Muonium fine structure: theory update, tests of Lorentz violation, and experimental prospects.

We review the status of the QED calculations for the muonium energy interval and provide the updated theoretical value of 9874.357(1) . Additionally, we present a model for probing Lorentz-violating coefficients within the Standard Model Extension framework using the fine structure measurement in the presence and absence of a weak external magnetic field, enabling novel tests of CPT and Lorentz symmetry. Using Monte Carlo simulations, we estimate that a precision of on the isolated transition could be achievable employing Ramsey's separate oscillatory fields (SOF) technique. Collecting the required statistics will become feasible with the upcoming High-Intensity Muon Beam (HiMB) at the Paul Scherrer Institute (PSI) in Switzerland. These advancements will enable precise tests of radiative QED corrections and nuclear self-energy contributions, while also providing tests of new physics and sensitivity to unconstrained coefficients for Lorentz violation within the Standard Model Extension framework.