The stationary product extends the local boost and circle to a complete collar

On the formal model define $\mathsf{T}={\mathrm{\partial }}_{\tau }-\mathrm{\Omega }{Z}_S$. Its orbit is $(r,\tau +t,{\psi }_{-\mathrm{\Omega }t}y)$. By [F1] it meets the reference graph once, at a smooth time $t_0(r,\tau ,y)$, with $\mathrm{\mid }\tau +t_0\mathrm{\mid }\le A$. Denote that actual reference point by $x_0\in C$ and define $\mathcal{F}(r,\tau ,y)={\varphi }_{-t_0}^T(x_0)$. The complete actual $T$ flow makes this defined for every real $\tau$. Shrink $r_0$ so all graph points and all segments between graph rapidities lie in the original tube: their rapidities have absolute value at most $A$, so their normal sizes are bounded by a constant times $r_0{e}^{\kappa A}$.

The map to stationary-product data $(r,\tau ,y)\mapsto (-t_0,x_0)$ is a diffeomorphism onto $\mathbb{R}\times Q$, where $Q$ is the reference collar. Indeed formal $\mathsf{T}$ flow recovers the original formal point uniquely from a graph point and elapsed time; the root is smooth and transverse. The actual stationary product is injective, so $\mathcal{F}$ is a diffeomorphism onto the open saturation $D_0={\bigcup }_t{\varphi }_t^T(Q)$. On a sufficiently small actual tube it agrees with the normal-coordinate map: its path to the graph has rapidity between the starting rapidity and the bounded graph value, so stays in a larger fixed tube; both the normal-coordinate path and the actual $T$-flow path are integral curves of the smooth field $T$ with the same reference value $x_0$. Local uniqueness from [F2], continued along their common time segment, gives equality. This also excludes inconsistent returns after leaving the local neighborhood, rather than assuming such returns absent.

On the formal model let $A_s(r,\tau ,y)=(r,\tau +s,y)$, $B_s(r,\tau ,y)=(r,\tau ,{\psi }_{s}y)$ and $H_t(r,\tau ,y)=(r,\tau +t,{\psi }_{-\mathrm{\Omega }t}y)$. These are commuting global flows, and $B_s$ has the supplied circle period. Uniqueness of the graph intersection gives $t_0(H_{t}p)=t_0(p)-t$ and $x_0(H_{t}p)=x_0(p)$, so $\mathcal{F}\circ H_t={\varphi }_t^T\circ \mathcal{F}$. Differentiating at zero identifies ${\mathcal{F}}_{\ast }\mathsf{T}=T$. Define the extended fields $K={\mathcal{F}}_{\ast }{\mathrm{\partial }}_{\tau }$ and $Z={\mathcal{F}}_{\ast }{Z}_S$ by [F4]. Differentiating the conjugated maps $\mathcal{F}\circ A_s\circ {\mathcal{F}}^{-1}$ and $\mathcal{F}\circ B_s\circ {\mathcal{F}}^{-1}$ shows they are their global flows. Thus the fields are complete, commute with each other and with $T$, and $Z$ retains the same circle period. By step 2.1 they agree with the original fields on an open tube about $Q$.

Fix a metric parameter $q$. Stationarity and [F5] give $({\varphi }_t^T{)}^{\ast }{g}_q=g_q$ for all real $t$: the complete flow has common domain $D$ for every time, or equivalently the local invariance identities compose along any finite time segment. Let $X$ be either extended field from step 3.1 and let ${\chi }_s$ be its global flow. Given $x\in D_0$, write $x=P(x_0)$ with $x_0\in Q$ and $P={\varphi }_t^T$. On a neighborhood $U$ of $x_0$ in the original tube, $X$ is the original Killing field. For sufficiently small $s$ and after restricting $U$, its flow stays in that tube, so [F5] gives ${\chi }_s^{\ast }{g}_q=g_q$ on $U$. Step 3.1 gives $P\circ {\chi }_s={\chi }_s\circ P$, hence near $x$ the flow is $P\circ {\chi }_s\circ P^{-1}$. The pullback composition formula gives ${\chi }_s^{\ast }{g}_q=(P^{-1}{)}^{\ast }{\chi }_s^{\ast }{P}^{\ast }{g}_q=(P^{-1}{)}^{\ast }{g}_q=g_q$ on $P(U)$. Differentiating at $s=0$ using [F3] proves ${\mathcal{L}}_X{g}_q=0$ near $x$. Since $x$ and $q$ were arbitrary, both extended fields are Killing for every $g_q$ throughout $D_0$. No extension beyond $D_0$ was used.