Corollary: Exact-model limiting front

For ${\tau }_e\sim \mathrm{Exp}(\lambda )$, the infinite-volume controlled CNOT front and every fixed annealed OTOC level $a\in (0,1)$ have macroscopic body $\mathcal{B}$. Finite boxes with $L=\lceil an\rceil$ and the explicit choice of $a$ following Corollary cor:box give the same limit at time $n{t}_0$.

The no-flat-face and $C^1$ conclusions are conditional on hyp:exp. For Gamma timers, only the $C^1$ conclusion follows conditionally from hyp:gamma.

Proof. Theorem thm:exact-otoc identifies the front exactly with the classical process. The escape estimate supplies finite-volume control, the classical shape theorem supplies the limiting body, and Corollary cor:regularity supplies only the stated conditional regularity.