Geodesics continue while velocity lifts remain compact

By [F1], $z$ is an integral curve of $S$. By [F2], for every $w\in TM$ one has $(0,w)\in \mathcal{D}$. Since $\mathcal{D}$ is open, [F3] gives an open set $U\subseteq TM$ containing $w$ and an $\epsilon >0$ such that $(-\epsilon ,\epsilon )\times U\subseteq \mathcal{D}$. Thus the set $\mathcal{A}:=\{(\epsilon ,U):\epsilon >0,U\subseteq TM\text{ open},(-\epsilon ,\epsilon )\times U\subseteq \mathcal{D}\}$ indexes an ambient-open cover $(U_{(\epsilon ,U)}{)}_{(\epsilon ,U)\in \mathcal{A}}$ of $TM$, and hence of $K$.

The tail hypothesis makes $K$ nonempty. By [F4], finitely many indices $({\epsilon }_0,U_0),\dots ,({\epsilon }_n,U_n)\in \mathcal{A}$ cover $K$. By [F5], $\delta :=\min \{{\epsilon }_0,\dots ,{\epsilon }_n\}>0.$ If $w\in K$, then $w\in U_i$ for some $i$, and $\mathrm{\mid }s\mathrm{\mid }<\delta \le {\epsilon }_i$ implies $(s,w)\in \mathcal{D}$. Therefore $(-\delta ,\delta )\times K\subseteq \mathcal{D}\mathrm{.}$ No pointwise family of choices was made: the index of the cover already contains both $U$ and its admissible $\epsilon$, and compactness returns a finite list of those pairs.

Assume the right-endpoint hypotheses. Put $c=\max \{t_0,b-\delta \mathrm{/}2\}<b$ and $t_1=(c+b)\mathrm{/}2$. Then $t_1\in I$, $t_1>t_0$, $z_1:=z(t_1)\in K$, and $b-t_1<\delta \mathrm{/}2$. Step 2.1 makes $s\mapsto \mathrm{\Phi }(s,z_1)$ an integral curve for $\mathrm{\mid }s\mathrm{\mid }<\delta$. By uniqueness in [F2], it agrees with $s\mapsto z(t_1+s)$ wherever both are defined.

Projecting the curve in step 3.1 to $M$ gives a geodesic by [F1]. It agrees with $\gamma$ on the overlap, so it glues smoothly to $\gamma$ and defines a geodesic on $I\cup (t_1-\delta ,t_1+\delta )=(a,t_1+\delta )\mathrm{.}$ Because $t_1+\delta >b$, this is the required extension past $b$. Notice that the new interval contains the formerly missing parameter value $b$; no value of $\gamma$ at $b$ was assumed.

For a finite left endpoint, put $c=\min \{t_0,a+\delta \mathrm{/}2\}>a$ and $t_1=(a+c)\mathrm{/}2$. Then $z(t_1)\in K$ and $t_1-a<\delta \mathrm{/}2$. The same maximal-flow curve, now using negative times, glues to $z$ and projects to a geodesic on $(t_1-\delta ,b)$, whose left endpoint is strictly less than $a$. This proves claim 2. If $\gamma$ were maximal, either extension would contradict maximality; contraposition gives the final consequence.

The empty manifold admits no geodesic with nonempty domain. In dimension zero the spray curves are stationary, and in dimension one the preceding argument is unchanged; no positive-dimensional coordinate was used. An empty $K$ cannot contain the nonempty tail, while a one-member finite subcover is allowed and gives $\delta ={\epsilon }_0$. Only finite endpoints are asserted, and steps 4.1 and 5.1 treat both endpoint directions. The sole choice principle is the stated ${\mathrm{A}\mathrm{C}}_{\omega }$ used through [F1]; the compact-cover argument itself is a ZF argument and makes no countable or arbitrary selection.