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

The final flow changes reports at every precision

Exact algebraic and resource checks › The final flow changes reports at every precision · lines 2398–2431

Let

L_{in} = r_{1} 1 - a_{1}^{2} = 2 a_{1}^{- 2} - 1 .

The proven coefficient bounds imply

\frac{3}{5} < a_{1} < \frac{1}{2}, 2 < L_{in} < \frac{8}{3} .

Since $B_{1} = r_{1} /2$ , the final persistent length is

L_{final} = (r_{1} - \frac{B _{1}}{2}) 1 - a_{1}^{2} = \frac{3}{4} L_{in},

and

\frac{3}{2} < L_{final} < 2.

For every integer $b \geq 0$ ,

⌊ 2^{b} L_{in} ⌋ \geq 2^{b + 1}, ⌊ 2^{b} L_{final} ⌋ \leq 2^{b + 1} - 1.

The outgoing germ has the same limiting persistent length $L_{in}$ as the installed incoming epoch. Therefore the final flow necessarily changes the $e_{i}$ reports, including at $b = 0$ .

Displayed formulas in this passage

unlabeled · line 2401
$L_{in} = r_{1} 1 - a_{1}^{2} = 2 a_{1}^{- 2} - 1 .$
unlabeled · line 2407
$\frac{3}{5} < a_{1} < \frac{1}{2}, 2 < L_{in} < \frac{8}{3} .$
unlabeled · line 2412
$L_{final} = (r_{1} - \frac{B _{1}}{2}) 1 - a_{1}^{2} = \frac{3}{4} L_{in},$
unlabeled · line 2418
$\frac{3}{2} < L_{final} < 2.$
unlabeled · line 2422
$⌊ 2^{b} L_{in} ⌋ \geq 2^{b + 1}, ⌊ 2^{b} L_{final} ⌋ \leq 2^{b + 1} - 1.$