Theorem C — Self-Interpretation

Prerequisites: Theorem B (door finiteness), Chapter 6 (Axiom A5).

Theorem C. Under Axiom A5, $\Sigma_\tau(F,t)$ and $\mathbf{e}(F,t)$ are determined by $\mathcal{D}(F,t)$ alone.

Proof. (Complete; reproduced from Chapter 6.)

$b_{F,t}(i)=d_t(i)-\sum_{j\in F}W_t(i,j)$. Both terms are in $\mathcal{D}(F,t)$. $\Sigma_\tau$ is a level set; $e_p=b_{F,t}(p)$. No exterior information is used. $\square$