Proposition: Supported lift

The graphs $t={\hat{f}}_s$ are continuous piecewise spacelike hypersurface markings with spatial Lipschitz constant at most $3/5$. They are strictly ordered:

They equal $t=2+s$ near $\partial \hat{K}$, and coincide with the transverse capped family for $|z|\le 1$.

They extend by the horizontal marking outside $\hat{K}$ on the regular conical product. The modification has compact spatial support disjoint from $\mathcal{W}$.

Proof. The affine pieces have spatial gradients among

Minima and maxima preserve the common Lipschitz bound. Their parameter slopes are at least $a=4/5$, proving strict ordering.

For $|z|\le 1$, $L_s=s-4<-3$, while $F_s\ge -4/5$. Thus the lift agrees with (eq:transverse-cap). For $|z|\ge 13$, $L_s\ge 2+s$, and the graph is horizontal. Near the lateral boundary of $K$, the transverse collar already has height $2+s$. This proves boundary matching and compact support.

Translation by $(20,20)$ places the entire box in a regular first-quadrant chart of the $\mu =\pi /2$ cone. The local isometry pulls the construction back. Outside the box the marking is unchanged.