## 2 Materials and Methods
### Simulation Dataset  
A Monte Carlo simulation was performed to create a dataset using Geant4 [28] with the same physical parameters as in [PERSON] et al. [8], except for the detector dimension. The Geant4 Livermore libraries were used for low-energy electromagnetic processes; both the photoelectric effect and Compton scattering were implemented according to [PERSON] et al. [29]. Geant4 has a maximum step length (MSL) parameter that balances the accuracy and computation time for particle tracking. For simulation with Cherenkov effects in the low-energy region, an appropriate setting of the MSL is important. We set the MSL to 1 \(\upmu\)m. The optical properties of PbF\({}_{2}\), for example, the transmittance, refractive index, and density, were determined based on the definition of [PERSON] et al. [24] and the Refractive Index Database [30]. Figure 1 shows a simulation setup of a paired Cherenkov detector and 125 training source positions. The two Cherenkov detectors were placed face-to-face, 50 mm apart. The origin of the coordinate system, \((x,y,z)=(0,0,0)\), was defined at the center of the detector plane for \((x,y)\) and at the center between the pair of the detectors at the \(z\)-axis, as shown in Figure 1. A total of 10\({}^{7}\) emissions of the \(\gamma\)-ray pair were performed at each source position, where \(x,y,z\) were scanned from 0 to 10 mm in intervals of 5 mm. Thus, the \(3\times 3\times 3=27\) source positions in the first quadrant were simulated, as shown in the red rectangle of Figure 1. By utilizing symmetry, 98 source positions were replicated from those of the 27 source positions, as shown by the green dots in Figure 1. For example, by flipping the sign of \(x\) for the detected photon information, the simulation data of the source position \((x,y,z)=(-5,0,0)\) can be generated from the simulation data of the source position \((x,y,z)=(5,0,0)\). Thus, 125 (\(5^{3}\)) source positions were simulated in total. The momentum direction of the \(\gamma\)-ray pair was randomly determined against the polar and azimuth angles, which were varied from \(-\pi\)/4 to \(+\pi\)/4 and 0 to \(2\pi\), respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Density (g cm\({}^{-3}\))** & **Refractive Index** & **Cutoff (nm)** \\ \hline
7.7 & 1.82 & 245–280 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optical properties of PbF\({}_{2}\) crystal. Figure 1: Paired Cherenkov detectors and 125 training source positions. Two Cherenkov detectors were placed face-to-face, 50 mm apart. The origin of the coordinate system, \((x,y,z)=(0,0,0)\), was defined at the center of the detector plane for \((x,y)\) and at the center between the pair of the detectors at the \(z\)-axis. A monolithic PbF\({}_{2}\) crystal with a \(40\times 40\times 10\) mm\({}^{3}\) Cherenkov radiator was coupled to a photodetector array through 300-\(\upmu\)m-thick protection resin (\(n=1.5\)) with grease (\(n=1.5\)). In all simulations, we recorded the following information event-by-event. * \((x_{s},y_{s},z_{s})\): true source position. * \(N,M,5\leq N,5\leq M\): number of photons reaching the photodetector plane of Cherenkov detectors 1 and 2, respectively. * \((x_{1n},y_{1n},t_{1n})\), \((x_{2m},y_{2m},t_{2m})\), \(1\leq n\leq N\), \(1\leq m\leq M\): \(xy\) position and time information for each photon reaching the photodetector plane of Cherenkov detectors 1 and 2, respectively, where the smaller of \(t_{11}\) and \(t_{21}\) is 0 s.  
As the photon detection threshold was assumed to be five, a total of 227,005 coincidence events in which both detectors detected more than five Cherenkov photons were obtained. This number was only 0.02 % of all \(\gamma\)-ray pair emissions. We split this set to assign 75% for training and 25% for testing. To increase the number of coincidence events in the training dataset, we extracted the \({}_{N}\)C\({}_{5}\times{}_{M}\)C\({}_{5}\) events from coincidence events that were \(6\leq N\) or \(6\leq M\).