Raul Jimenez, Carlos Peña Garay, Fergus Simpson, Licia Verde

Cosmological data have reached the precision needed to turn the neutrino mass ordering from a weak Bayesian preference into a decisive model-selection test. We compute the evidence for the Normal and Inverted Hierarchies by combining DESI DR2 clustering with NuFIT oscillation data. In baseline $Λ$CDM, DESI DR2 plus Planck CamSpec gives $Σm_ν<0.0642\,{\rm eV}$ at 95\% confidence, close to the normal-ordering floor, $Σm_ν^{\rm NH}\simeq0.059\,{\rm eV}$, but well below the inverted-ordering minimum, $Σm_ν^{\rm IH}\simeq0.099\,{\rm eV}$. Thus the inverted hierarchy lies in the tail of the cosmological likelihood. The Bayes factor $K=P(D|{\rm NH})/P(D|{\rm IH})$ exceeds $460$ even for a conservative reference prior, and remains strong, $K>40$, in baseline-model extensions. We show that this result is robust to the choice between a reference prior and a physically motivated logarithmic hierarchical prior, marking the transition from {\em prior-sensitive evidence} to {\em likelihood-dominated exclusion} of the inverted hierarchy within standard cosmology. Embedding these priors in the two-dimensional design space of measure (logarithmic versus linear in mass) and structure (hierarchical versus non-hierarchical), we find that all four prior constructions give decisive evidence under DESI DR2, with residual prior dependence governed mainly by the measure -- a factor $\sim\!10$ in $K$ -- rather than by the hierarchy assumption. At the prior-family level, the evidence favors the SJPV prior predictive over HS by a Bayes factor above $4,700$ across each matched-support variation tested. The favored normal ordering pushes the effective Majorana mass to the few-meV regime, with median $m_{ββ}=3.28\,{\rm meV}$ and 95\% credible interval $0.95<m_{ββ}<11.55\,{\rm meV}$, below the inverted-ordering target for upcoming neutrinoless double-beta decay experiments.