Lemma: Spacelikeness, embedding, and temporal ordering

For each fixed $\mu$, $F_s$ descends across all sector rays and the seam and is globally $\lambda$-Lipschitz in the intrinsic transverse cone distance.

The product graph is embedded. On each regular affine piece its induced metric satisfies

when the spatial cone metric in the parentheses is expressed in normalized units. The regular graph is acausal.

For $s^{\prime }<s^{\prime \prime }$,

The transverse singular intersection has normalized time

Proof. On a boundary ray, $\rho$ is its ray coordinate and $k=0$. Hence adjacent sectors and the seam agree.

The oblique spatial Gram matrix is

The squared physical norm of a covector $p$ is $p^{\mathsf{T}}{G_{\vartheta }}^{-1}p$. Consequently,

Minima and maxima preserve a common Lipschitz constant. Integrate along sectorwise paths and pass to their length infimum. Continuity handles rays and the apex limit. The axial derivative is zero.

Graph projection is a homeomorphism with the continuous inverse $(x,\zeta )\mapsto (F_s(x),x,\zeta )$. For a regular physical gradient $g$, the graph metric is ${\ell }^2(I-g{g}^{\mathsf{T}})$, bounded below by ${\ell }^2(1-{\lambda }^2)I={\ell }^{2}I/9$.

A future-causal regular curve has time increment at least its spatial path length, and thus at least intrinsic endpoint distance. Distinct points of the graph have a strictly smaller time difference, proving acausality. The same statement extends to the stipulated completion if its causal relation is defined by the corresponding ultrastatic length-space convention.

For fixed $x$, the capped term after the first $s$ in (eq:family) is nondecreasing and 1-Lipschitz in $s$. This proves (eq:ordering). Substitute $h=k=0$ at the apex.