1. Determining the signature of $\hat\sigma$
If any four points $A, B, J, K \in (\mathcal S^2 \setminus \mathcal G_P)$ can be found such that
$$\hat\sigma[ \, A, J \, ] \, \hat\sigma[ \, A, B \, ] \gt 0, \qquad \hat\sigma[ \, J, B \, ] \, \hat\sigma[ \, A, B \, ] \gt 0, \\ \hat\sigma[ \, A, K \, ] \, \hat\sigma[ \, A, B \, ] \gt 0, \qquad \hat\sigma[ \, K, B \, ] \, \hat\sigma[ \, A, B \, ] \gt 0, $$
and
$$\sqrt{\text{Sgn}[ \, \hat\sigma[ \, A, J \, ] \, ] \, \hat\sigma[ \, A, J \, ]} + \sqrt{\text{Sgn}[ \, \hat\sigma[ \, J, B \, ] \, ] \, \hat\sigma[ \, J, B \, ]} \lt \qquad \qquad \qquad \qquad \qquad \qquad \\ \sqrt{\text{Sgn}[ \, \hat\sigma[ \, A, B \, ] \, ] \, \hat\sigma[ \, A, B \, ]} \lt \\ \qquad \qquad \qquad \qquad \qquad \qquad \sqrt{\text{Sgn}[ \, \hat\sigma[ \, A, K \, ] \, ] \, \hat\sigma[ \, A, K \, ]} + \sqrt{\text{Sgn}[ \, \hat\sigma[ \, K, B \, ] \, ] \, \hat\sigma[ \, K, B \, ]}$$
then any pair of points $(U, V) \in (\mathcal S^2 \setminus \mathcal G_P)$ for which $\hat\sigma[ \, U, V \, ] \, \hat\sigma[ \, A, B \, ] \gt 0$ will in the following be called spacelike separated;
any pair $M, N \in (\mathcal S^2 \setminus \mathcal G_P)$ for which $\hat\sigma[ \, M, N \, ] = 0$ will be called lightlike separated;
and any pair $Q, Z \in (\mathcal S^2 \setminus \mathcal G_P)$ remaining will be called timelike separated.
For spacelike separated points $A, B$ and timelike separated points $Q, Z$ therefore $\hat\sigma[ \, A, B \, ] \, \hat\sigma[ \, Q, Z \, ] \lt 0$, of course.
2. Determining the signature of $(X, Y) \in \mathcal G_P$
Consider any and all (simple, invertible) curves $\gamma : [0 \ldots 1] \rightarrow (\mathcal S \setminus \{ P \}), \qquad \gamma[ \, 0 \, ] \mapsto X, \qquad \gamma[ \, 1 \, ] \mapsto Y$.
If among them there exist curves $\overline\gamma$ such that
$\forall \, r \in \mathbb R \, \mid \, 0 \lt r \lt 1 : $ the pair $(X, \overline\gamma[ \, r \, ]) \in (\mathcal S^2 \setminus \mathcal G_P)$ and timelike separated, and
$\forall \, s \in \mathbb R \, \mid \, 0 \lt s \lt 1 : $ the pair $(\overline\gamma[ \, s \, ], Y) \in (\mathcal S^2 \setminus \mathcal G_P)$ and timelike separated, and
$\forall \, r, s \in \mathbb R \, \mid \, 0 \lt r \lt s \lt 1 : $ the pair $(\overline\gamma[ \, r \, ], \overline\gamma[ \, s \, ]) \in (\mathcal S^2 \setminus \mathcal G_P)$ and timelike separated,
then the pair $(X, Y)$ is called timelike separated as well.
Vice versa, if there exists a curve $\overline\gamma$ whose points, with exception of the pair $(X, Y)$ of endpoints itself, are all pairwise spacelike separated, then the pair $(X, Y)$ is called spacelike separated, too.
All remaining pairs $(X, Y) \in \mathcal G_P$ are called lightlike separated.
