{"id":"44d1666f-71d0-4573-b4b1-0799afda64c1","arxiv_id":"2607.18621","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A volume-plus-surface-gradient charge-symmetry-breaking functional calibrated to four mirror pairs predicts charge radii of 40Ti, 42Ti, 46Cr, and 50Fe and shows MDEs constrain mainly one effective CSB combination.","lead":"The paper fits a two-term nuclear charge-symmetry-breaking correction to four measured mirror pairs and predicts the charge radii of four proton-rich nuclei that have not yet been measured. The result matters because it quantifies a correction that must be controlled before mirror charge radii can be used as clean neutron-skin or symmetry-energy probes.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Radius validation and predictions rely on linear-response extrapolation to couplings 5–15× larger than the tested finite-difference steps; direct HFB at the fitted couplings should be run to check whether the ~0.005 fm radius predictions survive.","rationale":"The central claim is that a single two-parameter class-III CSB functional, calibrated to four MDE plus radius anchors, yields quantitative mirror-radius predictions. For that claim to hold, the linear response used to move from zero-coupling slopes to the fitted operating point must be accurate. The supplementary material is unusually explicit that the reported values are extrapolated from slopes and are not direct HFB results at the fitted couplings; that is exactly where I looked. The fit moves to couplings far outside the step sizes used to certify linearity, and the quoted nonlinearity check only bounds the even part of the MDE at the steps, not the radius at the final couplings. Because the radius is surface-sensitive and the two large couplings cancel in the MDE but combine in the radius, the radius response is the most plausible place for nonlinearity to appear. The proposed rerun is cheap relative to the calibration campaign and decisive: if direct results match, the concern is retired; if not, the predictions need revision. The reader's weakest_assumption (residual attribution) is a real, broader concern and is explicitly acknowledged in the paper; my check does not address it. I therefore only partially agree with the reader's weakest_assumption. Since the reader already returned CONDITIONAL and the new concern reinforces rather than overturns that judgment, I recommend UNCHANGED.","tokens_in":14151,"tokens_out":6326,"duration_ms":73137,"concrete_test":"Re-run the full self-consistent HFB calculations at the exact joint-fit couplings of Supp. Eqs. (S10)/(S11) for the four anchor and four target pairs, and compare the resulting ΔB and ΔR_ch^mirr with the linear-response values used in Tables 2 and 3. Require agreement within the adopted floors (0.10 MeV and 0.005 fm). As a minimal version, run at 50%, 100%, and 150% of (t0,CΔ) and verify that the MDE and radius responses are linear across this interval; if they are not, the quoted predictions and uncertainties must be recomputed nonlinearly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing assumption is the linear-response construction in Supp. Eqs. (S4)/(S14): the reported residuals and predictions are O_i(t0,CΔ) ≈ O_i^(C) + S0 t0 + SΔ CΔ, with slopes evaluated at t0=CΔ=0 and central finite differences h0=5.6 MeV fm3, hΔ=1.0 MeV fm5. The joint-fit couplings, however, are t0=-25.82 MeV fm3 (SLy4; 4.6×h0) and CΔ=-14.71 MeV fm5 (14.7×hΔ); SkM* is similar. The only quoted nonlinearity test — the even finite-step remainder, max ~0.026 MeV in the MDE at the steps — does not cover the radius response at the final couplings. In the MDE the two large terms largely cancel along t_eff, but in the radius channel they combine additively; a 10% nonlinear correction to the individual radius responses would move the predictions by more than 0.005 fm, i.e., at the level of the quoted parenthetical errors. Since Table 3 and the validation residuals in Table 2 are both linear-response extrapolations, the central claim that the calibrated CSB response predicts radii to ~0.005 fm depends on a linearity assumption that is currently untested at the operating point.