Unfinished fabric of the three neutrino paradigm

In the current $3\ensuremath{\nu}$ paradigm, neutrino flavor oscillations probe three mixing angles $({\ensuremath{\theta}}_{12},{\ensuremath{\theta}}_{23},{\ensuremath{\theta}}_{13})$, one $CP$-violating phase $\ensuremath{\delta}$, and two independent differences between squared masses ${m}_{i}^{2}$, that can be chosen as $\ensuremath{\delta}{m}^{2}={m}_{2}^{2}\ensuremath{-}{m}_{1}^{2}>0$ $\mathrm{\ensuremath{\Delta}}{m}^{2}={m}_{3}^{2}\ensuremath{-}({m}_{1}^{2}+{m}_{2}^{2})/2$, where $\mathrm{sign}(\mathrm{\ensuremath{\Delta}}{m}^{2})=+(\ensuremath{-})$ for normal (inverted) mass ordering. Absolute $\ensuremath{\nu}$ probed by effective ${m}_{\ensuremath{\beta}}$ in beta decay, total $\mathrm{\ensuremath{\Sigma}}$ cosmology and---if neutrinos are Majorana---by another ${m}_{\ensuremath{\beta}\ensuremath{\beta}}$ neutrinoless double decay. Within an updated global analysis of oscillation nonoscillation data, we constrain these parameters, both separately selected pairs, highlight concordance or discordance among different constraints. Five parameters $(\ensuremath{\delta}{m}^{2},|\mathrm{\ensuremath{\Delta}}{m}^{2}|,{\ensuremath{\theta}}_{12},{\ensuremath{\theta}}_{23},{\ensuremath{\theta}}_{13})$ consistently measured, with overall accuracy ranging from $\ensuremath{\sim}1%$ $|\mathrm{\ensuremath{\Delta}}{m}^{2}|$ to $\ensuremath{\sim}6%$ ${\mathrm{sin}}^{2}{\ensuremath{\theta}}_{23}$ (due its persisting octant ambiguity). We find hints ordering (at $\ensuremath{\sim}2.5\ensuremath{\sigma}$), well ${\ensuremath{\theta}}_{23}<\ensuremath{\pi}/4$ $\mathrm{sin}\ensuremath{\delta}<0$ (both at 90% C.L.), discuss some tensions datasets. Concerning include recent KATRIN constraints on ${m}_{\ensuremath{\beta}}$, combine latest $^{76}\mathrm{Ge}$, $^{130}\mathrm{Te}$ $^{136}\mathrm{Xe}$ bounds ${m}_{\ensuremath{\beta}\ensuremath{\beta}}$, accounting nuclear matrix element covariances. also variants related cosmic microwave background (CMB) anisotropy lensing which may affect cosmological $\mathrm{sign}(\mathrm{\ensuremath{\Delta}}{m}^{2})$. The default option, including all Planck results, irrespective so-called anomaly, sets upper level $\ensuremath{\sim}{10}^{\ensuremath{-}1}\text{ }\text{ }\mathrm{eV}$, further favors up $\ensuremath{\sim}3\ensuremath{\sigma}$. An alternative includes ACT results plus other (from WMAP data) globally consistent standard lensing, is insensitive but prefers $\mathrm{\ensuremath{\Sigma}}\ensuremath{\sim}\mathrm{few}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}1}\text{ implications searches. general, unfinished fabric paradigm appears junction diverse searches particle physics, astrophysics cosmology, whose convergence will crucial achieve a convincing completion.