3. Determining $\sigma[ \, X, Y \, ]$ of points $(X, Y) \in \mathcal G_P$
3.1 $(X, Y)$ lightlike separated:
$$\sigma[ \, X, Y \, ] := 0.$$
3.2 $(X, Y)$ timelike separated:
For each of the completely timelike curves $\overline\gamma \in \overline \Gamma,$
$\overline\gamma : [0 \ldots 1] \rightarrow (\mathcal S \setminus \{ P \}), \qquad \overline\gamma[ \, 0 \, ] \mapsto X, \qquad \overline\gamma[ \, 1 \, ] \mapsto Y$ consider all its finite partitionings $t \in \mathcal T$, with $t_k \in [0 \ldots 1],$ integer indices $k \in [0, 1, \ldots n] \qquad t_0 = 0, \qquad t_n = 1, \qquad (j \lt k) \implies (t_j \lt t_k)$,
with $n \ge 2$ and such that $\forall k \in [0, \ldots (n - 1)] : (\overline\gamma[ \, t_k \, ], \overline\gamma[ \, t_{(k + 1)} \, ]) \in (\mathcal S^2 \setminus \mathcal G_P)$. Then
$$\sigma[ \, X, Y \, ] := -\text{Sgn}[ \, \hat\sigma[ \, A, B \, ] \, ] \left( \underset{\overline\gamma \in \overline\Gamma}{\text{Sup}} \! \! \left[ \, \underset{t \in \mathcal T}{\text{Sup}} \! \! \left[ \, \sum_{k = 0}^{(n[t] - 1)}\left[ \, \sqrt{ -\text{Sgn}[ \, \hat\sigma[ \, A, B \, ] \, ] \, \hat\sigma[ \, \overline\gamma[ \, t_k \, ], \overline\gamma[ \, t_{(k + 1)} \, ] \, ] } \, \right] \, \right] \, \right] \right)^{\! \! 2}.$$
3.3 $(X, Y)$ spacelike separated:
For each of the completely spacelike curves $\overline\gamma \in \overline\Gamma,$
$\overline\gamma : [0 \ldots 1] \rightarrow (\mathcal S \setminus \{ P \}), \qquad \overline\gamma[ \, 0 \, ] \mapsto X, \qquad \overline\gamma[ \, 1 \, ] \mapsto Y$ consider all its finite partitionings $t \in \mathcal T$, with $t_k \in [0 \ldots 1],$
integer indices $k \in [0, 1, \ldots n] \qquad t_0 = 0, \qquad t_n = 1, \qquad (j \lt k) \implies (t_j \lt t_k)$,
with $n \ge 2$ and such that $\forall k \in [0, \ldots (n - 1)] : (\overline\gamma[ \, t_k \, ], \overline\gamma[ \, t_{(k + 1)} \, ]) \in (\mathcal S^2 \setminus \mathcal G_P)$. Then
$$\sigma[ \, X, Y \, ] := \text{Sgn}[ \, \hat\sigma[ \, A, B \, ] \, ] \, \left( \underset{\overline\gamma \in \overline\Gamma}{\text{Inf}}\left[ \, \underset{t \in \mathcal T}{\text{Sup}}\left[ \, \sum_{k = 0}^{(n[t] - 1)}\left[ \, \sqrt{ \text{Sgn}[ \, \hat\sigma[ \, A, B \, ] \, ] \, \hat\sigma[ \, \overline\gamma[ \, t_k \, ], \overline\gamma[ \, t_{(k + 1)} \, ] \, ] } \, \right] \, \right] \, \right] \right)^{\! \! 2}.$$
4. Determining $\sigma[ \, X, P \, ] = \sigma[ \, P, X \, ]$
4.1 $X \equiv P$:
$$\sigma[ \, P, P \, ] = 0.$$
4.2 $\sigma[ \, X, P \, ]$ for $(X, Y) \in \mathcal G_P$ lightlike separated:
$$\sigma[ \, X, P \, ] = \sigma[ \, P, X \, ] = 0.$$
4.3 $\sigma[ \, X, P \, ]$ for $(X, Y) \in \mathcal G_P$:
Define set $\mathcal H_X \equiv \{ H \in (\mathcal S \setminus \{ P \}) \, \mid \, ((X, H) \in (\mathcal S^2 \setminus \mathcal G_P) \text{ and } $
$(\text{Sgn}[ \, \sigma[ \, X, Y \, ] \, ] \, \sqrt{ \text{Sgn}[ \, \sigma[ \, X, Y \, ] \, ] \, \sigma[ \, X, Y \, ] } = $
$\text{Sgn}[ \, \sigma[ \, X, H \, ] \, ] \, \sqrt{ \text{Sgn}[ \, \sigma[ \, X, H \, ] \, ] \, \sigma[ \, X, H \, ] } +
\text{Sgn}[ \, \sigma[ \, H, Y \, ] \, ] \, \sqrt{ \text{Sgn}[ \, \sigma[ \, H, Y \, ] \, ] \, \sigma[ \, H, Y \, ] })$
$\}.$
In terms of this:
$$\sigma[ \, X, P \, ] := \text{Sgn}[ \, \sigma[ \, X, Y \, ] \, ] \left( \underset{H \in \mathcal H_X}{\text{Sup}}\left[ \, \sqrt{ \text{Sgn}[ \, \sigma[ \, X, H \, ] \, ] \, \sigma[ \, X, H \, ] } \, \right] \right).$$
