{"id":"cca88094-ba89-4fad-ad02-65ddeec75097","arxiv_id":"2411.17112","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"A semi-empirical formula incorporating the quadrupole deformation of both parent and daughter nuclei is fitted to one-proton decay data and used to predict half-lives of new proton-emitter candidates.","lead":"This paper proposes a new empirical formula for one-proton decay half-lives that includes the deformation of both the parent and daughter nuclei. The authors fit it to 52 measured or upper-limit cases and argue it outperforms earlier formulas and can predict new proton emitters.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Comparative RMSE for Eq. (5) is computed on the fitted sample, so the claimed predictive superiority over prior formulas is not established and is likely smaller out-of-sample.","rationale":"The reader's weakest assumption was that the RMF beta inputs are accurate for drip-line nuclei. That is a genuine uncertainty that propagates into predictions for new candidates, but it is not the single most load-bearing issue for the paper's comparative claim, because the same betas enter only the new formula, and an error in them would be absorbed by the fitted coefficients. A more decisive problem is that the comparison table (Table 1), which is the quantitative evidence for 'lowest RMSE and chi^2', reports in-sample residuals for the 5-parameter formula and presumably fixed published coefficients for competing formulas. This comparison does not measure predictive accuracy and can be expected to favor the refitted formula through sheer parametric flexibility. Since the abstract and conclusions advertise the formula as accurate and 'reliable for forecasting,' the paper needs an honest out-of-sample or cross-validated comparison against the same formulas. The authors do run k-fold CV, but only for their own formula, and the numbers are not given in a table for external checking. The shape-coexistence section is explicitly preliminary and labeled as such; it is not central to the main formula claim. On the RMF beta point, I partially agree with the reader: it matters for the predicted candidates in Tables 3 and 4, but the main statistical claim can be tested independently of the true beta values by checking the CV comparison. Thus my verdict remains CONDITIONAL, with the condition being that the in-sample/in-fold comparison be replaced by a fair cross-validated comparison.","tokens_in":22330,"tokens_out":1623,"duration_ms":14947,"concrete_test":"Recompute Table 1 under a fair out-of-sample protocol: for each competing formula with k free parameters, use k-fold or leave-one-out cross-validation on the same 27 heavy-emitter points (refitting each formula's parameters within each training fold, or using the authors' published training set for each formula and testing on the held-out points). Report the average test-fold RMSE and chi^2 for Eq. (5) and for every listed formula, and quote the Fig. 1 CV numbers as a table. If Eq. (5)'s mean test-fold RMSE is not the lowest, the headline claim of predictive superiority should be softened to in-sample accuracy.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central empirical claim is that Eq. (5) outperforms all prior 1p half-life formulas, with RMSE 0.6998 vs 0.8854 for the next best in Table 1. But Table 1 reports in-sample residuals: the five coefficients a...e were least-squares fitted to the same 27 heavy-emitter points (or to the 52/34-point set), and the competing formulas are quoted with their published coefficients, fitted elsewhere or to different datasets. In-sample RMSE is a monotone function of fit flexibility, so a 5-parameter formula refitted to the evaluation set will almost always beat fixed formulas, regardless of predictive value. The k-fold cross-validation in Fig. 1 is described as 80/20 splits, but its RMSE values are not tabulated and are not compared with the same CV treatment for any competing formula, so it does not establish superiority either. The weakest physical assumption, that RMF beta_p and beta_d are accurate at the drip line, is acknowledged inside the paper and is secondary here; even with exact betas the headline superiority claim requires an out-of-sample comparison, which is missing. The result should be reported as 'best in-sample fit on the 2020 dataset' unless a fair comparison is supplied.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a new semi-empirical formula, Eq. (5), for one-proton radioactivity half-lives that includes the quadrupole deformations of both the parent and daughter nuclei, along with the usual Q-value and angular-momentum terms. The formula is fitted to 52 proton-emitting states (34 with accurate half-lives and 18 with upper limits) using a censored-data maximum-likelihood loss. The authors report the lowest RMSE and chi-squared among nine formulas on 27 heavy-emitter points, show a k-fold cross-validation plot, and use the formula to predict half-lives of candidates. They also discuss shape coexistence in selected parent-daughter pairs, considering G-G, G-S, S-G, and S-S transitions and their effect on half-lives and branching