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper calibrates a finite-density class-III charge-symmetry-breaking (CSB) functional, consisting of a volume term t0^III ρ0ρ1 and a surface-gradient term C_Δ^III (ρ0Δρ1 + ρ1Δρ0), against four measured mirror pairs (34Ar–34S, 36Ca–36S, 38Ca–38Ar, 54Ni–54Fe) using both mirror displacement energies (MDEs) and mirror charge-radius differences. Starting from Coulomb-only SLy4 and SkM* Skyrme EDF baselines, the authors compute local sensitivity coefficients and find that the MDEs constrain mainly the effective combination t_eff^III = t0^III − λ_cl C_Δ^III, with λ_cl ≈ 1.32 fm^{-2} (SLy4) and 1.20 fm^{-2} (SkM*). A joint mass–radius fit is then used to select a point on the MDE-compatible covariance band, and linear-response extrapolation yields predictions for the charge radii of 40Ti, 42Ti, 46Cr, and 50Fe, with claimed uncertainties of ~0.005 fm plus a two-EDF spread up to 0.021 fm. The paper argues that finite-density CSB corrections must be included before mirror charge-radius differences are used as clean neutron-skin or symmetry-energy probes.","tokens_in":14560,"tokens_out":2428,"duration_ms":27709,"significance":"If the central extraction is correct, the paper provides a useful framework for connecting MDE data to charge-radius predictions: it explicitly separates the energy-selected CSB direction from the radius-selected point on that direction, and it makes four concrete, falsifiable predictions for proton-rich charge radii. The identification of the effective combination and the compact λ_cl band across two EDFs is a genuinely useful diagnostic. The paper also honestly flags that the extracted couplings are reference-dependent effective parameters and that the absolute radius correction is EDF dependent. However, the strength of the quantitative claims — especially the ~0.005 fm precision of the predicted radii — currently rests on an untested linear-response extrapolation and on an in-sample 'validation' of the radius residuals, so the significance is conditional on resolving those issues.","major_comments":[{"comment":"The predicted radii and the Table 2 validation residuals are linear-response extrapolations O_i(t0,CΔ) ≈ O_i^(C) + S0 t0 + SΔ CΔ, where the slopes are evaluated at t0=CΔ=0 with finite-difference steps h0=5.6 MeV fm3 and hΔ=1.0 MeV fm5. The joint-fit couplings are t0 ≈ −25.8 MeV fm3 (SLy4) and CΔ ≈ −14.7 MeV fm5, i.e., about 4.6×h0 and 14.7×hΔ. The quoted nonlinearity test (even finite-step remainder ≤0.026 MeV in the MDE) is performed at the small steps, not at the operating point, and does not cover the radius channel where the two terms add rather than cancel. A 10% nonlinearity in the individual radius responses would shift the predictions by more than the quoted 0.005 fm floor. A direct HFB calculation at the fitted couplings is needed to verify the reported radii and residuals.","section":"Supplementary Eqs. (S2)–(S4), (S14); Table 3"},{"comment":"The joint calibration in Eq. (16) uses η_R=1, so the measured anchor charge radii enter the χ² directly. The subsequent presentation of Table 2 as 'pair-by-pair validation' is therefore in-sample: the residuals shown are reduced not because the CSB functional was independently confirmed by radii, but because those same radii were used to select the couplings. This is a legitimate calibration procedure, but it should be called a reproduction, not a validation. The genuinely out-of-sample statements are the target predictions in Table 3, and these rest on the linear-response issue noted above.","section":"Eq. (16) and Table 2"},{"comment":"The SkM* joint fit gives RMS_R = 0.0136 fm and χ²/dof = 3.9, with the radius residual barely improved from the MDE-only fit (0.0174 fm → 0.0136 fm) while the MDE RMS becomes worse than the SLy4 joint fit. This indicates that the same two-parameter functional cannot simultaneously reproduce the MDE and radius residuals in SkM* at the adopted error floors. The paper's claim of a 'common finite-density CSB response' is thus only partially supported; the SkM* result is more naturally read as a model tension. The authors should either lower the claim to 'SLy4 supports the common response; SkM* does not', or quantify why the SkM* tension does not undermine the target predictions.","section":"Table 1, SkM* row"},{"comment":"The entire calibration treats the residual R_B,i = ΔB_exp − ΔB^(C) as if it were dominated by the two adopted class-III CSB terms. The paper itself states that this residual 'may absorb omitted electromagnetic, surface, shell, pairing, and deformation effects.' Only a small pp contact term is varied as a check, and the check does not cover deformation, pairing, or beyond-mean-field