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

Identity example: corrected numerical interpretation

Supplied script execution report and its interpretation › Identity example: corrected numerical interpretation · lines 1327–1372

Oai265-B@2#eq:identityreported

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For all three identity settings, the report gives the outward Clopper–Pearson lower endpoint

p_{L} = \frac{1905242823}{2147483648} \approx 0.8871978255920112.

Therefore

F_{L} = \frac{3 p _{L} - 1}{2} = \frac{3568244821}{4294967296} \approx 0.8307967383880168.

The rational upper endpoint is

U_{rat} = \frac{28267292255}{34359738368} \approx 0.8226864812604617.

This corrects the report's coarser interpretation “approximately $0.8225$ ” without changing its exact fraction. Moreover,

4 (1 - F_{L}) = \frac{726722475}{1073741824} \approx 0.6768130464479327 > 0.01.

The exact acceptance test therefore fails.

For an arithmetic consistency check, the reported fractions satisfy

(\frac{28267292255}{34359738368} - 2^{- 39})^{2} < \frac{726722475}{1073741824} \leq (\frac{28267292255}{34359738368})^{2} .

The spacing $2^{- 39}$ is the spacing of twice a 40-step dyadic square-root bracket. This is a check of the recorded endpoint's algebraic role, not an independent execution of the script.

Displayed formulas in this passage

Oai265-B@2#eq:identityreported · line 1342
$U_{rat} = \frac{28267292255}{34359738368} \approx 0.8226864812604617.$
unlabeled · line 1331
$p_{L} = \frac{1905242823}{2147483648} \approx 0.8871978255920112.$
unlabeled · line 1336
$F_{L} = \frac{3 p _{L} - 1}{2} = \frac{3568244821}{4294967296} \approx 0.8307967383880168.$
unlabeled · line 1351
$4 (1 - F_{L}) = \frac{726722475}{1073741824} \approx 0.6768130464479327 > 0.01.$
unlabeled · line 1358
$(\frac{28267292255}{34359738368} - 2^{- 39})^{2} < \frac{726722475}{1073741824} \leq (\frac{28267292255}{34359738368})^{2} .$