{"id":"7dafb2b5-f60b-470f-aa3a-c025f522e63b","arxiv_id":"2412.01364","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Numerical tables of the relative para- and diamagnetic contributions to magnetic shielding constants are presented for low-lying states of Dirac hydrogenlike ions up to Z=137.","lead":"This paper calculates and tabulates the diamagnetic and paramagnetic parts of the magnetic shielding constant for relativistic one-electron atoms with nuclear charge Z from 1 to 137. The tables are intended as reference data for atomic theory and NMR-style calculations in the relativistic regime.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Table 8's rows for Z=70–137 violate Eq. (1.1): σd/σ + σp/σ ≠ 1, with σp/σ apparently copied from Table 4; 68 central data entries are invalid.","rationale":"The strongest claim is that Tables 1–14 contain accurate values of σd/σ and σp/σ. For that claim to hold, every row must satisfy σd/σ + σp/σ = 1. Table 8 fails this for 68 consecutive rows, and the σp/σ column in the failing range is identical to the 2p3/2, μ=±1/2 column of Table 4. This is the most load-bearing concern because it directly invalidates a substantial part of the data product without requiring any assumption about the correctness of the imported formulas from Ref. [8]. The reader's verdict was CONDITIONAL partly on this same observation; my stress test confirms it and makes it the primary objection. I do not see another equally concrete defect: other tables pass spot checks, and the external comparisons in Tables 15–16 are favorable. The fix is algorithmic, not conceptual: recompute the affected rows from Eqs. (1.2)–(1.4) and check normalization. Because the error is localized and detectable, UNCHANGED (i.e., still CONDITIONAL) is the appropriate verdict rather than REJECT. If the author supplies a corrected Table 8 and all rows sum to 1, the primary objection would be resolved.","tokens_in":54732,"tokens_out":18359,"duration_ms":156300,"concrete_test":"Recompute rows Z=70–137 of Table 8 by evaluating Eqs. (1.2)–(1.4) for n=1, κ=-2, μ=±1/2 with α^{-1}=137.035999177, and require σd/σ + σp/σ = 1. A minimal check is to sum the two printed columns: the Z=70 row already fails, so all 68 rows must be regenerated and checked against the 3p3/2 values rather than the 2p3/2 values of Table 4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central deliverable is the set of relative contributions in Tables 1–14, and Eq. (1.1) requires σd + σp = σ, so every row must satisfy σd/σ + σp/σ = 1. Table 8 (3p3/2, μ=±1/2) violates this for all rows Z=70–137. For example, at Z=70 the table gives σd/σ = 1.056872965322 and σp/σ = 1.138700966968; their sum is 2.195573932290, not 1. The σp/σ entries in this range reproduce the 2p3/2, μ=±1/2 values of Table 4 (e.g., Z=70: 1.138700966968), indicating a copy/paste error rather than a physics result. The same failure occurs at Z=137: 1.300435816068 + 2.243288648100 = 3.543724464168. Evaluating Eqs. (1.2)–(1.4) for 3p3/2 with n=1, κ=-2, μ=±1/2 gives σd/σ negative and σp/σ slightly above 1 in this Z region, matching the Z≤69 rows rather than the printed Z≥70 rows. Because the abstract claims tabulated data for the full range 1≤Z≤137, this is not a cosmetic typo: 68 rows of the main data table are unreliable until recomputed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper tabulates relative diamagnetic and paramagnetic contributions to the magnetic shielding constant of relativistic hydrogenlike atoms. The author evaluates closed-form expressions from her earlier work (Eqs. (1.2)-(1.6)) for the ground state and the first two excited-state manifolds, with Z from 1 to 137 and alpha^-1 = 137.035999177, and presents the ratios sigma_d/sigma and sigma_p/sigma in Tables 1-14. It also compares absolute sigma_d and sigma_p values with Pyper and Zhang (Tables 15-16) and lists total sigma for selected ions under three CODATA values of alpha (Table 17).","tokens_in":55030,"tokens_out":6725,"duration_ms":60300,"significance":"The paper fills a gap by providing a systematic tabulation of the separation of sigma into para- and diamagnetic parts for many states and Z values, using parameter-free analytical formulas and agreeing with published values where comparisons exist. The formulas are stated explicitly, and no parameters are fitted; the comparisons in Tables 15-16 provide independent numerical checks for a subset of cases. However, because the contribution of the paper is exclusively numerical data, internal