Manuscript En01-B@2 · En01-B-v2.tex · corollary · f226d1eb

Corollary: Exact three-separation signature

Beyond quadratic localization: an explicit control › Exact homogeneous-sphere calculation · lines 1613–1637 · proof included or linked

En01-B@2#cor:exactcontrast, En01-B@2#eq:exactcontrast

With $h_{*} = 5041/81920$ ,

B_{0} = - K_{G} h_{*}, C_{G} = - λ t K_{G} h_{*} .

Proof. The quadratic contribution cancels. The remaining moments are

i \sum w_{i} (d_{i} / R)^{3} = \frac{15}{32}, i \sum w_{i} (d_{i} / R)^{5} = \frac{435}{1024} .

Therefore

- \frac{3}{16} \frac{15}{32} + \frac{1}{160} \frac{435}{1024} = - \frac{5041}{81920} .

Displayed formulas in this passage

En01-B@2#eq:exactcontrast · line 1616
$B_{0} = - K_{G} h_{*}, C_{G} = - λ t K_{G} h_{*} .$
unlabeled · line 1626
$i \sum w_{i} (d_{i} / R)^{3} = \frac{15}{32}, i \sum w_{i} (d_{i} / R)^{5} = \frac{435}{1024} .$
unlabeled · line 1632
$- \frac{3}{16} \frac{15}{32} + \frac{1}{160} \frac{435}{1024} = - \frac{5041}{81920} .$