Nuclear-mass prediction steps out of the black box: an interpretable AI rediscovers forgotten Jaffe factorization and pushes WS4 error below 0.12 MeV

Binding-energy prediction gains an interpretable high-precision route: structure learned inside the model translates directly into physical terms rather than only into numbers. The most precise phenomenological microscopic-macroscopic model, WS4, has a root-mean-square error of about 0.28 MeV, while high-precision AI models reach the 0.13 MeV level but, being black boxes, are hard to judge when extrapolating into unmeasured regions. On 2325 nuclei selected from AME2020, each evaluated nucleus is excluded when the parameters entering its prediction are determined, and the 0.28 MeV WS4 and 0.13 MeV AI errors are computed on the same set of nuclei.

The model maps each proton number and neutron number to a 1024-dimensional learnable vector, and after training the leading principal components form a double helix, with links connecting the proton and neutron representations of the most stable nucleus in each isotopic chain. The liquid-drop model describes binding energy with symbolic volume, surface, Coulomb, asymmetry and pairing terms, and the asymmetry term contains (Z−N)², which is invariant under translations along the proton and neutron directions and so leaves a repeatable representation space for the helical geometry. Principal component analysis of the embeddings shows the first three components forming a double helix; within this regularized model family the helical geometry is robust across the hyperparameter variations explored in the code and earlier studies, whereas removing regularization yields an ordered representation with comparable performance but without the helix.

The binding-energy prediction decomposes in the penultimate layer into a macro-to-micro hierarchy: the first three principal components are approximately smooth and correspond to liquid-drop-like macroscopic terms, while the lesser components are discrete and approximately factorize into a function of Z alone plus a function of N alone. Microscopic terms in physics models are normally computed with mean-field many-body quantum mechanics, whereas the microscopic term learned here is a much simpler separable estimate. Fitting a higher-order principal component with a characteristic 100 keV scale to FZ(Z)+FN(N) leaves a residual root-mean-square of only 4 keV.

Using local Jaffe factorization as a correction, taking only nearest and next-to-nearest neighbors with the same proton or neutron number, works better than taking neighbors by geometric distance. Many models learn local corrections to WS4 from distances to observed neighboring nuclei, with WS4 itself reporting a root-mean-square error of 0.18 MeV, whereas local Jaffe factorization implies that distance in the nuclear plane is not the most important consideration. Corrections from same-number neighbors arranged in a plus outperform the remaining neighbors arranged in a cross, and the cross corrections give no improvement; local Jaffe corrections bring a simple model without physics microscopic terms to 0.18 MeV, reduce WS4 to 0.16 MeV, and after a linear regression that also corrects the neutronic and protonic components fall below 0.12 MeV, with the residual map giving an overall root-mean-square of 115 keV.

The demonstrated scope is 2325 nuclei from AME2020 with Z and N both at least 12, mass uncertainties no larger than 100 keV and masses not determined from mass-surface analysis, covering 12≤Z≤118 and 12≤N≤178; all errors are leave-one-out, so the evaluated nucleus does not enter the determination of its prediction parameters. Local Jaffe corrections do not use the target nucleus, only measured nearest and next-to-nearest isotopic and isotonic neighbors.

Local Jaffe factorization breaks down near double-magic nuclei such as 208Pb, requiring dedicated phenomenological terms for those regions; within the measured domain the method stays accurate along the neutron-rich edge, especially for Z and N above about 50, with larger residuals mostly near magic and double-magic nuclei at lower masses. How far local Jaffe factorization extends toward the neutron-rich nuclei needed for the r-process remains open, and a quantitative assessment requires input from nuclear-structure theory.