Joint Constraints on Neutrinos and Dynamical Dark Energy in Minimally Modified Gravity

The \(w_{\dagger}\)VCDM framework provides a theoretically well-controlled extension of \(\Lambda\)CDM within the class of minimally modified gravity theories, allowing for flexible cosmological background evolution and linear perturbation dynamics while remaining free of pathological instabilities. In this work, we have shown that this scenario remains robust when confronted with current cosmological observations, even in the presence of an extended neutrino sector. Combining \textit{Planck} CMB data with DESI DR2 BAO and DESY5 supernovae, we obtain stringent constraints on neutrino physics, including \(\sum m_\nu < 0.11~\mathrm{eV}\) (95\% CL) and \(N_{\rm eff} = 2.98^{+0.13}_{-0.14}\), fully consistent with Standard Model expectations. Crucially, the data exhibit a statistically significant preference for a late-time dark-energy transition, characterized by a robust quintessence--phantom crossing that remains stable across all dataset combinations and neutrino-sector extensions, including the presence of a sterile neutrino. The combined effects of modified late-time expansion and additional relativistic degrees of freedom systematically raise the inferred Hubble constant, substantially alleviating the \(H_0\) tension without invoking early dark energy or introducing theoretical instabilities. Overall, the \(w_{\dagger}\)VCDM scenario emerges as a compelling phenomenological framework that simultaneously accommodates current constraints on neutrino physics, provides an excellent fit to recent BAO and supernovae data, and offers a viable pathway toward resolving persistent tensions in the standard cosmological model.