Manuscript VIII-ST@1 · VIII-ST.tex · proposition · fe979f3b

Proposition: Exact finite evaluation

Source selection, affinity, and finite promises › An exact algebraic finite catalogue · lines 1386–1414 · proof included or linked

VIII-ST@1#prop:algebraic

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The ray and seam matrices of this table have rational entries. At dyadic $s$ , its vertex, adapted-chart, length, and area entries are real algebraic numbers. Their dyadic floor reports are exactly decidable by algebraic-number comparison, including equality cases. No floating-point angle is required.

Proof. The half-angle identity gives

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

Hence

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

These have rational components. All remaining formulas use finitely many arithmetic operations and positive square roots of positive algebraic numbers. Standard algebraic isolation and sign comparison determine each comparison with a dyadic threshold. The finite size bound in Section sec:encoding supplies a finite report range.

Displayed formulas in this passage

unlabeled · line 1398
$e^{2 i a r c t a n t} = \frac{( 1 + i t ) ^{2}}{1 + t ^{2}} .$
unlabeled · line 1402
$e^{i ϑ_{m}} = \frac{2 t _{m} + i ( 1 - t _{m} ^{2} )}{1 + t _{m} ^{2}}, e^{i μ_{m}} = i (\frac{( 1 + i t _{m} ) ^{2}}{1 + t _{m} ^{2}})^{3} .$

Proposition: Exact finite evaluation

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