Theorem: Finite half-diamond experiment certificate

Fix ${\alpha }_{\mathrm{s}}\in (0,1)$, and put

Define

If $F_L>F_U$, report an inconsistent interval and do not certify.

Otherwise, on an event of probability at least

the implemented target error obeys

A sufficient acceptance rule for threshold $\epsilon$ is

The probability of accepting while the actual target error exceeds $\epsilon$ is at most ${\delta }_{\mathrm{tree}}+{\alpha }_{\mathrm{s}}+\beta$.

The tree interval may be omitted. In that case its failure allowance is omitted as well.

Proof. Conditional Hoeffding inequalities give

A union bound and integration over the conditioning record give simultaneous statistical coverage with failure probability at most ${\alpha }_{\mathrm{s}}$. On this event and $G$, the experimental interval covers $F({\Lambda }_0)$. Lemma lem:treebell supplies the independent type of tree-law interval. On the intersection of the three valid events, their intersection is nonempty and

Apply Lemma lem:fidelity, then the reverse triangle inequality and $d_{\diamond }({\Lambda }_{\star },{\Lambda }_0)\le b$. The union bound proves joint coverage and the invalid-acceptance bound.