Manuscript VIII-C@3 · VIII-C-v3.tex · lemma · 3b070436

Lemma: Uniform coefficients and cap separation

Realization interfaces and explicit data › Particle-family geometry · lines 783–821 · proof included or linked

VIII-C@3#lem:bounds, VIII-C@3#eq:bounds

For every $μ \in I$ ,

0 < b_{μ} \leq \frac{3}{5} < a, \frac{129}{200} < a < \frac{137}{200}, \frac{1}{a - b _{μ} /2} < \frac{29}{10} < r < \frac{7}{2} .

For $0 < s \leq 1$ , the outgoing star is strictly inside the cap:

ρ_{c} (C_{i}) = 2 C s = \frac{s}{a - b _{μ} /2} < \frac{29}{10} < r .

Proof. Write $α /2 = π /4 + ε$ , where $∣ ε ∣ \leq π /120 < 11/420$ . Then

a = \frac{2}{3} (cos ε - sin ε) .

The inequalities $cos ε \geq 1 - ε^{2} /2$ and $∣ sin ε ∣ \leq ∣ ε ∣$ give

\frac{2}{3} (1 - \frac{1}{2} (11/420)^{2} - \frac{11}{420}) < a < \frac{2}{3} (1 + \frac{11}{420}) .

The left bound exceeds $129/200$ , and the right bound is below $137/200$ . Since $0 < α < π$ , $b_{μ} > 0$ , and $b_{μ} \leq 3/5 < 129/200 < a$ . Consequently,

\frac{1}{a - b _{μ} /2} < \frac{200}{69} < \frac{29}{10}, \frac{400}{137} < r < \frac{400}{129} < \frac{7}{2} .

The remaining claims follow from eq:scalars.

Displayed formulas in this passage

VIII-C@3#eq:bounds · line 786
$0 < b_{μ} \leq \frac{3}{5} < a, \frac{129}{200} < a < \frac{137}{200}, \frac{1}{a - b _{μ} /2} < \frac{29}{10} < r < \frac{7}{2} .$
unlabeled · line 793
$ρ_{c} (C_{i}) = 2 C s = \frac{s}{a - b _{μ} /2} < \frac{29}{10} < r .$
unlabeled · line 801
$a = \frac{2}{3} (cos ε - sin ε) .$
unlabeled · line 807
$\frac{2}{3} (1 - \frac{1}{2} (11/420)^{2} - \frac{11}{420}) < a < \frac{2}{3} (1 + \frac{11}{420}) .$
unlabeled · line 815
$\frac{1}{a - b _{μ} /2} < \frac{200}{69} < \frac{29}{10}, \frac{400}{137} < r < \frac{400}{129} < \frac{7}{2} .$

Lemma: Uniform coefficients and cap separation

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