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Using a marginalized DESI $\Sigma m_\nu$ posterior as a cosmological likelihood risks prior double counting and conceptual inconsistency (Sec. 2.1, Sec. 3.1). The manuscript states it extracts the 1D marginalized posterior for $\Sigma m_\nu$ from DESI DR2 MCMC chains and then uses it as $P(D_{\rm cosmo}|\Sigma m_\nu)$ in the evidence integral. But a chain-derived marginalized posterior is proportional to $L(\Sigma m_\nu,{\rm other})\times\pi_{\rm DESI}(\Sigma m_\nu,{\rm other})$ marginalized over other parameters using DESI priors, not a likelihood in $\Sigma m_\nu$. Treating it as a likelihood can silently import DESI priors and volume effects into the second-stage evidence, undermining the interpretation of differences between SJPV vs HS and potentially biasing $K$.
Recommendation: In Sec. 2.1 (and where Eq. (1)–(2) are defined), explicitly distinguish likelihood vs posterior and state the DESI priors relevant to $\Sigma m_\nu$ and the parameters being marginalized. Then do one of the following: (i) reconstruct an approximate likelihood in $\Sigma m_\nu$ (e.g., from a profile likelihood if available, or by deconvolving a known $\Sigma m_\nu$ prior and carefully discussing what remains after marginalization), or (ii) if you intentionally use a posterior-based surrogate, rewrite the evidence expression to avoid double counting and clearly state the approximation and its limitations. Provide a quantitative comparison of $K$ obtained from (a) a likelihood-based surrogate (profile-likelihood or equivalent) versus (b) your posterior-derived surrogate to demonstrate stability.
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The DESI cosmological information is overly compressed to a 1D truncated Gaussian in $\Sigma m_\nu$ without quantitative validation in the IH-critical tail and without accounting for degeneracies (Sec. 2.1, Sec. 3.1–3.2, Sec. 3.7). The Bayes factor is dominated by the likelihood/posterior tail near $\Sigma m_\nu \approx 0.09$–$0.12$ eV (around the IH minimum). Small mismodeling of the tail or ignoring correlations with parameters that broaden $\Sigma m_\nu$ (especially in $w_0 w_a{\rm CDM}$) can change $\ln K$ materially. The negative untruncated mean ($\mu_0 < 0$) further emphasizes that the effective distribution is boundary-dominated.
Recommendation: In Sec. 2.1 and Sec. 3.1, validate the 1D surrogate specifically in the range $0.06$–$0.12$ eV: provide goodness-of-fit diagnostics in the tail (not only bulk), and compare to at least one alternative surrogate (e.g., KDE/interpolated density used directly, skew-normal/mixture model). Report how $\ln K$ changes across these alternatives for both $\Lambda {\rm CDM}$ and $w_0 w_a{\rm CDM}$ (Sec. 3.7). Where feasible, include at least a low-dimensional approximation that retains the main $\Sigma m_\nu$ degeneracies (e.g., $\Sigma m_\nu$ with $\Omega_m$ and $w_0$) or reweight the original chains under NH/IH priors to avoid the 1D collapse.
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Oscillation information (NuFIT 6.0) and the hierarchy preference $\Delta\chi^2$ are incorporated ambiguously and may be double-counted or mis-applied (Sec. 2.1–2.2, Sec. 3.2). The manuscript mentions a NuFIT $\Delta\chi^2 \approx 6.1$ preference for NH and applies it via a multiplicative factor $\exp(\Delta\chi^2/2)$ to produce $K_{\rm full}$, but it is unclear whether $P(D_{\rm osc}|\theta)$ is also integrated over oscillation parameters in the evidence integral. The sign convention and the mapping from NuFIT’s global-fit $\Delta\chi^2$ to a Bayes factor between discrete hierarchies are not defined.
Recommendation: In Sec. 2.1–2.2 and Sec. 3.2: (i) define $\Delta\chi^2$ explicitly (e.g., $\Delta\chi^2 \equiv \chi^2_{\rm IH} - \chi^2_{\rm NH}$) and therefore the likelihood ratio mapping; (ii) state precisely what enters the evidence integral for each hierarchy (full $P(D_{\rm osc}|\theta)$ surface vs approximations vs none); and (iii) if you keep the $K_{\rm base}/K_{\rm full}$ decomposition, make it mathematically explicit and prove/justify that the $\exp(\Delta\chi^2/2)$ factor does not double-count any oscillation information already included. Ideally, implement a single end-to-end evidence calculation that multiplies $P(D_{\rm cosmo}|\Sigma m_\nu(\theta))\times P(D_{\rm osc}|\theta)$ and integrates over $\theta$ for NH and IH, and tabulate the resulting $\ln K$ alongside the decomposed values.
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Prior definitions (SJPV and HS) are insufficiently specified to reproduce results and to interpret prior dependence (Sec. 2.3, Sec. 3.4). The SJPV hierarchical log-normal prior needs explicit generative form, hyperpriors, bounds, and the exact procedure for mapping exchangeable draws to ordered eigenmasses under NH/IH (label switching/order constraints). The HS “reference” prior is given in a closed form (e.g., proportional to a polynomial in masses) without a transparent derivation, definition of the base measure, parameterization choices, or normalization domain; evidence values can depend strongly on those choices despite “reference” wording.
