Theorem: Embedded, uniformly spacelike ordered lift

The markings ${\hat{\Sigma }}_s$ are embedded graphs. Every regular affine piece is spacelike, with dimensionless spatial Lipschitz constant

Their induced metrics satisfy the lower bound

for dimensionless spatial tangent coordinates. For $s_2>s_1$,

Distinct points on one regular marking are not causally related. The extension to the conical stratum is understood in the stratified Lipschitz sense, not as a smooth hypersurface theorem at $\mathcal{W}$.

Proof. The affine spatial gradients, including the axial component, are

up to sector rotations. Taking minima and maxima preserves a common Lipschitz bound. On an affine graph of gradient $a$, the induced metric is ${\ell }^2(I-a{a}^{\mathsf{T}})$, whose least eigenvalue is at least ${\ell }^2(1-{\lambda }^2)={\ell }^2/9$.

Integrate the gradient bound along spatial paths and take the infimum of their lengths. The graph projection is a homeomorphism. The strict Lipschitz bound and Proposition prop:geometry exclude causal pairs of distinct graph points. For fixed $x$, the term after the initial $s$ in (eq:family) is nondecreasing and 1-Lipschitz in $s$, which gives (eq:ordering).