ratios.","tokens_in":22585,"tokens_out":4599,"duration_ms":46492,"significance":"If the central comparison were reliable, the formula would be a practically useful addition to the 1p-decay toolbox: it is explicit, easily reusable, and treats parent and daughter deformations simultaneously, which existing empirical formulas do not. The authors also deserve credit for using a censored-data loss and for reporting a k-fold cross-validation exercise; both are steps in the right direction. However, the key superiority claim is not yet established, because the comparison in Table 1 is in-sample for Eq. (5) while the competitors are evaluated with their published coefficients, and the cross-validation is not applied to the competing formulas. The deformation inputs are entirely theoretical (RMF) for highly exotic nuclei, and the predicted half-lives carry no uncertainty estimates. The shape-coexistence analysis is qualitative and post hoc. For these reasons the paper requires revision before the central claims can be accepted.","major_comments":[{"comment":"The headline comparison in Table 1 is not a predictive test. The coefficients a-e in Eq. (5) are least-squares fitted to the same 27 heavy-emitter points that are subsequently scored, whereas the competing formulas are evaluated with their published coefficients, which were obtained on other datasets. In-sample RMSE is therefore expected to favour the more flexible refitted formula, and the gap 0.6998 vs 0.8854 does not demonstrate predictive superiority. In addition, the text states that the deformation-term exponent p was scanned from 0 to 5 in steps of 0.5 and the best value selected on the same data; this model-selection step is an additional adaptive choice that is not counted in Np when computing chi-squared via Eq. (7). The k-fold CV in Fig. 1 is not tabulated and is not performed for any competing formula, so it does not repair the comparison. Please provide an out-of-sample comparison in which Eq. (5) and all formulas in Table 1 are fitted/selected on identical training folds and scored on identical test folds, or downgrade the claim to 'best in-sample fit on the 2020 dataset.'","section":"Table 1 and Theoretical Framework (Half-life calculations)"},{"comment":"All fits and predictions use RMF values of beta_p and beta_d for nuclei at or beyond the proton drip line, where there are no experimental deformations. Any systematic error in these RMF shapes is absorbed into the fitted coefficients, so the good in-sample RMSE cannot validate the deformation inputs. The extrapolated half-lives in Tables 3 and 4 are quoted to two decimals in log10T without propagating uncertainties from beta, Q, or l; for the least-bound candidates this can correspond to factors of order 10 in half-life. The authors state that WS4, HFB, and FRDM deformation tables were checked qualitatively, but no quantitative sensitivity test is reported. Please add a sensitivity analysis using alternative deformation sets for the fit and for the predicted candidates, and report approximate uncertainties on the predicted log10T values.","section":"Theoretical Framework (RMF deformations) and Table 3/Table 4"},{"comment":"The four-transition scheme (G-G, G-S, S-G, S-S) is applied post hoc: for each nucleus the transition that gives better agreement with experiment is identified after inspecting the data, without any model-comparison statistic or penalty for the additional transition channels. No wave-function overlaps or mixing amplitudes are computed, so the half-lives for the secondary-minimum paths should be regarded as schematic. The text itself acknowledges that this part is 'preliminary' and 'premature,' yet the conclusions and abstract present shape coexistence as a key outcome. Please either add a quantitative selection criterion (e.g., a likelihood comparison with a degrees-of-freedom penalty) or explicitly and consistently frame the shape-coexistence section as exploratory rather than as a validated prediction.","section":"Correlation between half-life and shape coexistence (Fig. 3, Table 5)"}],"minor_comments":[{"comment":"The abstract contains 'signicant' twice; the Introduction has 'performes'; the shape-coexistence section has 'inuenced'; Eq. (6) and the data-availability statement contain rendering artifacts ('vuut', 'left datasets'). The manuscript should be carefully proofread.","section":"Abstract and text"},{"comment":"The formula is difficult to parse as typeset: the exponents such as (-1)^(kappa_p + kappa_p kappa_d) need unambiguous parentheses, and the term preceding sqrt{l(l+1)} appears garbled (the factor d is separated from the radical). Please rewrite Eq. (5) with clear mathematical formatting.","section":"Eq. (5)"},{"comment":"The 'Decay Modes' column mixes probabilities and percentages (e.g., 'p=0.89; alpha=11'); please use a single convention throughout, preferably probabilities with stated uncertainties.","section":"Table 5"},{"comment":"The text states that average RMSE values across folds are shown, but no numerical values are given in the text or tables, making the cross-validation claim hard to reproduce. Please tabulate the per-fold and average RMSE for training and test splits.","section":"Figure 1 and