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

Finite particle lengths

Realization interfaces and explicit data › Finite particle lengths · lines 868–901

VIII-C@3#eq:length-coefficients, VIII-C@3#eq:lengths

Put

L_{α} W_{α} = 2 (1 - z_{α}), = (C - B)^{2} + C^{2} + 2 z_{α} C (C - B) - b_{μ}^{2} C^{2} .

The unrefined finite dictionary has three slots of each of eight types:

type e_{i} c_{i} = [A_{i}, A_{i + 1}] t_{i} = [A_{i}, D_{i}] o_{i} = [D_{i}, D_{i + 1}] u_{i} = [P, B_{i}] v_{i} = [P, C_{i}] w_{i}^{-} = [B_{i}, C_{i}] w_{i}^{+} = [C_{i}, B_{i + 1}] s < 0 r 1 - a^{2} r L_{α} 4 - r 4 L_{α} inactive inactive inactive inactive s > 0 (r - B s) 1 - a^{2} r L_{α} 4 - r 4 L_{α} B s C s 2 (1 + z_{α}) - b_{μ}^{2} W_{α} s W_{α} s

The persistent slot $e_{i}$ realizes $[P, A_{i}]$ before the event and $[B_{i}, A_{i}]$ after it.

Inactive slots are not incoming edges. At a germ, born-slot lengths have zero outgoing limit but the slots are metadata, not additional active zero-size cells. Their zero reports are coding conventions, not intrinsic incoming derivatives.

Displayed formulas in this passage

VIII-C@3#eq:length-coefficients · line 871
$L_{α} W_{α} = 2 (1 - z_{α}), = (C - B)^{2} + C^{2} + 2 z_{α} C (C - B) - b_{μ}^{2} C^{2} .$
VIII-C@3#eq:lengths · line 879
$type e_{i} c_{i} = [A_{i}, A_{i + 1}] t_{i} = [A_{i}, D_{i}] o_{i} = [D_{i}, D_{i + 1}] u_{i} = [P, B_{i}] v_{i} = [P, C_{i}] w_{i}^{-} = [B_{i}, C_{i}] w_{i}^{+} = [C_{i}, B_{i + 1}] s < 0 r 1 - a^{2} r L_{α} 4 - r 4 L_{α} inactive inactive inactive inactive s > 0 (r - B s) 1 - a^{2} r L_{α} 4 - r 4 L_{α} B s C s 2 (1 + z_{α}) - b_{μ}^{2} W_{α} s W_{α} s$