Theorem: Continuum domain of dependence

Assume C1–C3. Suppose two limit experiments have a common readout at $a$ and differ only by a compact modification footprint $K^{\prime }$ with

Then their readout laws agree.

Proof. Use C3(iii) to choose matched discrete realizations with terminal events $a_{\epsilon }\to a$. Uniform cone convergence and convergence of the modification footprints imply that, for sufficiently small $\epsilon$, every modification lies outside $J_{\epsilon }^{-}(a_{\epsilon })$, with a positive margin.

Because complete block footprints shrink, no changed block meeting the next cone slice is hidden outside the declared modification footprint. The discrete initial cone snapshots agree and the required inner block channels agree. Theorem thm:cone gives equal discrete readout laws. Their total-variation limits are equal by C3(i).