k-fold CV"},{"comment":"The caption refers to 'columns 8-27' for the compared models, but the printed table does not appear to contain that many model columns; please correct the column reference.","section":"Table 2 caption"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and the authors are transparent about many limitations, including the absence of experimental deformations and the preliminary nature of the shape-coexistence analysis. The main blocker is the lack of a fair out-of-sample comparison with existing formulas; this is fixable within the scope of a revision and does not require a wholesale rewrite. I would not reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this as a fitting paper, and you can learn something. The new piece is Eq. (5): a five-parameter semi-empirical formula that couples the quadrupole deformations of parent and daughter via the kappa sign structure, extending the earlier alpha-decay term to 1p decay. That is genuinely new, and the authors are careful to say no prior empirical formula did this. They also do some real work: 52-point dataset, MLE censoring for the 18 upper-limit values, k-fold cross-validation, and an explicit comparison against many prior formulas. Credit where due: the formula is concrete, the fitting procedure is described, and the paper avoids pretending it is a microscopic theory.\n\nThe soft spots are the usual ones for this genre, plus one that matters. The headline superiority claim in Table 1 is not a fair predictive comparison: the RMSE for the present formula is computed after fitting its five parameters to the same 27-point subset, while the competing formulas are quoted with published coefficients fitted elsewhere. In-sample RMSE favors the most flexible refitted formula, so this does not establish that Eq. (5) predicts better. The k-fold CV is good practice, but the CV RMSE values are not tabulated and no competing formula gets the same CV treatment, so it does not fix the comparison. The stress-test note is right on this. Second, the deformation inputs are pure RMF values; at drip-line nuclei there is no experimental check, and the fitted coefficients soak up systematic error in beta. The paper acknowledges this, but it means predictions for new candidates carry an unknown theory error. Third, the shape-coexistence section chooses among G-G, G-S, S-G, S-S pathways post hoc and reports the ones that match; that is suggestive, not a quantitative test. I would call this a minor-to-moderate concern; the authors label the analysis preliminary themselves.\n\nThe physics case is plausible. The formula's two-nucleus deformation dependence is reasonable, the sign structure is interpretable, and the improvement over one-sided deformation formulas is credible even if not proven out-of-sample. The paper is worth a serious referee, but the referee should require an honest predictive comparison: either train/test splits applied to all formulas, or a bootstrap/leave-one-out table. Also ask for the full dataset and fitted residuals; the data availability statement is thin.\n\nRead this with a grain of salt; it is a solid fitting contribution, not a breakthrough. If I worked in this niche I would cite it for the formula and the candidate-emitter list.","headline":"A serviceable new empirical formula for 1p-decay half-lives that includes both parent and daughter deformation; the predictive-superiority claim is weaker than advertised because the head-to-head RMSE is in-sample.","tokens_in":23107,"tokens_out":1922,"would_cite":true,"duration_ms":18060,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"One-proton decay half-lives are reproduced most accurately when the empirical formula includes the deformations of both parent and daughter nuclei.","keywords":["one-proton radioactivity","proton decay half-lives","nuclear deformation","quadrupole deformation","shape coexistence","semi-empirical formula","proton drip line","branching ratios"],"falsifier":"Refit Eq. (5) on the same 52 data points using deformation values from a second independent mass model instead of the relativistic mean-field values, and see whether the two-deformation formula still beats the other eight formulas in Table 1; if the RMSE advantage disappears, the claim that both-nucleus deformation is what drives the improvement fails. A complementary check is to measure the half-life (and ideally the shape) of a newly predicted emitter such as $^{83}$Tc, $^{47}$Co, or $^{71}$Rb, and compare with the formula's extrapolation.","tokens_in":22095,"feed_emoji":"⚛️","tokens_out":13057,"duration_ms":110701,"temperature":0.7,"pith_summary":"This paper argues that one-proton radioactivity half-lives are better predicted when the empirical decay formula includes the quadrupole deformation (prolate or oblate shape) of both the parent nucleus and the daughter nucleus, in addition to the usual decay energy and angular-momentum terms. The authors fit a five-parameter formula to 52 known proton-emitting states, treating the 18 half-life upper limits with a censored-data likelihood, and on the 27 heavy-emitter subset they report a root-mean-square error of $0.6998$ and a reduced $\\chi^2$ of $0.6010$, lower than