consistency of every table is essential. The error in Table 8 described below undermines a substantial block of the data until corrected.","major_comments":[{"comment":"These rows violate the identity sigma_d + sigma_p = sigma stated in Eq. (1.1). For example, at Z = 70 the entries sigma_d/sigma = 1.056872965322 and sigma_p/sigma = 1.138700966968 sum to 2.195573932290, not 1; at Z = 137 they sum to 3.543724464168. Every valid row must satisfy sigma_d/sigma + sigma_p/sigma = 1, and all other tables in the paper do. The sigma_p values in this block coincide with the 2p3/2 (mu = +/- 1/2) values in Table 4 (e.g., 1.138700966968 at Z = 70), rather than with a 3p3/2 evaluation. Since the abstract claims Tables 1-14 cover 1 <= Z <= 137, the 68 affected rows must be recomputed from Eqs. (1.2)-(1.4); this is a load-bearing data error, not a typographical issue.","section":"Table 8 (3p3/2, mu = +/- 1/2), rows Z = 70-137"}],"minor_comments":[{"comment":"The text contains several typographical errors: \"diamagentic\" in the abstract, \"analitycal\" in the Introduction, and \"theese\" in the Introduction; these should be corrected.","section":"Abstract and Introduction"},{"comment":"Several numerical entries contain stray spacing in the printed mantissas (for example, \"1.13870096696 8(0)\" in Table 8 and \"2.010634177 934(0)\" in Table 4); the final typeset version should ensure each number is printed as one continuous token.","section":"Tables 4, 8, 13, 14"},{"comment":"The author states that the results were obtained with the formulas from Eqs. (1.1)-(1.6), but the paper does not identify which numerical precision or rounding convention was used in the tables; a brief statement of the estimated numerical accuracy would help readers judge the reliability of the last digits.","section":"Section 2"},{"comment":"References [18] and [20] are given as web addresses without full bibliographic details; they should be completed with the standard citation information for the CODATA reports.","section":"References [18] and [20]"}],"recommendation":"major_revision","confidential_remarks":"The Table 8 error appears to be a block transcription/copying error rather than a failure of the underlying formulas, since the neighboring tables and the comparisons with Pyper and Zhang are internally consistent. A major revision requesting recomputation and an explicit sum check is appropriate. I would also suggest asking the author to deposit machine-readable tables or the generating code, since the paper's value is its data and one undetected block error slipped through."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a table dump: it evaluates the author's earlier closed-form formulas (Eqs. 1.2-1.6) for sigma_d/sigma and sigma_p/sigma across Z for eight low-lying states. The external comparisons in Tables 15-16 agree with Pyper and Zhang, so the underlying formulas are probably right. But Table 8 is not right. For the 3p3/2 (mu=+/-1/2) state, all rows from Z=70 to 137 violate the defining identity sigma_d + sigma_p = sigma. At Z=70 the two columns sum to about 2.196, not 1, and the sigma_p/sigma entries are word-for-word identical to Table 4's 2p3/2 (mu=+/-1/2) column. That is a copy-paste error, not a physics effect. The stress-test check via Eqs. (1.2)-(1.4) for n=1, kappa=-2 gives negative sigma_d in that region, matching the Z<=69 rows. So 68 central data rows are invalid.\n\nWhat the paper does well: it gives a systematic, wide-Z compilation of relative dia- and paramagnetic contributions for states that have not been tabulated this way before, it identifies crossing points Zc where the relative weights change, and it includes a careful CODATA 2014/2018/2024 comparison for total sigma. The text is honest that the formulas come from Ref. [8] and reduce to known results for low-lying states. The comparisons with Pyper and Zhang are genuinely reassuring for the states they cover.\n\nThe soft spots are localized but real. One corrupted table is a big deal in a data-only paper because the tables are the entire contribution; there is no independent derivation or code release to check against. The paper would also be stronger with machine-readable data, especially since the print tables are the only artifact. On novelty, this is a parameter scan of existing analytical results, not new physics. If the error were absent, I would call it a useful reference but not a conceptual advance.