Proposition: Two distinct extraction-error regimes

For an ideal target input with geometric marginal:

1. Under the global bounded model above, implemented extraction with a correctly prepared register differs from ${\mathcal{B}}_N$ by at most

   $${t_{N,K}}^{\mathrm{glob}}=\min \{1,{\zeta }_{\mathrm{glob}}+{q_h}^{K+1}\}.$$

2. If instead an implementation is certified only on inputs supported in $P_K$, with referenced error ${\zeta }_{\mathrm{low}}$, and its behavior outside that domain is arbitrary, then a valid bound is

   $${t_{N,K}}^{\mathrm{low}}=\min \{1,{\zeta }_{\mathrm{low}}+2\sqrt{{q_h}^{K+1}}\}.$$

The sharp probability tail is not available merely from a support-only control certificate.

Proof. For (i), compare the implemented channel with ${\mathcal{E}}_{N,K}$ using Lemma lem:Duhamel, then use Theorem thm:extraction.

For (ii), project and normalize the referenced input on $P_K$. Its distance from the original is at most the square root of the discarded probability, by purification. The implementation and ${\mathcal{B}}_N$ each contract this distance. On the projected input, ${\mathcal{E}}_{N,K}={\mathcal{B}}_N$, and the support certificate costs ${\zeta }_{\mathrm{low}}$. Add the two projection costs.