Theorem: Reference-stable finite-volume tail

For disjoint nonempty $A,B$, let $d=d_G(A,B)\ge 1$. Then

For every $\gamma >0$, the series is at most

The estimate is independent of the untouched reference dimension, but is not uniform in the number of edges.

Proof. Pass to the interaction picture for the onsite propagator, a product of onsite unitaries. It preserves support and all edge-term norms.

For a Hermitian output observable $O_{BE}$ of norm at most one, expand the Heisenberg evolution in its norm-convergent ordered nested-commutator series. At order $m$, summing edge choices and ordered integrals bounds the norm by $(2\Lambda {)}^m/m!$. Each nonzero commutator must meet the current site support, and can extend its graph distance from $B$ by at most one. Every term of order below $d$ is therefore identity on $A$, also when its coefficients are arbitrary operators on $E$.

The two intervention adjoints agree on that partial sum. They are operator-norm contractions, so their difference on the remainder is bounded by twice the series in (eq:tail). Trace-norm duality and the $1/2$ normalization in Definition def:signal give (eq:tail). Insert $1\le e^{\gamma (m-d)}$ for $m\ge d$ and sum the exponential series for the final bound.