Theorem: Resource-dependent tolerance schedule

Fix $h,\vartheta ,C,T$ and a desired expectation tolerance $\alpha >0$. Within the family permitting the stated growing shells, compact couplings, future tests, and squeezed probe preparations, finite parameters meeting ${ϵ}_{\mathrm{local}}\le \alpha$ exist.

A sufficient choice is:

1. choose a finite integer $R$ with $H_C{\Delta }_R\le \alpha /2$;

2. for that fixed $R$, choose $ϵ>0$ satisfying

   $$ϵ\le 1,{ϵ}^2{\Lambda }^2\le 1/2,$$
   $$(H_C{T}_R+N_R/4){ϵ}^2\le \alpha /2,$$

   and the nonzero-denominator condition for $p_{ϵ}$.

Proof. Theorem thm:packetbridge proves ${\Delta }_R\to 0$. The ordinary derivative integrals in (eq:ordinarytailbound) provide a terminating search criterion on increasing integer $R$.

For fixed $R$, all displayed Green, support, and covariance constants are finite. The inequalities in the second step are therefore satisfied by sufficiently small positive $ϵ$. Equation (eq:simplifiedmain) proves the result.