Manuscript Oai227-A@1 · Oai227-A.tex · section · 7f45f607

A direct finite arithmetic check

A direct finite arithmetic check · lines 1675–1708

The certificate's numerical inequality uses no approximation to $tanh (J / 2)$ . Its only size-dependent algebraic comparisons are

8 T^{3} - 27 = 32 - 27 = 5 > 0, 64 - 27 v^{3} = 64 - 54 = 10 > 0.

The exponential comparison uses

1 + 3 + \frac{9}{2} + \frac{27}{6} + \frac{81}{24} = \frac{131}{8}, 131 - 128 = 3 > 0.

Finally,

1 6^{4} = 65536, \frac{2}{3} 1 6^{- 4} = \frac{1}{98304} < \frac{1}{65536} .

The proof is therefore a finite analytic certificate with explicit rational comparisons. No floating-point agreement, numerical root finder, or simulation is used as a substitute for an inequality.

Displayed formulas in this passage

unlabeled · line 1680
$8 T^{3} - 27 = 32 - 27 = 5 > 0, 64 - 27 v^{3} = 64 - 54 = 10 > 0.$
unlabeled · line 1686
$1 + 3 + \frac{9}{2} + \frac{27}{6} + \frac{81}{24} = \frac{131}{8}, 131 - 128 = 3 > 0.$
unlabeled · line 1694
$1 6^{4} = 65536, \frac{2}{3} 1 6^{- 4} = \frac{1}{98304} < \frac{1}{65536} .$