Manuscript VIII-B@3 · VIII-B-v3.tex · section · 3aa6baa1

Algebraic packet entries

A finite algebraic table and reversible realization › Algebraic packet entries · lines 1784–1840

VIII-B@3#eq:packet-table, VIII-B@3#eq:algebraic-table

Fix an integer $q \geq 6$ and define

K_{q} t_{m} = ⌊ \frac{2 ^{q}}{40} ⌋, = m 2^{- q}, T_{q} μ_{m} = {- K_{q}, \dots, K_{q}}, = μ_{0} + 6 arctan t_{m} . T_{q} = 2 K_{q} + 1,

Since $∣ t_{m} ∣ \leq 1/40$ ,

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

so every table entry lies in $I$ .

Its geometric coefficients are exactly algebraic:

α_{m} z_{α_{m}} a_{m} = \frac{π}{2} - 2 arctan t_{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}} .

Indeed,

e^{2 i a r c t a n t} = \frac{( 1 + i t ) ^{2}}{1 + t ^{2}} .

Thus $e^{i α_{m}}$ has rational real and imaginary parts, as do the powers giving the rays. Also

e^{i μ_{m}} = i (\frac{( 1 + i t _{m} ) ^{2}}{1 + t _{m} ^{2}})^{3}

has rational components, so the seam matrix is rational.

The angle label $μ_{m}$ need not be an algebraic number. The geometric data reconstructed from its symbolic identity are algebraic. At dyadic $s$ , gradients, boosts, vertices, lengths, and areas are real algebraic.

Exact report floors are therefore decidable by algebraic-number comparison with dyadic thresholds, including equality. The finite length bound below gives a finite threshold range. These comparisons are a mathematical construction of the table; no executed table generation is claimed.

Displayed formulas in this passage

VIII-B@3#eq:packet-table · line 1787
$K_{q} t_{m} = ⌊ \frac{2 ^{q}}{40} ⌋, = m 2^{- q}, T_{q} μ_{m} = {- K_{q}, \dots, K_{q}}, = μ_{0} + 6 arctan t_{m} . T_{q} = 2 K_{q} + 1,$
VIII-B@3#eq:algebraic-table · line 1805
$α_{m} z_{α_{m}} a_{m} = \frac{π}{2} - 2 arctan t_{m}, = \frac{2 t _{m}}{1 + t _{m} ^{2}}, = \frac{2}{3} \frac{1 + t _{m}}{1 + t _{m} ^{2}}, S_{α_{m}} b_{m} = \frac{1 - t _{m} ^{2}}{1 + t _{m} ^{2}}, = \frac{3}{5} \frac{1 - t _{m} ^{2}}{1 + t _{m} ^{2}} .$
unlabeled · line 1798
$∣ μ_{m} - μ_{0} ∣ \leq 6 arctan (1/40) < \frac{3}{20} < \frac{π}{20},$
unlabeled · line 1818
$e^{2 i a r c t a n t} = \frac{( 1 + i t ) ^{2}}{1 + t ^{2}} .$
unlabeled · line 1824
$e^{i μ_{m}} = i (\frac{( 1 + i t _{m} ) ^{2}}{1 + t _{m} ^{2}})^{3}$