Manuscript Oai265-B@2 · Oai265-B-v2.tex · theorem · 94b6f42f

Theorem: Three-setting binomial certificate

Exact binomial alternatives · lines 807–844 · proof included or linked

Oai265-B@2#thm:binomial, Oai265-B@2#eq:cpF

Conditional on the training and calibration record, assume iid binary indicators within each setting, with pass probability

p_{j}^{obs} = \frac{1 + s _{j} m _{j}}{2} .

Set $ζ = α_{s} /6$ , and

p_{j, L} = ℓ (n_{j}, k_{j}; ζ), p_{j, U} = u (n_{j}, k_{j}; ζ) .

Define

F_{L} F_{U} = [\frac{p _{x, L} + p _{y, L} + p _{z, L} - 1}{2} - b_{cal}]_{[0, 1]}, = [\frac{p _{x, U} + p _{y, U} + p _{z, U} - 1}{2} + b_{cal}]_{[0, 1]} .

The recovery interval and coverage conclusions of Theorem thm:main remain valid with these endpoints.

Proof. The six one-sided binomial failure probabilities sum to at most $α_{s}$ , and

F_{obs} = \frac{p _{x} ^{obs} + p _{y} ^{obs} + p _{z} ^{obs} - 1}{2} .

Combine the pass-probability intervals, calibration event, and channel conversion.

Displayed formulas in this passage

Oai265-B@2#eq:cpF · line 820
$F_{L} F_{U} = [\frac{p _{x, L} + p _{y, L} + p _{z, L} - 1}{2} - b_{cal}]_{[0, 1]}, = [\frac{p _{x, U} + p _{y, U} + p _{z, U} - 1}{2} + b_{cal}]_{[0, 1]} .$
unlabeled · line 811
$p_{j}^{obs} = \frac{1 + s _{j} m _{j}}{2} .$
unlabeled · line 815
$p_{j, L} = ℓ (n_{j}, k_{j}; ζ), p_{j, U} = u (n_{j}, k_{j}; ζ) .$
unlabeled · line 838
$F_{obs} = \frac{p _{x} ^{obs} + p _{y} ^{obs} + p _{z} ^{obs} - 1}{2} .$