Manuscript En03-A@1 · En03-A.tex · proposition · 5398614e

Proposition: Incoming edge

External conical geometry and certified local moves › Four affine faces · lines 1543–1563 · proof included or linked

En03-A@1#prop:R2, En03-A@1#eq:incoming-edge, En03-A@1#eq:incominglength

For $s < 0$ , $E, W$ share exactly

x = 0, ∣ y ∣ \leq \frac{Δ ( - s )}{q}, T = a s .

Its endpoints are $E W N, E W S$ , and its length is

ℓ_{-} (s) = \frac{2Δ}{q} (- s) = \frac{16}{25} (- s) .

Proof. Equality gives $x = 0$ . Dominance requires $a s \geq q ∣ y ∣ + b s$ , hence $q ∣ y ∣ \leq - Δ s$ . The two equalities at its endpoints add $N$ or $S$ . The displacement is purely in the $y$ -direction.

Displayed formulas in this passage

En03-A@1#eq:incoming-edge · line 1546
$x = 0, ∣ y ∣ \leq \frac{Δ ( - s )}{q}, T = a s .$
En03-A@1#eq:incominglength · line 1552
$ℓ_{-} (s) = \frac{2Δ}{q} (- s) = \frac{16}{25} (- s) .$

Nearby in this section

section: Four affine faces
proposition: Proposition: Spacelike graph and time order
proposition: Proposition: Outgoing edge
proposition: Proposition: A junction, not a patch collapse
proposition: Proposition: Transversality and isolation
section: Four affine faces
proposition: Proposition: Exact ordered products
section: Four affine faces