Proposition: Exact local inner simulation can fail

An operation compactly embedded in an open region at one coarse scale need not have an exactly equal fine-scale realization inside the same region, even on nested increasing-stencil sites.

Proof. At a fixed scale $j$, choose a generator edge $\{a,b\}$. Let $O$ be the union of two small disjoint spacetime cylinders about $a,b$, including the coarse firing tick and the required static output dwell segments. Keep their spatial shadows separated by a positive gap.

A coarse exceptional-to-the-fine-description pair block prepares a Bell pair on registers at $a,b$. It is an ordinary admitted generator block at scale $j$, and its endpoint footprints lie inside $O$.

For all sufficiently fine $m$, one-step reach is smaller than the gap. No block of an operation confined to $O$ can meet both components. Starting from the product default, the complete fine operation factors across the components and cannot prepare a Bell pair. Exact equality is impossible.