effects. Given that the residuals are at the 0.1–2 MeV level and the target radii are claimed to 0.005 fm, the authors should provide a more explicit estimate of how large such omitted effects would have to be to change the target predictions, or restrict the conclusions to the level allowed by this ambiguity.","section":"Sec. 1, Eq. (5)"}],"minor_comments":[{"comment":"The fit labels 'mass' in Table 1 are confusing; these are MDE-only fits, not fits to nuclear masses. Please rename to 'MDE' or 'MDE-only'.","section":"Table 1 caption"},{"comment":"Table 2 refers to 'More detailed numbers are illustrated in Fig. 2', but the figure is numbered Fig. 2 in the main text while the reference in Sec. 2.3 says 'illustrated in Fig. 1'. Please harmonize the cross-references.","section":"Fig. 2 caption / Table 2 caption"},{"comment":"The symbol Rp is used for the point-proton rms radius, which is standard, but the text should explicitly distinguish Rp from R_p^ch to avoid confusion in Eq. (19) where R_p^ch denotes the predicted charge radius.","section":"Eq. (3)"},{"comment":"The parenthetical errors in Table 3 explicitly exclude coupling-covariance and λ_B-band assignment uncertainties. This should be stated directly in the main text (it is only in the Supplementary Material) so that readers do not mistake the parentheses for total theoretical uncertainties.","section":"Table 3 and Sec. 3"},{"comment":"The data availability statement says data are available 'upon reasonable request'. Given that the paper is built on a specific numerical procedure, consider providing the fitted couplings, slopes, and HFBTHO inputs in a small data file for full reproducibility.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely and important question, and the effective-combination analysis is a useful contribution even if the absolute radius predictions remain model dependent. The main risk is that the quantitative radius predictions are presented with parenthetical uncertainties of ~0.005 fm that do not yet account for the untested linear-response extrapolation at the operating point or for the in-sample nature of the 'validation'. I believe the issues are fixable with a direct HFB recalculation at the fitted couplings and a more careful framing of what is validated versus what is calibrated, hence major_revision rather than reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, what you should know: this is a serious EDF calibration paper with a clean diagnostic (the λB band and the effective combination t_eff) and four genuinely testable radius predictions. But the two main claims—that the radius residuals validate the CSB spatial content, and that the predictions are good to ~0.005 fm—need to be read with caution. The radius validation is in-sample, and the predictions are linear-response extrapolations to couplings 5–15 times larger than the tested steps.\n\nWhat the paper does well: it frames the problem correctly—integrated MDEs cannot fix the spatial form of the CSB field, so radius data are needed to pick the direction. The compact λB band among the four anchors in both SLy4 and SkM* is a real observation. The authors are also unusually candid: they say the residual may absorb Coulomb, shell, pairing, deformation effects; they only claim t_eff is robust; and they report the poor SkM* radius fit openly. The supplement includes force parameters and numerical settings, which is more than many EDF papers do.\n\nNow the soft spots. First, the linear-response concern is not a nitpick. The slopes are evaluated at t0=CΔ=0 with steps h0=5.6, hΔ=1.0, then applied to couplings -25.8 and -14.7. In the MDE channel the two terms cancel along t_eff, but in the radius channel they add. The even-remainder test at the small steps says nothing about the radius response at the operating point. Direct HFB at the fitted couplings is the obvious check, and the paper doesn't do it. Until that's done, the sub-0.01 fm predictions are not yet demonstrated.\n\nSecond, calling Table 2 a 'validation' is misleading. The same four anchor radii are used in the joint fit to select t⊥; of course the residuals shrink. A true validation would be a leave-one-out test or a prediction for a pair not in the calibration set (they mention 30S and 32Ar only as λB checks, not radius predictions). The authors don't hide this—the equations show η_R=1—but the framing oversells.