A proper stationary projection with the correct end has one sheet

Use the stationary product coordinates $(x,t-u_{\ast }(x))\in C\times \mathbb{R}$ from [F1]. The projection ${\mathrm{\Pi }}_{\ast }$ is $j$ composed with first-coordinate projection, and $T$ spans each vertical fibre's tangent line. Since ${\mathrm{\Sigma }}_w^{\circ }$ is three-dimensional and $T_p\mathrm{\notin }{T}_p{\mathrm{\Sigma }}_w$, that tangent space and the vertical line span the four-dimensional ambient tangent space. Thus [F5] applies at every point: the continued embedded graph is locally a smooth graph over $C$, and ${\mathrm{\Pi }}_{\ast }{\mathrm{\mid }}_{{\mathrm{\Sigma }}_w^{\circ }}$ is a local diffeomorphism. By [F2], the differential of $F_w$ into its embedded image is an isomorphism between three-dimensional tangent spaces; [F5] gives local smooth inverses. Since $F_w$ is already a bijection onto its image, these inverses agree, so $F_w:{\mathrm{\Sigma }}_{\ast }^{\circ }\to {\mathrm{\Sigma }}_w^{\circ }$ is a diffeomorphism. Hence $P$ is a smooth local diffeomorphism.

Apply [F3] to the explicitly supplied graph, return and compact-truncation data to obtain properness of $P$. Its end restriction $P(x)=x$ on $E_{R_0}$ is the exact end hypothesis in the Statement, also used in [F3].

Conjugate by $j$ to obtain $f=j^{-1}\circ P\circ j:C\to C$. It is smooth with invertible differential by step 1.1. It is proper by step 1.2: for compact $L\subset C$, $f^{-1}(L)=j^{-1}(P^{-1}(j(L)))$ is compact, using [F6] for the two continuous coordinate maps. The nonempty Euclidean exterior $C$ is connected, so [F4] supplies surjectivity and finite evenly covered fibres. Their cardinality $n$ is locally constant: on an evenly covered neighborhood every sheet contributes exactly one point to each fibre. Each level set of $n$ and its complement are open; connectedness in [F6] therefore makes $n$ constant. Transporting back by $j$ gives the same assertion for $P$.

Shrink the source collar $U$ if necessary so $P(U)\subset V$, and choose $R\ge R_0$ sufficiently large. The middle set $K={\mathrm{\Sigma }}_{\ast }^{\circ }\setminus (U\cup E_R)$ is compact by the supplied truncation hypothesis. If $K$ is nonempty, the continuous function $p\mapsto \mathrm{\mid }{j}^{-1}(P(p))\mathrm{\mid }$ has a finite maximum $M$ on $K$ by [F6]; if $K$ is empty take $M=0$. Choose $z\in C$ with $\mathrm{\mid }z\mathrm{\mid }>\max \{B,M,R\}$ and put $y=j(z)$. No preimage of $y$ lies in $U$ or $K$, by the bounds on their images. All preimages therefore lie in $E_R$, where $P$ is the identity by step 1.2; since $y\in E_R$, its fibre is exactly $\{y\}$. The constant fibre cardinality from step 2.1 is consequently one. Thus $P$ is bijective and its smooth local inverses from step 1.1 agree to give a smooth global inverse. Finally ${\mathrm{\Pi }}_{\ast }{\mathrm{\mid }}_{{\mathrm{\Sigma }}_w^{\circ }}=P\circ F_w^{-1}$ is also a diffeomorphism.