Proposition: Spacelike ordering and central exchange

The graph $T=F_s(x,y)$ is continuous, piecewise affine, and $p$-Lipschitz. Every face is spacelike, and for $s_2>s_1$,

For $s<0$, the central adjacency is $EW$, with edge

For $s>0$, the central adjacency is $NS$, with edge

The central incidence changes as

Proof. Each affine function has spatial gradient of norm at most $p<1$. Their maximum has the same Lipschitz bound. On a face of slope $v$, a nonzero tangent vector has squared norm

Every affine function increases in $s$ at rate at least $a$, proving the ordering bound.

For $EW$, equality of the two affine functions forces $x=0$. Their common value $as$ dominates the other two precisely when $q|y|\le -\delta s$. This gives eq:incoming and its triple endpoints. The endpoint difference has constant $T,x$, so its spacelike length is the displayed $y$-distance.

For $NS$, equality forces $y=0$, and dominance is $p|x|\le \delta s$, giving eq:outgoing.

At $s=0$, $F_0=\max \{p|x|,q|y|\}$. All four faces meet only at the origin, and each contains an open strict-dominance cone. Thus the central edge collapses, but the two-dimensional patch does not.