\n\nWho benefits: anyone needing quick relativistic shielding ratios for hydrogenic ions and willing to spot-check the numbers. As is, I would not cite Table 8, and I would warn colleagues to recalculate that case. If the author fixes the copy-paste and reissues, this becomes a decent data note.\n\nRecommendation: yes, send it to peer review, but with a clear mandate for major revision. The error is fixable and the rest of the data appears consistent, so a referee can push for corrected Table 8 rows and a note on why they changed. I would not desk-reject, but I would not accept it in this state.","headline":"Table 8's 3p3/2 (mu=+/-1/2) columns are corrupted for Z=70-137: the sum sigma_d+sigma_p exceeds 1, the sigma_p column is copied from Table 4, and 68 rows must be recomputed before the paper's central data can be trusted.","tokens_in":747,"tokens_out":1871,"would_cite":false,"duration_ms":35503,"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":"This paper presents tabulated values of the relative paramagnetic and diamagnetic contributions to the magnetic shielding constant of relativistic hydrogenlike atoms for all nuclear charges Z from 1 to 137.","keywords":["magnetic shielding constant","Dirac one-electron atom","paramagnetic contribution","diamagnetic contribution","Gordon decomposition","hydrogenlike ions","relativistic atomic data","fine-structure constant"],"falsifier":"Recompute the ratios by an independent method, such as solving the Dirac equation with a magnetic perturbation numerically for a few representative cases (for example $1s_{1/2}$ with $Z = 80$ and $2p_{1/2}$ with $Z = 118$), and compare against the tabulated values at the stated precision. Any disagreement beyond rounding would falsify the formulas or the evaluation.","tokens_in":54475,"feed_emoji":"⚛️","tokens_out":6368,"duration_ms":51699,"temperature":0.7,"pith_summary":"This paper provides the numerical values of the relative diamagnetic and paramagnetic contributions to the magnetic shielding constant of a Dirac one-electron atom, for the ground state and the first two sets of excited states, across the full range of nuclear charge $Z = 1$ to $137$. The values are direct evaluations of closed-form formulas previously derived by the author using the Gordon decomposition of the Dirac current. If the formulas are sound, the tables constitute a compact reference dataset for relativistic hydrogenlike ions, including comparison points where the paramagnetic part overtakes the diamagnetic part (or becomes the entire shielding).","feed_headline":"Data splits magnetic shielding for hydrogenlike ions, Z=1..137","feed_subtitle":"Closed-form Dirac results give first compact table set for nine low-lying states.","key_machinery":"The central object is the Gordon decomposition of the Dirac current, which separates the magnetic interaction into a convection (diamagnetic-like) and a spin (paramagnetic-like) piece, giving $\\sigma = \\sigma_d + \\sigma_p$. The paper works with the explicit closed forms (1.2)–(1.6), expressed through the radial quantum number $n$, the Dirac quantum number $\\kappa$, the magnetic quantum number $\\mu$, and $\\gamma_\\kappa = \\sqrt{\\kappa^2 - (\\alpha Z)^2}$. These formulas, imported from the author's earlier derivation that used a Sturmian expansion of the Dirac–Coulomb Green function, are what the tables evaluate.","core_discovery":"The paper's central claim is that for any discrete energy eigenstate of a relativistic hydrogenlike atom, the magnetic shielding constant splits into a diamagnetic part $\\sigma_d$ and a paramagnetic part $\\sigma_p$ whose relative sizes are fixed by the three quantum numbers $(n, \\kappa, \\mu)$ through the closed expressions in Eqs. (1.2)–(1.6). The tabulated ratios $\\sigma_d/\\sigma$ and $\\sigma_p/\\sigma$ are the numerical content of those expressions. The data exhibit regular trends: for states with $\\kappa < 0$ at maximal $\\mu$, $\\sigma_d$ dominates; for $\\kappa > 0$, $\\sigma_p$ dominates; for $s$-states there is a crossover $Z_c$ (67 for $1s$, 79 for $2s$, 86 for $3s$) after which the paramagnetic term is larger. For $p_{1/2}$ states, the paramagnetic share has a minimum at a high $Z_c$ (99 for $2p_{1/2}$, 105 for $3p_{1/2}$). The same formulas are used to compile total shielding constants for three CODATA values of the fine-structure constant.","pith_inferences":["The one-electron results are the natural zero-order reference for many-electron heavy atoms, so these ratios could be used to isolate electron-correlation and finite-nuclear-size effects on shielding.","The observed $Z_c$ pattern