Theorem: Quantitative local packet-effect correspondence

Fix the normalized outgoing s-wave packet $h$, the nonzero quadrature parameter $\vartheta$, and $C\ge 1$. Use the compact tests of Section sec:packetbridge, the local coupling and future probe effect of Specification spec:local, and the independent preparation of Section sec:probe.

For every $\omega \in {\mathfrak{S}}_C$, and every allowed $R,T,ϵ$,

Thus one may take

For $z_{ϵ}\le 1/2$ and $ϵ\le 1$, define

Then the completely displayed upper bound is

The additional smallness condition ensuring the definition of $p_{ϵ}$ is retained.

Proof. The exact induced effect is (eq:inducedeffect). Replacing $c_{ϵ}$ by one changes its expectation by at most $(1-c_{ϵ})/2$.

The coefficient of its system Weyl operator is $k(f_{ϵ})$. Theorems thm:packetbridge and thm:transfer give

Apply Lemma lem:Weylcomparison directly with target $k_{\ast }$, and then cap the difference of two effect expectations at one. This proves (eq:mainbound).

When $z_{ϵ}\le 1/2$, $d_{\mathrm{trf}}\le T_R{ϵ}^2$. Equation (eq:noisebound) supplies the probe term, proving (eq:simplifiedmain).