Simulation-calibrated Bayesian inference for progenitor properties of the microquasar SS 433

SS$\,$433 is one of the most extreme Galactic X-ray binaries, launching semi-relativistic jets and showing clear signs of super-critical accretion onto what is likely a black hole. Yet the properties of the binary system that produced it remain uncertain. To solve the inverse problem of inferring the progenitor properties of binaries that evolve into SS$\,$433-like systems, we use an iterative, simulation-based calibration framework that combines Bayesian inference with the isolated binary-evolution code COSMIC. Using six measured properties of SS$\,$433 and the dynamic nested sampler $\texttt{dynesty}$, we explore a ten-dimensional space of possible progenitor masses, orbits, mass-transfer histories, and natal-kick velocities. This approach identifies the regions of parameter space most consistent with SS$\,$433 and allows us to iteratively refine the resulting progenitor distributions. We find 90% confidence intervals for the progenitor initial primary mass of (8, 11) M$_\odot$, secondary mass of (32, 40) M$_\odot$, orbital period of (136, 2259) days, eccentricity of (0.26, 0.6), common-envelope efficiency of (0.44, 0.76), accreted fraction during stable mass transfer of (0.22, 0.6), and black-hole natal-kick magnitude of (5, 68) km/s. These results show that direct probabilistic inference of X-ray binary progenitors can yield new constraints on the formation of extreme accretion systems like SS$\,$433, which has important implications for theoretical expectations of the population of SS$\,$433-like systems in the Galaxy and their connection with cosmic ray observations.