Lemma 3.3: common spacelike reference collar

Extend the original section smoothly through $S$ as already provided by its smooth embedding; maximality of this extension is not needed. At $S$, $T\in TS$, so the invertible Harrison tangent map $A_q$ preserves $T{\mathrm{\Sigma }}_1$. Its restriction there is invertible. The formula $g_q=(D_q\mathrm{/}4)A_q^{\ast }g$ from [F3] therefore makes the section tangent form positive definite at $S$ for every $q$. The uniform-family lemma in [F8] applies: the zero field makes ${\mathrm{\nabla }}^{(q)}K$ connection independent at $S$, while [F2]–[F3] fix its quotient eigenvalues. It supplies one common spacelike collar, smooth uniformly controlled induced normal data, the positive boost derivative, both uniform Taylor expansions and future timelikeness.

Use the original metric and [F5] to obtain the invariant frame $(e_0,e_1)$ and the exact local boost/circle coordinates, with the exterior as the outward wedge. The original section is smooth through $S$. Its original exterior-pointing unit normal ${\bar{\nu }}_1$ to $S$ is orthogonal to $TS$, hence has the form ${\bar{\nu }}_1=\beta e_0+\alpha e_1$. The outward orientation of this frame is the one toward the actual exterior $C$ in [F1]–[F2], so $\alpha >0$; unit spacelike length gives ${\alpha }^2-{\beta }^2=1$. Thus the side hypothesis of [F6] holds, allowing the tilt $\beta \mathrm{\ne }0$. The positive original stationary lapse in [F1] gives future transversality of $T$, and step 1.1 gives future-timelike $K$. Apply [F6] to obtain a smooth graph $\tau =a(r,y)$ with $\mathrm{\mid }a\mathrm{\mid }\le A$, $1+{\mathrm{\Omega }}_H{Z}_{S}a>0$, and the unique formal stationary root. No preservation of the original section by the circle action is assumed. Shrink the common collar once to fit these constructions.

The coefficient ${\mathrm{\Omega }}_H$ is nonzero: otherwise $T=K$ would vanish identically on $S$, contrary to [A1]. Since each $g_q$ is stationary and locally $K$-invariant, ${\mathcal{L}}_Z{g}_q={\mathrm{\Omega }}_H^{-1}({\mathcal{L}}_K{g}_q-{\mathcal{L}}_T{g}_q)=0$ in the common local tube. Thus every hypothesis of the stationary-saturation helper [F7] is met by the original complete product and the bounded reference graph. It explicitly constructs $D_0$ and the same-field all-time extension by transporting unique local graph crossings through the complete $T$ product. This step supplies the missing completeness; it is not inferred merely from timelikeness or a local Taylor estimate.

Apply the complete reference-product helper in [F9]. In its global model $K={\mathrm{\partial }}_{\tau }$ and the map is $(s,r,y)\mapsto (r,a(r,y)+s,y)$ with the displayed smooth global inverse. It gives the required all-real-time diffeomorphism and identifies its image as the stationary collar saturation. Its comparison with the original Cauchy section is legitimate because the original stationary time diverges at both ends of each orbit. Only original-metric Cauchy data enter that comparison. All uniform estimates remain the local compact-collar bounds proved in step 1.1; no uniform control at unbounded flow times is claimed.