## Appendix A Projection Algorithm for Physically Constrained Particle-Distribution Prediction
In this context, only regions that are inherently characterized by low number density--namely, areas where an abrupt local drop in number density occurs adjacent to high-density regions, reflecting a dramatic local change--are considered to be under-estimated regions. Then, binary dilation is applied to the high-density areas until they extend to the dead zone boundaries. After this dilation, the low-density regions that fall within the expanded high-density areas are isolated by taking their intersection, which constitutes the valid region. This valid region, defined as the intersection of the expanded high-density area and the original low-density zone, represents the potential area for reallocating invalid prediction points that have erroneously fallen within the dead zone. Figure 16 illustrates this procedure: the left image highlights the initial identification of high-density regions, while the right image shows the result after expanding the high-density regions to the dead zone boundary and intersecting them with the low-density areas. A key point to emphasize is that although the procedure described above effectively identifies regions of under-estimation, it does not inherently quantify the degree of under-estimation within each region. Thus, the subsequent phase provides a framework for assessing the relative severity of under-estimation across these regions, thereby determining which areas contain a greater or lesser number of invalid predictions that require reallocation. As previously mentioned, in the absence of a mask, the deposited single-track profile or the probability density function of particle distribution on the substrate should exhibit a smooth, continuous decay [36]. The highest particle number density should be at the center, gradually decreasing with increasing radial distance. In other words, all the regions outside the dead zone are expected to follow this trend. This decay phenomenon is typically modeled by an exponential function, as such functions possess smooth, continuous, and monotonically decreasing properties that align well with the anticipated physical distribution. Based on this assumption, the data from the current valid prediction points are used to fit a target density function that characterizes the decay of particle number density as a function of the radial distance R; that is, the density function is assumed to follow an exponential form. By applying a logarithmic transformation, this exponential relationship can be converted into a linear one, thereby facilitating parameter estimation and fitting. Subsequently, the fitted function is used to assign weights to each region within the valid zone identified in the earlier stage. To be more specific, initially, radial statistics are computed for the predicted points within the deposition (effective) region. Specifically, the radial distance for each valid predicted point is calculated, as defined in  
\[\text{R}_{\text{i}}=\sqrt{{y_{i}}^{2}+{z_{i}}^{2}} \tag{17}\]  
where R is the radial distance from the substrate center (\(y=0\) and \(z=0\)), and \(y\) and \(z\) are the coordinates of the current point (location of each particle). To quantify the number of particles in distinct radial intervals, the radial distance--from 0 to the maximum value--is partitioned into n intervals/bins:  
\[\text{bins}=\{0,\text{R}_{1},\text{R}_{2},\ldots,\text{R}_{\text{max}}\} \tag{10}\]  
For each interval, the number of particles N is first computed, and the corresponding number density is then determined based on the annular area (Equation (11)), yielding the number density (\(\rho\)) as defined in (Equation (12)). \[\text{Area}_{i}=\pi\Big{(}\text{R}_{i+1}^{2}-\text{R}_{i}^{2}\Big{)} \tag{11}\]  
\[\rho_{i}=\frac{\text{N}}{\text{Area}_{i}} \tag{12}\]  
Subsequently, the logarithm of these number density values is calculated, and a linear regression is performed to establish the exponential relationship between radius and number density (Equation (13)).