_0,{\delta }_1$, proving the reverse inequality.