Manuscript VII-E-ST@1 · VII-E-ST.tex · lemma · d28d9c5d

Lemma: Central exchange

A four-face certificate and its supported 3+1 lift › Transverse faces, edge laws, and affine charts · lines 764–800 · proof included or linked

VII-E-ST@1#lem:central, VII-E-ST@1#eq:central-edges

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The graph $t = F_{s} (x, y)$ is piecewise spacelike and $p$ -Lipschitz. For $s_{2} > s_{1}$ ,

F_{s_{2}} - F_{s_{1}} \geq a (s_{2} - s_{1}) .

Its central edge is

s < 0 : s > 0 : x = 0, ∣ y ∣ \leq d (- s) / q, t = a s, ℓ_{-} (s) = \frac{16}{25} (- s), y = 0, ∣ x ∣ \leq d s / p, t = b s, ℓ_{+} (s) = \frac{16}{39} s .

At $s = 0$ the central edge becomes one four-face junction. No transverse face disappears there.

Proof. Each affine spatial gradient has norm at most $p < 1$ . Their maximum has the same Lipschitz bound. A tangent to a face of slope $v$ has squared norm

∣ u ∣^{2} - (v \cdot u)^{2} > 0.

All four functions increase with $s$ at rate at least $a$ .

For $E = W$ , equality forces $x = 0$ , and dominance over $N, S$ is $q ∣ y ∣ \leq - d s$ , possible precisely on the incoming side. For $N = S$ , equality forces $y = 0$ , with $p ∣ x ∣ \leq d s$ . The edge tangents have constant $t$ , so their lengths are the displayed Euclidean coordinate intervals. At $s = 0$ , each strict-dominance region contains an open cone.

Displayed formulas in this passage

VII-E-ST@1#eq:central-edges · line 772
$s < 0 : s > 0 : x = 0, ∣ y ∣ \leq d (- s) / q, t = a s, ℓ_{-} (s) = \frac{16}{25} (- s), y = 0, ∣ x ∣ \leq d s / p, t = b s, ℓ_{+} (s) = \frac{16}{39} s .$
unlabeled · line 768
$F_{s_{2}} - F_{s_{1}} \geq a (s_{2} - s_{1}) .$
unlabeled · line 789
$∣ u ∣^{2} - (v \cdot u)^{2} > 0.$

Lemma: Central exchange

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