Baryonification IV: Constraining baryonic feedback with X-ray gas fractions

Baryonic feedback redistributes gas around dark matter halos, suppressing the matter power spectrum at scales now probed by weak lensing surveys. X-ray observations directly trace this hot gas, and are one of the main probes of its distribution and properties. We present a forward-modelling framework, built on the baryonification model, linking the three-dimensional gas density and temperature profiles of groups and clusters to observed X-ray surface brightness and luminosity profiles on one side, and to matter power spectrum suppression on the other. We validate the model against independent three-dimensional density reconstructions from the literature, and examine our temperature and metallicity treatment in the group-scale regime. Applying this framework to the SZ-selected CHEX-MATE and X-ray-selected eFEDs samples, we measure gas fractions across the group-to-cluster mass range while accounting for X-ray selection effects, with the first published gas fractions based on CHEX-MATE data. Combining both samples, we derive a joint constraint on the hot gas fraction retained by groups and clusters as a function of mass and on the baryonic suppression of the matter power spectrum. We find f gas = 0.029 plus or minus 0.006 at M 500c = 3 times 10 13M, f gas = 0.078 plus or minus 0.004 at M 500c = 3 times 10 14M, and suppression of 6% at k=1,h/ Mpc and 23% at k=5,h/ Mpc. Our findings are consistent with recent kinematic Sunyaev-Zel'dovich results, hinting at strong feedback. We also show that the L X - M relation is degenerate with feedback strength, and that different feedback scenarios produce distinct X-ray profile shapes that map onto the same L X - M point. This work is a first step toward extending the framework to forward-model diffuse X-ray emission at the map level for simulation-based inference in upcoming wide-area X-ray surveys such as eROSITA.