I'd like to check my understanding of the notions "geodesic path" and "arc length of a geodesic path" in the context of GTR and "Lorentzian manifolds".
Considering a set of "spacetime events", $\mathcal S$, such that for any two distinct events $A, B \in \mathcal S$ and for any subset $\mathcal G \subset \mathcal S$ it is known
whether $\mathcal G$ is a "geodesic path between $A$ and $B$", or not, and (if so 1)
whether $\mathcal G$ is a "geodesic path between $A$ and $B$ of minimal arc length"2 (which I'll denote as "$s[~A, B~]$" for reference below), or not,
is it true that
(1):
for any two distinct events $A, B \in \mathcal S$ and for any "geodesic path $\mathcal G$ of minimal arc length between $A$ and $B$" (if there is one at all) then for each event $X \in \mathcal G$ which is distinct from events $A$ and $B$ (if there is one at all)
there is a "geodesic path $\mathcal P$ of minimal arc length between $A$ and $X$" and a "geodesic path $\mathcal Q$ of minimal arc length between $B$ and $X$" which are both subsets of $\mathcal G$, and such that
the arc length of $\mathcal G$ equals the sum of the arc length of $\mathcal P$ and the arclength of $\mathcal Q$, and either
$\mathcal G$, $\mathcal P$, and $\mathcal Q$ are all "null paths", i.e. $s[~A, B~] = s[~A, X~] = s[~B, X~] = 0$, or
$\mathcal G$ is not a "null path",
the ratio of the arc length of $\mathcal P$ to the the arc length of $\mathcal G$ is a positive real number, $\frac{s[~A, X~]}{s[~A, B~]} \gt 0$, and
the ratio of the arc length of $\mathcal Q$ to the the arc length of $\mathcal G$ is a positive real number, $\frac{s[~B, X~]}{s[~A, B~]} \gt 0$, too; and
(2):
for any two events $J, K \in \mathcal S$ with a non-null "geodesic path of minimal arc length between $J$ and $K$"
either for any three distinct events $A, B, X \in \mathcal S$ with "geodesic paths of minimal arc length between" any two of them, and with $$\frac{s[~A, B~]}{s[~J, K~]} \gt 0, \qquad \frac{s[~A, X~]}{s[~J, K~]} \ge 0, \qquad \frac{s[~B, X~]}{s[~J, K~]} \ge 0,$$ holds $$\scriptsize{1 + \left(\frac{s[~A, X~]}{s[~A, B~]}\right)^4 + \left(\frac{s[~B, X~]}{s[~A, B~]}\right)^4 \ge 2~\left(\frac{s[~A, X~]}{s[~A, B~]}\right)^2 + 2~\left(\frac{s[~B, X~]}{s[~A, B~]}\right)^2 + 2~\left(\frac{s[~A, X~]}{s[~A, B~]}\right)^2~\left(\frac{s[~B, X~]}{s[~A, B~]}\right)^2},$$
or for any three distinct events $A, B, X \in \mathcal S$ with "geodesic paths of minimal arc length between" any two of them, and with $$\frac{s[~A, B~]}{s[~J, K~]} \lt 0, \qquad \frac{s[~A, X~]}{s[~J, K~]} \le 0, \qquad \frac{s[~B, X~]}{s[~J, K~]} \le 0,$$ holds likewise $$\scriptsize{1 + \left(\frac{s[~A, X~]}{s[~A, B~]}\right)^4 + \left(\frac{s[~B, X~]}{s[~A, B~]}\right)^4 \ge 2~\left(\frac{s[~A, X~]}{s[~A, B~]}\right)^2 + 2~\left(\frac{s[~B, X~]}{s[~A, B~]}\right)^2 + 2~\left(\frac{s[~A, X~]}{s[~A, B~]}\right)^2~\left(\frac{s[~B, X~]}{s[~A, B~]}\right)^2}$$
?
Notes (in response to comments)
1: The fact that a certain subset $\mathcal G \subset \mathcal S$ is indeed a "geodesic path between $A$ and $B$, with respect to the given set of events $\mathcal S$" may be expressed by more explicit notation. I'd prefer: $\mathcal G \equiv \gamma_{\mathcal S}[~A, B~]$.
2: Here "arc length" of a given "geodesic path $\mathcal G$" is understood as "proper arc length" (a "signed quantity"), formally:
$\int_{\mathcal G} \text{sgn}[~\text{ds}^2~]~\sqrt{\text{sgn}[~\text{ds}^2~]~\text{ds}^2} \equiv \int_{\mathcal G} \text{sgn}[~\text{ds}^2~]~\sqrt{~|~\text{ds}^2~|}$,
where the "spacetime interval differential $\text{ds}^2$" can be positive, zero or negative.
(This understanding of "arc length" may however be at variance with the notion "arc length" as described in Wikipedia.)
Therefore the required "geodesic path between $A$ and $B$ of minimal arc length" refers to the (or any) "geodesic path between $A$ and $B$" with the absolutely smallest, possibly negative, arc length of all "geodesic paths between $A$ and $B$", according to a suitable sign convention. (However, in order to be independent of any particular choice of such a convention, consider the requirement instead as "geodesic path between $A$ and $B$ of extremal arc length, according to any one consistently applied sign convention".)
