SIMBAD references

We introduce a simulation-based inference framework to constrain the origins of individual ultra-high-energy cosmic rays by combining realistic three-dimensional propagation modeling with Bayesian parameter estimation. Our method integrates CRPropa 3 simulations, including all relevant interactions and magnetic deflections in both Galactic and extragalactic fields, with approximate Bayesian computation to infer posterior distributions over key parameters such as source position, distance, energy, and magnetic field properties. This approach allows joint constraints from the observed energy and arrival direction to be applied simultaneously, naturally incorporating their correlations in addition to relevant modelling uncertainties. We demonstrate our method by applying it to the Amaterasu particle detected by the TA observatory, the second-highest-energy cosmic ray ever detected. The resulting posterior distributions quantify the regions of space consistent with its reconstructed properties under different energy and composition assumptions, revealing a broader set of nearby source candidates than found in previous analyses. This application highlights the framework's ability to translate individual ultra-high-energy cosmic-ray observations into directly interpretable source constraints and provides a foundation for future simulation-based analyses of cosmic rays at the highest energies.