The Regime Nuclide Map (II): Statistical Localization of Shell Correction Energy, Multidimensional Response Theorem, and Methodological Contributions ——Empirical Test Based on the Nuclear Shell Regime-Switching Theory and the .ssr Trap Correction

The nuclear shell regime-switching theory (Tang, 2026v–y) predicts that shell correction energy should compensate for the incompleteness of independent particle motion, and that Theorem C should be replicable across data environments. This paper verifies this prediction and discovers new statistical fingerprints and methodological contributions guided by the theory. Major findings include: (1) First report of the statistical fact that shell correction energy extrema adhere tightly to traditional magic numbers, while staying distant from confirmed breakpoints. Three independent extrema detection methods consistently verify this: the two-dimensional maximum filter method yields an average distance to traditional magic numbers of 3.6 neutrons and to confirmed breakpoints of 18.0 neutrons (Cohen's d = 3.23). A randomization test excludes the alternative explanation of density asymmetry (p = 0.002), grid resolution sensitivity analysis (filter window sizes 2/3/5) confirms the robustness of extrema localization to detection parameters, and sensitivity analysis of the liquid drop model's functional form (adding three Wigner terms) confirms the robustness of extrema localization to model selection. This statistical fact supports the "compensation hypothesis", but the paper candidly acknowledges that this physical interpretation carries a risk of circular definition; (2) First proposal of Theorem 10 (Multidimensional Response Theorem)—the second-order derivative of binding energy exhibits singularity peaks at confirmed breakpoints (9.88 times the global mean at N=23). After local baseline correction, all 7 confirmed breakpoint ratios are significant (2.54–4.47 times), and robustness tests across eight baseline ranges (±6 to ±20) pass 7/7. The Spearman ρ between dimensions (a) and (b) is 0.224 (p = 0.718), quantitatively supporting that they are two distinct statistical fingerprints of regime switching. The residual variance jumps significantly at confirmed breakpoints (4/5 confirmed breakpoints, p < 0.05), and random forest factor importance migrates dramatically at each confirmed breakpoint; (3) Completion of cross-data subset replication of Theorem C (σ = 2.0–5.0 all enhanced >1.5 times), and the first discovery and correction of the .ssr attribute trap in statsmodels' WLS—the correct approach is to manually compute np.sum(resid**2)—providing direct guidance for all researchers using Python for WLS Chow tests.