A preprint reports lower prediction errors for two global models of nuclear charge radii after applying ARCUS, a correction built from earlier model errors, with uncertainty estimates that remained broadly aligned with later measurements. In the temporally blind test, the root-mean-square error, an overall measure of prediction misses, fell from 17.6 to 13.1 millifemtometres for WS* and from 26.0 to 18.6 millifemtometres for HFB-25.
The study asks whether residuals, or model errors, from global charge-radius models contain usable local structure for extrapolating to nearby unmeasured nuclei while retaining reliable uncertainty. The document is an arXiv version 1 preprint dated 26 August 2026.
The errors carried a local pattern
The analysis used residuals for 885 nuclei with proton number Z of at least 8. Those errors changed much more smoothly along isotopic chains than across a change in proton number.
At a one-step separation, WS* semivariance, a measure of how much neighbouring errors differ, was 70 square millifemtometres along chains and 408 square millifemtometres across chains, compared with total residual variance of 476 square millifemtometres. Those figures represented 15% and 86% of the total variance. HFB-25 showed the same directional pattern in normalized terms, with 19% along chains and 75% across them.
The two models' residuals were also substantially correlated on common nuclei. The Pearson correlation was 0.73, while element-by-element offsets alone produced 0.77. A common-component model assigned about 400 square millifemtometres of shared variance, equal to 85% of WS* variance and 62% of HFB-25 variance. The study cannot determine whether that shared pattern reflects missing nuclear physics, element-dependent experimental structure, or both.
A test against later measurements
ARCUS first removes parity-class and per-element trends from the residuals. It then uses nearby measured residuals in a directional kernel correction and calibrates neighbour-scatter uncertainty separately for parity classes through isotonic regression.
The evaluation combined five-fold cross-validation on the 885-nucleus training set with a fixed temporally blind test of 129 nuclei measured later. The blind set was fixed before modelling choices were made. In the cross-validation results, RMSE fell from 21.8 to 10.3 millifemtometres for WS* and from 25.4 to 13.2 millifemtometres for HFB-25.
On the blind test, intervals labelled 68% contained 69.8% of WS* cases and 63.6% of HFB-25 cases. With 129 nuclei, the reported binomial standard error was plus or minus 4.1 percentage points. Across 45 reruns, median 68% coverage was 73.6% for WS* and 66.7% for HFB-25, while median 95% coverage was 95.3% for both.
The directional comparison
The study also compared ARCUS with an isotropic kernel that treats the neighbourhood as directionally uniform. The isotropic version was recalibrated to the same out-of-fold coverage target. In the blind test, it had an RMSE of 14.2 millifemtometres, compared with 13.1 for ARCUS. Its median 68% uncertainty was 23.2 millifemtometres, versus 12.0 for ARCUS, and its intervals were 1.9 times wider. Its blind-test coverage was 86.0% at the 68% level and 99.2% at the 95% level.
The improvement was not uniform across the nuclear chart. In localized out-of-fold summaries, WS* improvement factors were about 5.4 to 5.6 in deformed rare-earth and trans-Pb regions. In the calcium-to-nickel region, WS* was the only case showing deterioration, with a factor of 0.94, while HFB-25 improved by 1.31. These summaries do not establish uniform performance across the chart.
A useful pattern with a clear limit
One blind-test case showed the limit of local correction. For calcium-52, the ARCUS error was 35.7 millifemtometres for WS*. HFB-25's uncorrected prediction was within about 2 millifemtometres, but its ARCUS-corrected error was 18.4. The standardized residuals were 1.9 and 1.5, respectively, and both predictions remained inside calibrated 95% intervals. The authors attribute the miss to the absence of a corresponding local signal in the training residuals.
Other examples were more favourable but narrower in scope. Along strontium, WS* RMSE fell from 24.1 to 10.4 millifemtometres and HFB-25 RMSE from 28.5 to 9.9 millifemtometres. In the blind cadmium example, the respective RMSEs were 2.6 and 9.6 millifemtometres, and all predictions were inside calibrated 95% intervals. These are chain-specific examples and do not guarantee similar performance where local residual support is absent.
A catalogue with uneven confidence
Using the same approach, the researchers generated an uncertainty-ranked catalogue of 1,008 nearby nuclei not measured in the listed compilations. Selection required proton and neutron numbers of at least 8 and a six-neutron locality window. Of the candidates, 648, or 64%, had a reported 95% uncertainty below 50 millifemtometres. The paper warns that point predictions are not equally reliable and that intervals widen when extrapolation support is weaker.
The reported gains persisted in a chain-block bootstrap that preserved along-chain correlation. The lower bound for out-of-fold improvement was 1.64-fold for WS* and 1.53-fold for HFB-25. In the blind test, the estimated improvement was 1.35-fold for WS*, with an interval from 1.11 to 1.64, and 1.40-fold for HFB-25, with an interval from 1.15 to 1.63. The blind sample remains limited, and catalogue predictions are not equally reliable.
What the evidence leaves unresolved
Taken together, the results support predictive and calibration claims under the study's stated splits, but they do not identify whether the shared residual pattern reflects missing nuclear physics or element-dependent experimental structure. Nor do they establish uniform accuracy for every catalogue entry. Public experimental and theoretical sources are cited. The catalogue is supplied as supplementary data, and the ARCUS implementation and trained configurations are available from the corresponding author on reasonable request.
Paper data and sources
Original title: Directional correlations in nuclear charge-radius model residuals enable extrapolation with calibrated uncertainties
Authors: Debodyuti Kar, Soumya Bagchi, Timo Dickel, Rituparna Kanungo
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text