Manuscript VII-F-ST@1 · VII-F-ST.tex · proposition · 9eee7a44

Proposition: Complete finite-edge table

Cellulation, exact measures, and the unique event parameter › Edges and lengths · lines 557–593 · proof included or linked

VII-F-ST@1#prop:lengths

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Each row has multiplicity three. The physical lengths are

edge e_{i} c_{i} t_{i} o_{i} u_{i} v_{i} w_{i}^{-} w_{i}^{+} s \leq 0 ℓ 5 3 ℓ 2 ℓ 4 ℓ 2 inactive inactive inactive inactive s > 0 ℓ 5 (1 - s /2) 3 ℓ 2 ℓ 4 ℓ 2 3 ℓ s /2 3 ℓ 41 s /11 3 ℓ 65 s /22 3 ℓ 65 s /22

At $s = 0$ , the born labels are germ metadata, not regular zero-length edges.

Proof. For a developed segment, $L^{2} = ℓ^{2} (∣Δ x ∣^{2} - (Δ t)^{2})$ . On $e_{i}$ , the spatial length is $3$ or $3 - 3 s /2$ , with time slope $2/3$ , giving the first row. The collar rows have $Δ t = 0$ . For $u_{i}$ , both endpoint times equal $2 s$ . For $v_{i}$ , put $b = 15 s /11$ ; its displacement is $(3 b /5, b, b)$ , whose squared norm is $41 b^{2} /25$ . For $w_{i}^{-}$ ,

(Δ t, Δ ξ, Δ η) = \frac{s}{22} (18, - 3, 30),

whose squared norm is $585 s^{2} /484$ . The remaining edge follows by symmetry.

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unlabeled · line 560
$edge e_{i} c_{i} t_{i} o_{i} u_{i} v_{i} w_{i}^{-} w_{i}^{+} s \leq 0 ℓ 5 3 ℓ 2 ℓ 4 ℓ 2 inactive inactive inactive inactive s > 0 ℓ 5 (1 - s /2) 3 ℓ 2 ℓ 4 ℓ 2 3 ℓ s /2 3 ℓ 41 s /11 3 ℓ 65 s /22 3 ℓ 65 s /22$
unlabeled · line 587
$(Δ t, Δ ξ, Δ η) = \frac{s}{22} (18, - 3, 30),$

Proposition: Complete finite-edge table

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