Connecting Primordial Star Forming Regions and Second Generation Star Formation in the Phoenix Simulations.

We introduce the {\em Phoenix Simulations}, a suite of highly resolved cosmological simulations featuring hydrodynamics, primordial gas chemistry, Population III and II star formation and feedback, UV radiative transfer, and saved outputs with $\Delta t$=200 kyr. The suite samples 73,523 distinct primordial star formation events within \npiii distinct regions, forming \ngii second-generation enriched star clusters by $z \geq 12$ within a cumulative 156.25 Mpc$^3$ volume. The regions that lead to enriched star formation contain up to $167$ primordial stars, with 78.7 \% of regions having experienced multiple types of primordial supernovae. The extent of a primordial region, measured by its metal-rich surrounding cloud, is highly variable: the average region has radius $\sim 3$ kpc, with 95 \% confidence limit on the distribution of measured radii is $\sim 5-7$ kpc. For continuing star formation, we find that the metallicity distribution of second generation stars is similar to that of subsequent Population II star formation, with both distributions spanning hyper metal-deficient ([Z/H]$\sim-7$) to super-solar ([Z/H]$\sim0.8$). We find that the metallicity of second generation stars has no strong dependence on the configuration of progenitor supernovae, with the mean metallicity of second-generation stars having $-1.73 < $[Z/H]$<-2.15$. Finally, we create an interpretable regression model to predict the radius of metal-rich influence of \piii star systems within the first 7-18 Myr after the first light. The model predicts the radius with $R_2 \gtrsim 0.4$ and mean squared error $\leq 0.06$. The probability distribution function of predicted radii compares well to that of observed radii with Jensen-Shannon distance $\lesssim 0.2$ for all modelled times.