Expressing curvature invariants ($K_1, I_1, ...$), at any one event, through Synge's WF $\sigma$ (given of each event pair, in a suitable region)

Considering a set $$\mathcal S$$ of events such that for each pair $$p, q \in \mathcal S$$ Synge's world function $$\sigma$$ is defined and the corresponding value $$\sigma[ ~ p, q ~ ]$$ is given, and such that a subset $$\mathcal D_0 \subseteq \mathcal S$$ constitutes a causal diamond, then surely the values of curvature invariants, such as the Ricci scalar $$R$$, the Kretschmann scalar $$K_1$$, among others, are in an abstract sense defined; "at", or "referring to", each event $$x$$ of (at least: the interior of) causal diamond $$\mathcal D_0$$.

My question:

Are there specific expressions for values of curvature invariants, at event $$x \in (\mathcal D_0 \setminus \partial \mathcal D_0)$$, explicitly in terms of values of Synge's world function $$\sigma[ ~ p, q ~ ]$$ of (some, or all) event pairs $$p, q \in \mathcal D_0$$, and (surely) in particular involving values $$\sigma[ ~ p, x ~ ]$$ of (some, or all) events $$p \in \mathcal D_0$$ and (the "target event") $$x$$ ? (And if so: Which are these expressions?, of course.)

Presumably, the sought expressions would be "in the limit of vanishing size of causal diamond $$\mathcal D_0$$", i.e. symbolically

$${\large \underset{\text{Size}[ ~ \mathcal D_0 ~ ] \rightarrow 0 }{\text{Lim}}\Big[ ~ ... ~ \Big] }$$

where $$\text{Size}[ ~ \mathcal D_0 ~ ]$$ as "the size of causal diamond $$\mathcal D_0$$" would itself be explicitly expressed as

$${\large \text{Size}[ ~ \mathcal D_0 ~ ] := \underset{p, q ~ \in ~ \mathcal D_0}{\text{Sup}}\Big[ \{ \sqrt{ 2 ~ \text{Abs}[ ~ \sigma[ ~ p, q ~ ] ~ ] } \} ~ \Big] }.$$

• I've noticed the (1) immediate "drive-by downvote" to my OP question; canonically indicating that it "does not show any research effort; it is unclear, or not useful". I took that as motivation to right away do a corresponding web search (which, admittedly, I hadn't done in a while). And, indeed, we find prominently: arxiv.org/abs/2304.00088 (A. Nasiri, "Synge's world function applied to causal diamonds and causal sets"). Quote: "Of great importance is the recovery of curvature." Sep 28, 2023 at 20:40
• As motivational example for my question consider timelike path segment $\gamma \subset \mathcal D_0$ with event $x \in \gamma$. Then the expression $$\Large \underset{\sigma[~ p,q ~] \rightarrow 0; ~~ x \text{ between } p,q,\in \gamma}{\text{Lim}}\Big[ \\ \left( \frac{\sigma[~ p,q ~]}{\sigma[~ p,x ~] ~ \sigma[~ x,q ~]} + \frac{\sigma[~ p,x ~]}{\sigma[~ p,q ~] ~ \sigma[~ x,q ~]} + \frac{\sigma[~ x,q ~]}{\sigma[~ p,x ~] ~ \sigma[~ p,q ~]} -\frac{2}{\sigma[~ p,q ~]} -\frac{2}{\sigma[~ p,x ~]} -\frac{2}{\sigma[~ x,q ~]}\right)^{(-1/2)}~ \Big]$$ is a.k.a. curvature radius $\rho_{\gamma}$ at $x$. Sep 29, 2023 at 7:01