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

Finite packet table

An explicit promised sourced history › Finite packet table · lines 1411–1442

VIII-C@3#eq:table, VIII-C@3#eq:algebraic

For $q \geq 6$ , define

K_{q} = ⌊ \frac{2 ^{q}}{40} ⌋, T_{q} = {- K_{q}, \dots, K_{q}}, t_{m} = m 2^{- q}, μ_{m} = μ_{0} + 6 arctan t_{m} .

This is the promised table of VIII-B CheungVIIIB (Theorem thm:finite). Every entry lies in $I$ , since

∣ μ_{m} - μ_{0} ∣ \leq 6 arctan (1/40) < 3/20 < π /20.

At $q = 6$ , $T_{6} = {- 1, 0, 1}$ .

For $α_{m} = (2 π - μ_{m}) /3$ ,

z_{α_{m}} a_{m} = \frac{2 t _{m}}{1 + t _{m} ^{2}}, = \frac{2}{3} \frac{1 + t _{m}}{1 + t _{m} ^{2}}, S_{α_{m}} b_{m} = \frac{1 - t _{m} ^{2}}{1 + t _{m} ^{2}}, = \frac{3}{5} \frac{1 - t _{m} ^{2}}{1 + t _{m} ^{2}} .

The rays and seam matrices have rational entries; the finite coordinates, charts, and lengths are algebraic. The real number $μ_{m}$ itself is specified symbolically, not approximated by a floating-point angle. Exact derivations appear in app:checks.

Displayed formulas in this passage

VIII-C@3#eq:table · line 1414
$K_{q} = ⌊ \frac{2 ^{q}}{40} ⌋, T_{q} = {- K_{q}, \dots, K_{q}}, t_{m} = m 2^{- q}, μ_{m} = μ_{0} + 6 arctan t_{m} .$
VIII-C@3#eq:algebraic · line 1431
$z_{α_{m}} a_{m} = \frac{2 t _{m}}{1 + t _{m} ^{2}}, = \frac{2}{3} \frac{1 + t _{m}}{1 + t _{m} ^{2}}, S_{α_{m}} b_{m} = \frac{1 - t _{m} ^{2}}{1 + t _{m} ^{2}}, = \frac{3}{5} \frac{1 - t _{m} ^{2}}{1 + t _{m} ^{2}} .$
unlabeled · line 1424
$∣ μ_{m} - μ_{0} ∣ \leq 6 arctan (1/40) < 3/20 < π /20.$