across $n$ suggests a nearly linear relation that an empirical formula could capture; one could test $Z_c(n)$ against a larger set of $n$ values.","For states with $|\\kappa|=1$, the restriction $Z < \\alpha^{-1}\\sqrt{3/2} \\approx 118.67$ means the tables stop before the Dirac critical charge; extending to the supercritical regime would require a different boundary condition.","The sharp alpha-dependence seen near high $Z$ (e.g., $1s$ at $Z=118$) implies these ratios could in principle be used to constrain $\\alpha$ if measured in highly charged ions."],"forward_implications":["The tables give benchmarks for approximate relativistic quantum-chemical calculations of magnetic shielding in heavy ions.","The crossover charges $Z_c$ provide a simple diagnostic for when paramagnetic shielding overtakes diamagnetic shielding in $s$-states.","Table 17 quantifies how sensitive the total shielding constant is to the fine-structure constant across CODATA 2014, 2018, and 2024 values.","Because the formulas hold for arbitrary discrete states, the same evaluation can be extended to higher excited states and to arbitrary $\\mu$ values without further derivation."],"supporting_citations":[{"why":"Supplies the closed-form expressions for $\\sigma_d$ and $\\sigma_p$ that every table entry is evaluated from.","marker":"[8]"},{"why":"Provides the Sturmian expansion of the Dirac–Coulomb Green function used in the derivation of the formulas.","marker":"[1]"},{"why":"Introduces the Gordon decomposition technique that separates the magnetic response into diamagnetic and paramagnetic parts.","marker":"[15]"},{"why":"Gives the earlier numerical results used for comparison of $\\sigma_d$ and $\\sigma_p$ for selected ions.","marker":"[17]"},{"why":"Source of the CODATA 2024 fine-structure constant value used for the main tables.","marker":"[18]"}],"fun_headline_variants":["Magnetic shielding split for hydrogenlike atoms up to Z=137","Para vs diamagnetic shielding in relativistic H-like ions","Tabulated shielding ratios for hydrogenlike ions, Z=1–137","Closed-form ratios for magnetic shielding in hydrogenic atoms","Shielding constants: new tables for relativistic one-electron atoms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The correctness of every table rests on the imported closed-form expressions for $\\sigma_d$ and $\\sigma_p$ being valid for all discrete states and for all $Z$ up to 137; the paper does not re-derive them here.","fun_headline_variants_meta":{"raw":{"variants":["Magnetic shielding split for hydrogenlike atoms up to Z=137","Para vs diamagnetic shielding in relativistic H-like ions","Tabulated shielding ratios for hydrogenlike ions, Z=1–137","Closed-form ratios for magnetic shielding in hydrogenic atoms","Shielding constants: new tables for relativistic one-electron atoms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000758,"raw_usage":{"total_tokens":3475,"prompt_tokens":1159,"completion_tokens":2316,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":775,"completion_tokens_details":{"reasoning_tokens":2232}},"tokens_in":775,"tokens_out":2316,"duration_ms":15955,"temperature":1.0,"reasoning_tokens":2232,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:25:25.373086+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the ratios by an independent method, such as solving the Dirac equation with a magnetic perturbation numerically for a few representative cases (for example $1s_{1/2}$ with $Z = 80$ and $2p_{1/2}$ with $Z = 118$), and compare against the tabulated values at the stated precision. Any disagreement beyond rounding would falsify the formulas or the evaluation.","supporting_citations":[{"cited_title":"Stefa´ nska, Phys","cited_arxiv_id":null,"evidence_quote":"Supplies the closed-form expressions for $\\sigma_d$ and $\\sigma_p$ that every table entry is evaluated from."},{"cited_title":"Szmytkowski, J","cited_arxiv_id":null,"evidence_quote":"Provides the Sturmian expansion of the Dirac–Coulomb Green function used in the derivation of the formulas."},{"cited_title":"Gordon, Z","cited_arxiv_id":null,"evidence_quote":"Introduces the Gordon decomposition technique that separates the magnetic response into diamagnetic and paramagnetic parts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the earlier numerical results used for comparison of $\\sigma_d$ and $\\sigma_p$ for selected ions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the CODATA 2024 fine-structure constant value used for the main tables."}],"review_version":1}