DESI DR2 Constraints on Coupled Hu--Sawicki f(R) Gravity with Interacting Neutrinos: Implications for the Hubble and S 8 Tensions

We investigate the cosmological implications of viable Hu--Sawicki f(R) gravity models and their extension through an effective interaction with the neutrino sector. Motivated by the persistent H 0 and S 8 tensions, we explore whether the combined effects of modified gravity, neutrino free-streaming, and dark-sector interaction can alleviate these discrepancies while remaining consistent with current cosmological observations. Starting from the metric formulation of f(R) gravity, we derive the modified background and perturbation equations and implement the resulting cosmological scenario within the MontePython / hi class framework. A comprehensive Bayesian MCMC analysis is performed using combinations of Planck 2018 CMB data, CMB lensing, DESI DR2 BAO measurements, Cosmic Chronometers, and Pantheon supernova observations. Our results show that the interacting Hu--Sawicki scenario provides a noticeable improvement over both the standard Λ CDM cosmology and the non-interacting f(R) framework. In particular, the interacting model shifts the inferred Hubble constant toward larger values, yielding H 0 = 70.21 plus or minus 1.60 km s-1Mpc-1, while simultaneously lowering the clustering amplitude to S 8 = 0.796 plus or minus 0.009. Consequently, the residual tensions with SH0ES R22 and DES-Y6 are reduced to nearly the 1σ level. From the statistical perspective, the interacting model provides the best overall fit according to the Akaike Information Criterion, with Δχ squared-45 relative to Λ CDM. The obtained neutrino mass constraints remain fully consistent with current cosmological bounds, with sum m ν< 0.122 eV. These results suggest that interacting modified-gravity scenarios offer a viable route to moderately easing both late-time tensions at once.