\n\nThird, the quoted errors on the predicted radii are not the total uncertainty. They only include the experimental radius of the known partner and the 0.005 fm floor. The coupling uncertainties (t0 ±7.9, CΔ ±6.0) and the λB assignment uncertainty are not propagated. The interesting number is the two-EDF spread, up to 0.021 fm, which appears in a separate column but not in the parenthetical errors. For 46Cr, the spread alone is four times the quoted 0.0055 fm.\n\nWho should read this: anybody working on mirror radii, neutron skins, or ISB in EDFs. The method is a useful template even if the current implementation is incomplete. I would send it to peer review, but I'd ask for direct HFB runs at the fitted couplings, propagate errors properly, and demote the 'validation' language. With those changes it could be a solid contribution; as is, the predictions are interesting but not yet at the claimed precision.","headline":"The paper identifies a useful effective CSB combination and makes testable radius predictions, but the predictions rest on an untested linear-response extrapolation and the 'validation' is in-sample; worth a serious referee with revisions.","tokens_in":15005,"tokens_out":3723,"would_cite":true,"duration_ms":43468,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Mirror displacement energies from four measured mirror pairs constrain mainly one effective charge-symmetry-breaking combination, and the same calibrated finite-density functional predicts the charge radii of 40Ti, 42Ti, 46Cr, and 50Fe—so m","keywords":["mirror nuclei","charge radii","charge-symmetry breaking","mirror displacement energy","neutron skin","energy-density functional","isospin symmetry breaking","symmetry energy"],"falsifier":"Measure the charge radius of 42Ti (or 50Fe) with an uncertainty below roughly 0.006 fm and compare with the predicted 3.5780(54) fm (or 3.7110(82) fm); a deviation beyond the combined experimental error and the two-functional spread would falsify the calibrated CSB correction. On the energy side, an ab initio calculation of the four anchor-pair MDE residuals that reproduces the residual without any surface-gradient term would falsify the claim that a common surface-gradient-dominated class-III response is required.","tokens_in":14037,"feed_emoji":"⚛️","tokens_out":5749,"duration_ms":59794,"temperature":0.7,"pith_summary":"The paper argues that the binding-energy differences between mirror nuclei (mirror displacement energies, MDEs) and their charge-radius differences are governed by the same finite-density charge-symmetry-breaking (CSB) effect, described by a volume term and a surface-gradient term. The measured MDEs pin down almost entirely one effective combination of the two coupling strengths, not each coupling separately. Adding measured charge radii of the same four calibration pairs selects the point on that weakly constrained direction, and the resulting functional predicts charge radii for four unmeasured proton-rich nuclei. If the predictions hold, mirror charge-radius differences must be corrected for finite-density CSB before being interpreted as neutron-skin or symmetry-energy probes. The work matters because it supplies concrete, testable radii and a method for separating isospin-symmetry-breaking effects from the physics of neutron skins.","feed_headline":"One nuclear force asymmetry, set by 4 mirror pairs, predicts 4 radii","feed_subtitle":"A calibrated charge-symmetry-breaking correction now predicts 40Ti, 42Ti, 46Cr, and 50Fe radii—and must be subtracted before neutron-skin pr","key_machinery":"The central objects are the volume term (1/2)t0^III ρ0ρ1, which probes the bulk-weighted neutron–proton density imbalance, and the surface-gradient term C_Δ^III(ρ0Δρ1+ρ1Δρ0), which after integration by parts acts as -2C_Δ^III[(∇ρ_n)^2-(∇ρ_p)^2] and shifts the radial region where CSB acts. Their MDE responses are captured by self-consistent local sensitivity coefficients S_0^B and S_Δ^B, whose ratio λ_B=-S_Δ^B/S_0^B is nearly constant across the four mirror pairs. This near-degeneracy means the MDE data constrain mostly the combination t_eff^III=t0^III-λ_cl C_Δ^III, while the complementary direction t_⊥^III is weakly constrained and is fixed by the measured mirror charge radii in the joint fi","core_discovery":"The central claim is that a class-III charge-symmetry-breaking functional—volume term (1/2)t0^III ρ0ρ1 plus surface-gradient term C_Δ^III (ρ0Δρ1+ρ1Δρ0)—added to a Coulomb-only Skyrme energy-density functional