Manuscript Oai265-A@2 · Oai265-A.tex · proposition · c0a34a45

Proposition: Forty-shot nonacceptance for this procedure

Finite memory certification is not entropy calibration › A separate, explicitly defined qubit procedure · lines 1721–1750 · proof included or linked

Oai265-A@2#prop:forty

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For the procedure of Theorem thm:statistics, $N = 40$ cannot certify a half-diamond threshold $0.1$ , even with $b_{cal} = b = 0$ and perfect observed oriented correlations. This is a limitation of the stated confidence procedure, not a universal impossibility theorem for every test.

Proof. Since every observed mean is at most one, $f \leq 1$ . The smallest possible upper endpoint is therefore

min {1, 2 min {1, 3 t_{N} /4}} .

For $0 < α < 1$ ,

t_{N} > \frac{2 lo g 6}{40} > \frac{1}{5} .

Thus the squared uncapped endpoint is greater than $3/5$ , far above $0. 1^{2}$ ; capping at one does not help.

More generally, with zero bias and transfer allowance, this procedure requires at least

N \geq \frac{18 lo g ( 6/ α )}{τ ^{4}}

for perfect data even to permit acceptance at threshold $τ < 1$ .

Displayed formulas in this passage

unlabeled · line 1733
$min {1, 2 min {1, 3 t_{N} /4}} .$
unlabeled · line 1738
$t_{N} > \frac{2 lo g 6}{40} > \frac{1}{5} .$
unlabeled · line 1746
$N \geq \frac{18 lo g ( 6/ α )}{τ ^{4}}$

Proposition: Forty-shot nonacceptance for this procedure

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