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

Installed packet and final lengths

An explicit promised sourced history › Installed packet and final lengths · lines 1484–1542

VIII-C@3#eq:installed, VIII-C@3#eq:ray, VIII-C@3#eq:final-lengths

After source replacement,

μ_{1} a_{1} r_{1} = \frac{π}{2} + 6 arctan (1/64), = \frac{130}{3 4097}, = \frac{3 4097}{65}, b_{1} B_{1} = \frac{3}{5} \frac{4095}{4097}, = \frac{3 4097}{130}, C_{1} = \frac{1}{2 a _{1} - b _{1}} .

The ray generator is

e^{i α_{1}} = \frac{128 + 4095 i}{4097} .

Every subsequent face, chart, overlap, and report uses these installed coefficients rather than the reference table.

At $s = 1/2$ ,

T (P) = 1, B_{i} = \frac{B _{1}}{2} e_{i}, C_{i} = \frac{C _{1}}{2} (e_{i} + e_{i + 1}), T (A_{i}) = T (D_{i}) = \frac{5}{2},

and

ℓ (e_{i}) ℓ (v_{i}) = \frac{3}{4} r_{1} 1 - a_{1}^{2}, = \frac{C _{1}}{2} 2 (1 + z_{α_{1}}) - b_{1}^{2}, ℓ (u_{i}) ℓ (w_{i}^{\pm}) = \frac{B _{1}}{2}, = \frac{W _{α_{1}}}{2} .

Step	Command	$s$	Packet	$(ν_{G}, ν_{src})$	Certificate and intersection
0	Initial	$- 1$	$μ_{0}$	$(1, 1)$	Reference realization; $P = (- 1, 0, 0)$
1	$G$	$- 1/2$	$μ_{0}$	$(1, 1)$	Regular source input; $P = (- 1/2, 0, 0)$
2	$S_{tab}$	$- 1/2$	$μ_{1}$	$(1, 0)$	Complete new family; $P = (- 1/2, 0, 0)$
3	$G$	$0^{-}$	$μ_{1}$	$(1, 0)$	Incoming germ; $P = (0, 0, 0)$
4	$E_{3 - 6}$	$0^{+}$	$μ_{1}$	$(0, 0)$	Outgoing germ, frozen packet; $P = (0, 0, 0)$
5	$G$	$1/2$	$μ_{1}$	$(0, 0)$	Outgoing realization; $P = (1, 0, 0)$

Displayed formulas in this passage

VIII-C@3#eq:installed · line 1487
$μ_{1} a_{1} r_{1} = \frac{π}{2} + 6 arctan (1/64), = \frac{130}{3 4097}, = \frac{3 4097}{65}, b_{1} B_{1} = \frac{3}{5} \frac{4095}{4097}, = \frac{3 4097}{130}, C_{1} = \frac{1}{2 a _{1} - b _{1}} .$
VIII-C@3#eq:ray · line 1497
$e^{i α_{1}} = \frac{128 + 4095 i}{4097} .$
VIII-C@3#eq:final-lengths · line 1511
$ℓ (e_{i}) ℓ (v_{i}) = \frac{3}{4} r_{1} 1 - a_{1}^{2}, = \frac{C _{1}}{2} 2 (1 + z_{α_{1}}) - b_{1}^{2}, ℓ (u_{i}) ℓ (w_{i}^{\pm}) = \frac{B _{1}}{2}, = \frac{W _{α_{1}}}{2} .$
unlabeled · line 1505
$T (P) = 1, B_{i} = \frac{B _{1}}{2} e_{i}, C_{i} = \frac{C _{1}}{2} (e_{i} + e_{i + 1}), T (A_{i}) = T (D_{i}) = \frac{5}{2},$