Proposition: Spacelike cap and fixed boundary

The capped graph is continuous, piecewise flat, $p$-Lipschitz, and strictly time-ordered with lower increment $a(s_2-s_1)$. It equals $T=H+s$ on a neighbourhood of $\partial K$. The central edge laws in eq:incoming,eq:outgoing are unchanged.

Proof. Minimum and maximum preserve a common Lipschitz bound. Both arguments of the minimum increase by at least $a(s_2-s_1)$, so the minimum does also. The strict margins in eq:margins give the resting boundary neighbourhood.

On the incoming central edge, the cap-height margin is $2+s/5\ge 9/5$; on the outgoing edge it is $2+s/13\ge 25/13$. Hence neither central edge is clipped.