Corollary: Explicit finite-time epsilon certificate

Fix $d\ge 2$, $c>0$, and $0<\epsilon <1$. Choose

and

For the prescribed packets, if

then ${\epsilon }_{\mathrm{implemented}}\le \epsilon$. A separate accessible collection allowance is included in the second budget if needed.

Nonzero autonomous perturbations are allowed; for example,

The same packet count gives the independent predictive allowance ${\zeta }_N(T)\le \epsilon /4$.

Proof. The survival estimate gives

Since $M\ge 2ez$,

Since $M\ge {\log }_2(16/\epsilon )$,

The ideal term is at most $\epsilon /2$ and the implementation terms at most $\epsilon /2$. Apply Theorem thm:C8.

The constant source shift is bounded, autonomous, and excitation preserving, with ${\Delta }_T=T\delta$. For positive $\delta$, positivity is retained. Theorem thm:C7 gives ${\zeta }_N(T)\le 42^{-M}\le \epsilon /4$. This predictive allowance is separate from the direct recovery calculation.