Manuscript VII-J@1 · VII-J.tex · section · d8c01b04

The exact slicing family

The fixed cone and VII-F's slicing event › The exact slicing family · lines 1369–1395

VII-J@1#eq:F

In each of VII-F's three developed quadrants, write $x = ξ v_{i} + η v_{i + 1}$ , $ξ, η \geq 0$ , and set

ρ (x) = ξ + η, H (x) = \frac{2}{3} (ξ + η), K (x) = \frac{3}{5} min {ξ, η} .

The functions agree on identified rays. For $- 1 \leq s \leq 1$ , define

F_{s} (x) = s + min {2, max {H (x), s + K (x)}}, Σ_{s} = {(F_{s} (x), x) : x \in C_{π /2}} .

VII-F proves the complete geometric certificate, including its affine holonomy, collar, and three-to-six particle link CheungVIIF (thm:certificate). For self-containment of the causal argument, the required estimates are verified here.

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VII-J@1#eq:F · line 1380
$F_{s} (x) = s + min {2, max {H (x), s + K (x)}}, Σ_{s} = {(F_{s} (x), x) : x \in C_{π /2}} .$
unlabeled · line 1373
$ρ (x) = ξ + η, H (x) = \frac{2}{3} (ξ + η), K (x) = \frac{3}{5} min {ξ, η} .$

The exact slicing family

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