the nine formulas they compare. If the formula is correct, it provides a simple predictive tool for half-lives of proton emitters that have not been measured yet, and it indicates that the shapes of both partners matter for the decay rate. The paper further argues that shape coexistence—two nearly degenerate shapes in the same nucleus—can open alternative decay pathways and change branching ratios.","feed_headline":"Parent and daughter shapes together set proton-decay lifetimes","feed_subtitle":"A five-parameter fit with deformation terms for both nuclei beats existing formulas on 27 heavy proton emitters.","key_machinery":"The central object is Eq. (5), a five-parameter empirical formula for $\\log_{10}T_{1/2}$: $$\\log_{10}T_{1/2}=a+\\frac{b\\sqrt{\\mu}}{Z_d $A^{{1/3}}$}+\\frac{c\\sqrt{\\mu}\\,Z_d}{\\sqrt{Q}}+d\\sqrt{l(l+1)}+e\\left[(-1)^{\\kappa_p+\\kappa_p\\kappa_d}(\\kappa_p\\beta_p)^{1/2}+(-1)^{\\kappa_d+\\kappa_p\\kappa_d}(\\kappa_d\\beta_d)^{1/2}\\right]\\frac{Z_d}{\\sqrt{Q}}.$$ The first three terms are the usual decay-law structure, the fourth is the centrifugal-barrier hindrance, and the last term carries the new physics: it adds the deformation-weighted contributions of parent and daughter, with the $\\kappa$ convention ($2$ prolate, $-1$ oblate) and the sign factor enforcing constructive addition when both shapes match. The deformation values $\\beta_p,\\beta_d$ come from relativistic mean-field calculations, and the constants $a$--$e$ are determined by a least-squares fit with a maximum-likelihood loss that only penalizes predictions above the upper-limit data. This term is what lets the formula connect half-lives to the shapes of both partner nuclei.","core_discovery":"The paper's central claim is that Eq. (5), a semi-empirical relation with five fitted constants, captures one-proton decay lifetimes across a wide mass range ($21\\le A\\le 185$) by adding deformation terms for both the emitting and the residual nucleus to the standard $\\log_{10}T_{1/2}\\sim Z_d/\\sqrt{Q}$ systematics. For each nucleus the deformation enters as $(\\kappa\\beta)^{1/2} Z_d/\\sqrt{Q}$, where $\\beta$ is the quadrupole deformation and $\\kappa=2$ for prolate, $-1$ for oblate shapes; a sign factor $(-1)^{\\kappa_p+\\kappa_p\\kappa_d}$ makes same-shape parent-to-daughter transitions add constructively while damping shape-changing transitions. Fitted to 34 true experimental half-lives plus 18 upper limits by maximum likelihood, the formula yields RMSE $0.6998$ and $\\chi^2=0.6010$ on the 27 heavy emitters used for Table 1, the lowest of the compared formulas, and its predicted half-lives for new candidates fall in an experimentally accessible range ($\\log_{10}T_{1/2}\\approx -8.9$ to $-6.3$ s). The authors read the negative fitted coefficient $e$ as deformed nuclei decaying faster than spherical ones, and they use potential-energy surfaces to show that in shape-coexisting nuclei transitions from a second minimum can sometimes agree with measured half-lives better than the ground-to-ground transition.","pith_inferences":["A cleaner test of the physical content of the deformation term would be to refit Eq. (5) with the deformation coefficients set to zero and compare RMSE; the paper does not report this ablation, so part of the improvement could in principle come from the two extra fit parameters rather than from the shapes themselves.","Because the fitted deformation values come from one theoretical model, the extrapolated half-lives are conditional on that model's shapes; an independent check would be to compare the formula's predictions for candidates whose deformations have since been measured or constrained by other observables.","The same same-shape sign structure could be exported to other decay modes; if it improves $\\alpha$-decay or two-proton formulas as well, it would suggest a common empirical rule that tunneling lifetimes depend on whether parent and daughter shapes are similar."],"forward_implications":["New proton emitters can be screened quickly: the formula turns measured Q-values and computed deformations into half-life predictions, as done for the 24 candidates in Table 3, without full barrier-penetration calculations.","The comparison in Table 1 implies that the deformation terms, not just extra parameters, improve the fit; formulas without both-nucleus deformation terms have RMSEs of about $0.89$--$2.38$, so shape effects are not a negligible correction.","Because the formula predicts $\\log_{10}T_{1/2}$ between roughly $-8.9$ and $-6.3$ s for the candidates in Table 3, several of these nuclei should be detectable as proton emitters with existing techniques.","For shape-coexisting nuclei, the calculated ground-to-ground, ground-to-second-minimum, second-minimum-to-ground, and second-minimum-to-second-minimum transition half-lives differ; matching an observed lifetime to one of these variants can identify which shape state participates in the decay."],"supporting_citations":[{"why":"Supplies the 52 experimental half-lives, Q-values, spins, and parities (18 of them as upper limits) that define the dataset fitted by Eq. (5).","marker":"[69]"},{"why":"Supplies the evaluated Q-values used to predict half-lives for