Radial comparison gives proper normal projection onto a fixed target collar

Fix a supplied candidate. In the reference interior $P_q$ is smooth and locally invertible by the spacelike graph hypothesis and F2. The set $U_q$ is open there because it is the inverse image of the open target collar. It is nonempty: fix $y\in S$ and take $0<\rho <\min ({\rho }_0,r_{\delta }\mathrm{/}{c}_2)$; then $0<c_1\rho \le R_q(\rho ,y)\le c_2\rho <r_{\delta }$. Via the specified identification $S\cong S^2$, use $(t,y)\mapsto ty$ to identify each positive radial collar with an open annulus in ${\mathbb{R}}^3$. Under these diffeomorphisms $U_q$ is a nonempty Euclidean open set and $P_q$ is a smooth local diffeomorphism.

Let $B\subset V_q$ be compact. The empty case has empty compact preimage, so assume $B\mathrm{\ne }\mathrm{\varnothing }$. Continuous inclusion of its annulus model into ${\mathbb{R}}^3$ preserves compactness, and the radial coordinate is continuous; F4 gives attained radial extrema $0<a\le b<r_{\delta }$. Every point of $P_q^{-1}(B)$ therefore has $a\mathrm{/}{c}_2\le \rho \le b\mathrm{/}{c}_1<{\rho }_0\mathrm{/}2$. The closed reference subannulus $A=\{a\mathrm{/}{c}_2\le \rho \le b\mathrm{/}{c}_1\}\times S$ lies strictly inside the reference interior. Its Euclidean annulus model is closed and bounded, hence compact by F4, and the inverse collar identification preserves compactness. The target set $B$ is closed in the Euclidean target model by F4, so continuity of $P_q{\mathrm{\mid }}_A$ makes $P_q^{-1}(B)$ closed in $A$. A closed subset of this compact metric space is compact; intrinsic compactness and continuous inclusion give compactness in $U_q$. This proves properness. The outer source boundary cannot contribute: at $\rho ={\rho }_0$ its target radius is at least $c_1{\rho }_0>2r_{\delta }$.

The target annulus is path-connected: for $(r,y)$ and $(r^{\mathrm{\prime }},y^{\mathrm{\prime }})$ choose a continuous path $\gamma :[0,1]\to S^2$ joining $y$ to $y^{\mathrm{\prime }}$ by F5; then $t\mapsto ((1-t)r+t{r}^{\mathrm{\prime }})\gamma (t)$ remains in that annulus and joins the corresponding points. It is connected by F5. The source is nonempty and the derivative is invertible by step 1.1; the map is proper by step 2.1, with compactness preserved by the collar diffeomorphisms in F4. All hypotheses of F3 now hold in dimension three. Thus $P_q$ is surjective and each target point has a neighborhood whose preimage is a finite disjoint union of diffeomorphic sheets. The same $r_{\delta }$ works for every candidate because only the background constants $c_1,c_2,{\rho }_0$ and common height bound determined it. Compactness and smoothness were used separately for each fixed candidate, so no uniform height derivatives or inverse-function radius were needed.