Shell structure, a Wigner-energy feature and the role of network capacity in statistical learning of nuclear masses below A approximately 60

Contatto e-mail: multiversou@gmail.com A minimal heteroscedastic neural network is trained on AME2016 masses using the twelve bulk-and-valence features of Lovell et al., with no explicit shell-gap, Wigner or deformation terms. Under stratified five-fold cross-validation, with early stopping confined to the training data and eight independent repetitions, the baseline (M12) reaches a global RMS of 0.84 MeV, but 1.79 MeV for A below 60 against 0.54 MeV for A of 60 and above. A single smooth Wigner-energy feature, 47 times the absolute value of N minus Z divided by A, lowers the RMS by 17 percent for A below 60 and by 27 percent for light N approximately equal to Z nuclei; a literature odd-odd term adds nothing. A wider network without this feature, however, achieves a comparable global gain, so part of the light-nucleus degradation reflects model capacity rather than missing physics, and the two effects are additive. Continuous valence distances carry the shell signal, while discrete shell indices are largely redundant. The Wigner feature gives no advantage in two extrapolation tests, and the two-neutron shell gap at N = 82 is underestimated by about 22 percent. Technical note on the nature of this work This manuscript is a methodological study, not the report of a new physical result. Although the title and abstract describe the behaviour of a Wigner-energy feature, the work does not establish a new nuclear observable, a new mass formula, or a new constraint on nuclear forces. Its contributions are instead of a methodological nature: (1) the demonstration that a deliberately small network with generic features degrades sharply below A approximately 60; (2) the controlled test of whether a single literature Wigner-energy feature repairs that degradation; (3) the demonstration that a comparable share of the global improvement is recovered by capacity alone, so the two effects are additive rather than competing; (4) the finding that continuous valence distances carry the shell-closure information while discrete shell indices are largely redundant; and (5) the negative result that the Wigner feature provides no advantage in out-of-domain extrapolation. Readers should interpret the manuscript accordingly, as an ablation and capacity study rather than a discovery paper. Note on the FRDM2012 benchmark The value 0.457 MeV quoted for the macroscopic-microscopic model FRDM2012 on the A of 60 and above subset should be recalculated. In the present work it is compared against the mean of single training runs and against ensemble averages of a small network, and it is not directly comparable to either. A consistent recalculation of FRDM2012 on the exact same subset of nuclei, with the same mass table and the same error definition, is required before any quantitative statement of superiority or parity can be made. The author therefore does not claim to outperform FRDM2012; the comparison is reported only to place the network errors in context. Data and code availability The code (NumPy implementation, training and cross-validation scripts, ablation and extrapolation scripts), the AME2016 table and the analysis outputs are available at the Zenodo DOI to be assigned upon publication. The AME2016 evaluation is publicly available. In addition, the author releases the full set of network files used in this work. These are numerous and are therefore provided as supplementary files attached below the PDF, rather than embedded in the manuscript. The list is organised by function in the section that follows. The AME2016 evaluation is publicly available. Released files, grouped by function Group 1. Core models and network infrastructure These files contain the mathematical architecture of the neural network (mixture density network) and the rules for extracting physical properties from the data. They are the core of the project. mdn_np.py: the main numerical engine. Implements the neural network, the Adam optimiser, manual gradient computation and the validation protocols (leaky versus nested) in pure NumPy. common.py: shared utility functions for loading predictions and computing basic statistics; used by the analysis and plotting scripts. common_stats.py: computes the root-mean-square error (RMS) broken down by the specific nuclear mass regions. Group 2. Extrapolation scripts and referee response (high priority) Developed in PyTorch, these scripts implement the out-of-domain stress tests requested specifically by Referee 1 to demonstrate the physical robustness of the Wigner term. extrapolation_isospin.py: trains the network on neutron-rich nuclei and forces it to extrapolate into the critical N approximately equal to Z region in order to test the effectiveness of the Wigner feature. extrapolation_mass.py: trains the network on heavy nuclei (A of 60 and above) and tests its ability to predict entirely unseen light nuclei (A below 60). Group 3. Automation of production runs (data generators) Scripts written in NumPy for large-scale execution of the calculations. They perform the bulk CPU work, generating the .npz and .json files that contain the final results. run_main.py: the general coordinator. Launches the 36 cross-validation sessions by combining models, repetitions (up to 8) and validation protocols. run_extrap.py: runs the extrapolation tests by combining the four main models (M12, M13, M13s, M12w) and saves everything to extrap_results.json. run_ablation.py: automatically runs the ablation study (selective removal of features) over 5 different repetitions to check its statistical stability. run_sweep.py: manages the dimensional sweep of the network and the learning curve based on the fraction of data used. ablation_monolithic.py: a standalone PyTorch script that performs the ablation study and the permutation tests (random shuffling of columns) with five-fold cross-validation. Group 4. Statistical analysis and graphics (results translators) These take the raw data computed by the models and turn them into mathematical p-values, LaTeX tables or high-resolution figures ready for the journal. generate_numbers.py: the analytical core. Computes physical averages, runs paired statistical tests (t-test and Wilcoxon), estimates the energy anomalies of the tin isotope chain and produces the final tables. plot_main.py: generates the main figures of the paper (Figures 1 to 5 and 7), including the residual plots, the behaviour of the tin chain and the anomaly distributions. plot_capacity.py: produces the comparison plots for the ablation study (Figure 6) and the violin plots for the extrapolation tests (Figure 7). plot_capacity_sweep.py: draws Figure 8, mapping the behaviour of the RMS error both as a function of the number of network parameters and along the learning curve. check_sweep.py: analyses the residuals and the spatial autocorrelation of neighbouring light nuclei for the different network sizes. check_results.py: performs a quick comparison between protocols, computes the effectiveness of individual models against the FRDM2012 theoretical benchmark and maps the percentage improvements. Group 5. Automatic compilation and editorial pipeline (closing the loop) The last mile of the project, where the code meets the writing of the scientific text. compile_paper.py: takes all the mathematical data computed and saved in numbers.json and inserts them automatically into the LaTeX source of the text, generating the final updated file for submission.