candidate proton emitters in Tables 3 and 4.","marker":"[88]"},{"why":"The earlier deformation-dependent formula for two-proton radioactivity that Eq. (5) extends to one-proton decay.","marker":"[52]"},{"why":"Provides the $(\\kappa\\beta)^{1/2}Z/\\sqrt{Q}$ deformation term and the convention $\\kappa=2$ (prolate), $\\kappa=-1$ (oblate) adopted in the last term.","marker":"[48]"},{"why":"Source of the relativistic mean-field deformation values used for $\\beta_p$ and $\\beta_d$ in the fit.","marker":"[65]"},{"why":"The companion calculation that supplies deformation values for the proton emitters and their daughters.","marker":"[70]"},{"why":"Provides the macroscopic-microscopic potential-energy surfaces used to identify second minima and shape coexistence in parent/daughter pairs.","marker":"[58]"},{"why":"An existing empirical one-proton formula used as a primary baseline in Table 1; the comparison defines the claimed RMSE improvement.","marker":"[44]"},{"why":"The closest other formula in Table 1, used to demonstrate the margin of the new formula's RMSE and $\\chi^2$.","marker":"[79]"}],"fun_headline_variants":["Five-parameter fit with both nuclear shapes beats proton decay models","Parent and daughter deformation drive proton decay lifetimes","Shape of both nuclei key to accurate proton-decay half-lives","New semi-empirical formula links proton decay to nuclear shapes","Proton decay prediction improves when both parent and daughter deform"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole prediction stands on the assumption that the shapes (quadrupole deformations) of the parent and daughter nuclei computed by relativistic mean-field theory are the true shapes; for most drip-line nuclei there is no experimental deformation to check against, so if the calculation is wrong for a candidate nucleus, the predicted half-life inherits that error in a way that the five fitted constants cannot reveal.","fun_headline_variants_meta":{"raw":{"variants":["Five-parameter fit with both nuclear shapes beats proton decay models","Parent and daughter deformation drive proton decay lifetimes","Shape of both nuclei key to accurate proton-decay half-lives","New semi-empirical formula links proton decay to nuclear shapes","Proton decay prediction improves when both parent and daughter deform"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000396,"raw_usage":{"total_tokens":2444,"prompt_tokens":1059,"completion_tokens":1385,"prompt_tokens_details":{"cached_tokens":1024},"prompt_cache_hit_tokens":1024,"prompt_cache_miss_tokens":35,"completion_tokens_details":{"reasoning_tokens":1303}},"tokens_in":35,"tokens_out":1385,"duration_ms":39023,"temperature":1.0,"reasoning_tokens":1303,"cache_read_input_tokens":1024,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:30:20.200392+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Refit Eq. (5) on the same 52 data points using deformation values from a second independent mass model instead of the relativistic mean-field values, and see whether the two-deformation formula still beats the other eight formulas in Table 1; if the RMSE advantage disappears, the claim that both-nucleus deformation is what drives the improvement fails. A complementary check is to measure the half-life (and ideally the shape) of a newly predicted emitter such as $^{83}$Tc, $^{47}$Co, or $^{71}$Rb, and compare with the formula's extrapolation.","supporting_citations":[{"cited_title":"& Audi, G","cited_arxiv_id":null,"evidence_quote":"Supplies the 52 experimental half-lives, Q-values, spins, and parities (18 of them as upper limits) that define the dataset fitted by Eq. (5)."},{"cited_title":"J., Kondev, F","cited_arxiv_id":null,"evidence_quote":"Supplies the evaluated Q-values used to predict half-lives for candidate proton emitters in Tables 3 and 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The earlier deformation-dependent formula for two-proton radioactivity that Eq. (5) extends to one-proton decay."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the $(\\kappa\\beta)^{1/2}Z/\\sqrt{Q}$ deformation term and the convention $\\kappa=2$ (prolate), $\\kappa=-1$ (oblate) adopted in the last term."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the relativistic mean-field deformation values used for $\\beta_p$ and $\\beta_d$ in the fit."},{"cited_title":"& Aggarwal, M","cited_arxiv_id":null,"evidence_quote":"The companion calculation that supplies deformation values for the proton emitters and their daughters."},{"cited_title":"Coexisting shapes with rapid transitions in odd-z rare-earth proton emitters","cited_arxiv_id":null,"evidence_quote":"Provides the macroscopic-microscopic potential-energy surfaces used to identify second minima and shape coexistence in parent/daughter pairs."},{"cited_title":"& Balasubramaniam, M","cited_arxiv_id":null,"evidence_quote":"An existing empirical one-proton formula used as a primary baseline in Table 1; the comparison defines the claimed RMSE improvement."},{"cited_title":"& Naderi, D","cited_arxiv_id":null,"evidence_quote":"The closest other formula in Table 1, used to demonstrate the margin of the new formula's RMSE and $\\chi^2$."}],"review_version":1}