Lemma: Uniform realization bounds

The graphs ${\Sigma }_s$ are seam compatible, embedded, and globally acausal. Each regular face is uniformly spacelike, with

For $s_2>s_1$,

Moreover,

The family has a common uniform limit at $s=0$.

Proof. The nonzero affine branch gradients in a sector are

Their Euclidean norms are bounded by $\lambda$. Minimum and maximum of real functions with a common Lipschitz constant retain that constant. Shared-ray values give continuity across all rays and the seam.

Integrating the sectorwise bound along spatial paths, including paths through shared rays or the apex, gives

Graph projection is a homeomorphism onto the cone, so the graph is embedded. On a face with gradient $a$, the induced metric is $I_2-a{a}^{\mathsf{T}}$, and

A causal curve in the static product has elapsed absolute time at least the length of its spatial projection. Distinct graph points have smaller absolute time separation than their spatial distance, because $\lambda <1$. They cannot be causally related.

For fixed $x$, the capped inner expression is nondecreasing and $1$-Lipschitz in $s$. Adding $s$ gives A:eq:ordering. Since $\min \{\xi ,\zeta \}\le \rho /2$,

If $\rho \ge 3$, then $h\ge 2$, so the outer minimum equals $2$. The upper bound in A:eq:ordering also gives uniform convergence to $F_0$ as $s\to 0$.