Manuscript En01-C@1 · En01-C.tex · corollary · 00a2161c

Corollary: Required precision and optimal clipping scale

A bounded complete-record statistical certificate › Tail and implementation control · lines 1733–1758 · proof included or linked

En01-C@1#cor:optimal

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For fixed $G_{T} > 0$ , $B > 0$ , and $γ > 0$ , the nominal gap before complete-law errors is maximized at

L_{*} = \frac{2 B}{G _{T}},

with value

g_{nom, max} = \frac{γ G _{T} ^{2}}{96 B} .

Consequently a positive certificate of this form exists exactly when

e_{0} + e_{1} < \frac{γ G _{T} ^{2}}{96 B},

provided the clipping scale is unconstrained. If hardware limits the record range, optimize only over admissible $L$ .

Proof. Maximize $γ (G_{T} / L - B / L^{2}) /24$ over $L > 0$ . Its derivative vanishes at $2 B / G_{T}$ , where the unique positive maximum has the stated value.

Displayed formulas in this passage

unlabeled · line 1737
$L_{*} = \frac{2 B}{G _{T}},$
unlabeled · line 1741
$g_{nom, max} = \frac{γ G _{T} ^{2}}{96 B} .$
unlabeled · line 1745
$e_{0} + e_{1} < \frac{γ G _{T} ^{2}}{96 B},$

Corollary: Required precision and optimal clipping scale

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