simultaneously describes the residual MDEs and mirror charge-radius differences of the four pairs 34Ar–34S, 36Ca–36S, 38Ca–38Ar, and 54Ni–54Fe. The response is surface-gradient-dominated: the sensitivity ratio λ_B = -S_Δ^B/S_0^B is nearly the same for all four anchors (1.32±0.04 fm^-2 with SLy4, 1.20±0.05 fm^-2 with SkM*), so the MDEs determine chiefly the effective combination t_eff^III = t0^III - λ_cl C_Δ^III ≈ -6.4 MeV fm^3, rather than t0^III and C_Δ^III separately. Measured anchor","pith_inferences":["A precise measurement of R_ch(42Ti) or R_ch(50Fe), with uncertainty comparable to the quoted parenthetical errors, would directly test whether the calibrated CSB functional captures the real spatial content of the MDE residual.","The λ_B-ratio method could be applied to other isospin-breaking observables, such as triplet displacement energies, to identify which combinations of class-II and class-III couplings are actually constrained before interpreting individual coupling strengths.","Extending the same volume-plus-surface-gradient CSB calibration to a relativistic mean-field framework with ω-ρ meson mixing would test whether the surface-gradient-dominated pattern survives a different many-body expansion.","If the predicted radii are confirmed, the same calibration can be reused to assign CSB corrections to other proton-rich mirror pairs, effectively converting mirror-radius measurements into probes of isospin-symmetry breaking rather than direct neutron-skin measurements."],"forward_implications":["Mirror displacement energies do not separately determine the volume and surface-gradient CSB couplings; only the combination t_eff^III is pinned down by the energy data alone.","If the calibration is right, 42Ti and 50Fe are the cleanest near-term tests, with SLy4–SkM* spreads of about 0.002 fm in the predicted radii.","46Cr is the strongest test of model dependence: its predicted radius differs by 0.0212 fm between the two functionals.","Calibrated CSB corrections shift proton-rich mirror neutron skins at the 10^-2 fm level, so mirror charge-radius differences used as neutron-skin or symmetry-energy probes require the same CSB subtraction.","The surface-gradient-dominated response pattern is more robust across the two functionals than the absolute radius correction, which remains energy-density-functional dependent."],"fun_headline_variants":["One force asymmetry explains mirror energies and charge radii","A single symmetry-breaking force links mirror energies and radii","Mirror charge radii reveal a common symmetry-breaking response","One CSB term links mirror energies and radii","One asymmetry predicts radii of 40Ti, 42Ti, 46Cr, 50Fe"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The calibration assumes that the experiment–theory difference left after the adopted Coulomb-only calculation is entirely due to the two-term class-III CSB functional; if omitted electromagnetic, surface, shell, pairing, or deformation effects contribute at the 0.1 MeV or 0.005 fm level, the extracted couplings and predicted radii are biased.","fun_headline_variants_meta":{"raw":{"variants":["One force asymmetry explains mirror energies and charge radii","A single symmetry-breaking force links mirror energies and radii","Mirror charge radii reveal a common symmetry-breaking response","One CSB term links mirror energies and radii","One asymmetry predicts radii of 40Ti, 42Ti, 46Cr, 50Fe"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001073,"raw_usage":{"total_tokens":4497,"prompt_tokens":1080,"completion_tokens":3417,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":824,"completion_tokens_details":{"reasoning_tokens":3333}},"tokens_in":824,"tokens_out":3417,"duration_ms":26218,"temperature":1.0,"reasoning_tokens":3333,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T01:51:15.244432+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the charge radius of 42Ti (or 50Fe) with an uncertainty below roughly 0.006 fm and compare with the predicted 3.5780(54) fm (or 3.7110(82) fm); a deviation beyond the combined experimental error and the two-functional spread would falsify the calibrated CSB correction. On the energy side, an ab initio calculation of the four anchor-pair MDE residuals that reproduces the residual without any surface-gradient term would falsify the claim that a common surface-gradient-dominated class-III response is required.","supporting_citations":[],"review_version":2}