Recommendation: Expand Sec. 2.3 (preferably with an Appendix) to fully define both priors: for SJPV, write the full hierarchical model (including hyperparameter priors and explicit bounds), clarify whether masses are treated as unordered then ordered, and show how NH vs IH mapping is done. For HS, provide a step-by-step derivation from the chosen observable set/Fisher information to the stated mass-space density, including the Jacobian, the measure with respect to which the density is defined, and the normalization domain (upper mass bounds). In Sec. 3.4, add sensitivity checks showing how $\ln K$ changes under reasonable variations of hyperprior bounds and HS normalization bounds.
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Evidence computation methodology and numerical uncertainty are not documented at the level required for a paper that claims very large Bayes factors (Sec. 2.2, Sec. 3.2–3.4). The manuscript reports $K$ values with high apparent precision (e.g., $K=10231.4$), but does not specify the integral dimensionality, parameterization (three masses vs $m_{\rm lightest}+$splittings vs hyperparameters), numerical method (grid/MC/nested sampling/importance sampling), convergence checks, or uncertainty on $\ln K$. It is also unclear whether the “likelihood” surrogate is treated as normalized or up to a constant, which matters for evidences.
Recommendation: In Sec. 2.2 and/or an Appendix, specify: (i) the exact integration variables for each hierarchy/prior; (ii) the numerical integration/sampling method; (iii) sample sizes / grid resolution; (iv) convergence diagnostics; and (v) an estimated numerical error bar on $\ln Z$ and $\ln K$ (report $\ln K \pm \sigma$). Reduce reported significant figures accordingly. Also state explicitly whether your $P(D_{\rm cosmo}|\Sigma m_\nu)$ surrogate is normalized and how any normalization constants are handled consistently across hierarchies.
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Cosmological-model and dataset/systematic dependence is not explored or discussed in proportion to the strength of the “decisive” claim (Sec. 1, Sec. 3.1–3.3, Sec. 3.7, Conclusion). Only $\Lambda {\rm CDM}$ and $w_0 w_a{\rm CDM}$ are analyzed directly. However, $\Sigma m_\nu$ constraints can weaken in standard extensions (e.g., non-flat $\Lambda {\rm CDM}$, $\Lambda {\rm CDM}+N_{\rm eff}$, early dark energy, modified gravity), and $K$ is highly sensitive to any broadening of the $\Sigma m_\nu$ tail above the IH threshold. Additionally, the paper should more clearly separate “DESI+Planck (CamSpec) under a specific pipeline” from a general statement about cosmology.
Recommendation: In Sec. 3.7 and the Conclusion, add a structured robustness discussion: (i) explicitly list which cosmological extensions are expected to most impact $\Sigma m_\nu$ and why; (ii) either run at least one or two additional representative extensions (e.g., $\Lambda {\rm CDM}+N_{\rm eff}$ and non-flat $\Lambda {\rm CDM}$) and report $\ln K$, or (if infeasible) use published constraints in those models to bound how much $\ln K$ could plausibly decrease; and (iii) provide a table of $\ln K$ across CMB likelihood choices (CamSpec vs alternatives) and dataset combinations (DESI-only, DESI+Planck, DESI+Planck+SN) using a consistent methodology. Soften global statements (“definitive”/“decisive”) to be explicitly conditional on the assumed cosmological model class and dataset combination.
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$0\nu\beta\beta$ implications are presented too definitively without propagating key nuclear/theory uncertainties and mechanism dependence (Sec. 3.6, Conclusion). Translating $m_{\beta\beta}$ to half-life requires isotope choice, phase-space factors, and nuclear matrix elements (NMEs), which carry sizable spreads; moreover, the link assumes standard light Majorana neutrino exchange dominance.
Recommendation: In Sec. 3.6 and the Conclusion, specify the isotope(s), phase-space factors, and NME set(s) used, and propagate NME uncertainties into a band of predicted $T_{1/2}$ for a given $m_{\beta\beta}$ (e.g., show ranges across multiple standard NME calculations). State explicitly that results are conditional on the light-Majorana exchange mechanism and that alternative mechanisms can decouple $m_{\beta\beta}$ from the rate. Rephrase detection-probability statements to make these conditions explicit.
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Scholarly grounding and references need strengthening, and the framing risks overstating novelty/authority (Sec. 1, Sec. 4/References, Conclusion). Key primary references for DESI DR2 cosmology outputs, Planck likelihoods, NuFIT 6.0, prior Bayesian hierarchy studies, and $0\nu\beta\beta$ experimental sensitivities should be clearly and reliably cited. Any non-standard, placeholder, or hard-to-verify references undermine credibility, especially for a “decisive evidence” claim.
Recommendation: Revise Sec. 4/References to ensure all inputs are traceable to primary, citable sources (DESI DR2 cosmology/likelihood papers, Planck likelihood documentation, NuFIT 6.0 global-fit papers/data release, and key prior work on cosmological hierarchy inference). Update the Introduction/Conclusion phrasing to position the contribution relative to that literature (what is new: DESI DR2-specific update, prior comparison, etc.), and remove/replace any references that are not verifiable or not appropriate for the venue.