Regime-Switch Network and the True Structure of the Nuclear Shell: Internal Self-Consistency of the Confirmed-Breakpoint Network, Completeness Audit of the Shell Model, and Fifteen New Discoveries ——Transition-Zone Width Theorem, Cascade Skeleton, Liquid-Drop Masking Effect, and Shell-Correction Compensation Traces Based on Multi-Tool Aggregation

What is the true structure of the nuclear shell? The traditional shell model holds that the nuclear shell is marked by a few isolated magic numbers—N = 2, 8, 20, 28, 50, 82, 126. This paper finds that the nuclear shell is not a collection of points but a multi-level network composed of blind-detected breakpoints obeying precise topological regularities. Through rigorous permutation tests (500 permutations each), all 15 single-tool hit breakpoints fail the significance test (p = 1.0000), and only 5 multi-tool hit breakpoints—[23, 89, 108, 129, 145]—are confirmed as independent regime-switch points through cross-validation. The argument of this paper is organized into two tiers. The core tier comprises three theorems (Theorems 7–9) and two key pieces of evidence (shell-model coverage of only 15% and the liquid-drop masking effect), constituting the main line of the paper. The supplementary tier includes the odd-even effect jump, the N=Z line as a regime boundary from isotope-chain Chow tests, and independent verification from HMM and kernel change-point analyses—these are not independent discoveries but cross-validations of the same core discovery (the real existence of regime switches in nuclear physics) across different dimensions. The core findings unfold in five tiers. Tier 1—Transition-Zone Width Theorem (nuclear-physics version): establishes the internal self-consistency of the confirmed-breakpoint network—the confirmed breakpoints fall exactly on the shell-region boundaries defined by one another, organized according to precise topological regularities. Random noise cannot automatically arrange itself into a structure with zero deviation (1000 random breakpoint sets never reproduced 7 exact hits, p < 0.001). When externally defined shell-region boundaries based on traditional magic numbers are used, the alignment accuracy is not significantly different from random expectation (p = 0.4197); the self-consistency of Theorem 7 relies primarily on the internal consistency of the network rather than on external independent benchmarks. Tier 2—New structures unexplained by the shell model: shell-model coverage is only 15% (only 3 of the 20 blind-detected breakpoints fall within ±2 of traditional magic numbers); if only the 5 confirmed breakpoints are considered, coverage is 20% (1/5), but the sample size is too small, and the coverage analysis is positioned as an exploratory assessment. The coverage analysis tests whether "magic numbers are not the optimal description of the statistical structure" rather than whether "the shell model is wrong." The odd-even effect undergoes systematic jumps at magic numbers (halving at N=20 from 90 to 40, doubling at N=28 from 29 to 72, reversing at N=82 from +8 to −13). Chain-by-chain Chow tests on 111 isotope chains find that optimal breakpoints in the light-nucleus region tightly follow the N=Z line, identifying the N=Z line itself as a regime boundary. Of the 20 breakpoints, 17 are confirmed as direction-reversal type, with the intercept/slope ratio satisfying a monotonic increasing trend (Spearman ρ = 0.9489); Theorem 9 applies only to these 17 confirmed breakpoints; the 3 candidate breakpoints at N=14, 15, and 29 cannot exclude the rule-reset type. Tier 3—Disappearance of regime-switch signals under high-precision fitting of the liquid-drop model: the liquid-drop model achieves R² = 0.96, but its residual Chow signal is 4–9 times weaker in the light-nucleus region, and the residuals are not significantly different from random positions at the blind-detected breakpoints (p = 0.5749). In the heavy-nucleus region, the signal-to-noise ratio of the liquid-drop residuals is significantly higher than that of the OLS residuals (mean ratio 5.62), while the Chow F remains systematically lower (mean ratio 0.31), ruling out the alternative explanation of "reduced test power due to low signal-to-noise ratio." Hierarchical regression shows that R² jumps from 0.70 in light nuclei to 0.94 in medium nuclei to 0.99 in heavy nuclei, with the N_over_Z coefficient jumping by a factor of 24. The segmented model reduces the prediction RMSE by 30–66% near the blind-detected breakpoints (global Wilcoxon test p = 0.0036). The disappearance of regime-switch signals in the liquid-drop model residuals is a statistical fact; the shell-correction compensation traces in Section 6 support the "absorption" hypothesis—the success of the shell-correction term precisely demonstrates that it captures and absorbs the shell-closure signals. High-precision models may systematically mask structural signals outside their framework—this is a methodological caution. Tier 4—Compensation traces of the shell-correction term: three independent tools (PELT, Bai-Perron, and Bayesian segmentation) consistently avoid the shell-closure positions (p = 0.0487, marginally significant; the consistent avoidance by three tools is more critical evidence than the single p-value); the no-shell-correction liquid-drop model residual test rules out alternative explanations; the cross-regime rupture ratio of the mass covariance is 22.31 (same-side vs. cross-boundary); reaction-energy errors show systematic jumps across all 7 columns × 9 boundaries; all four decay-mode interaction effects have p < 10⁻²³. Together, these lines of evidence point to the success of the shell correction = direct evidence that independent particle motion requires external compensation. Tier 5—Cascade skeleton: based on the 5 confirmed breakpoints, N=20 and N=28 share N=23 as a cascade node (gap = 0), establishing exact cascade alignment. N=89 simultaneously clamps N=28, N=50, and N=82—three traditional magic numbers—with the local Chow signal attenuating with distance (F = 0.21 at N=28, F = 1.17 at N=50, F = 1.51 at N=82). The cascade degenerates from "precise transmission between adjacent magic numbers" to "one confirmed breakpoint simultaneously serving as the statistical boundary for multiple magic numbers," reflecting the fundamental difference between the continuity of regime factors in nuclear physics and the discrete symmetries in particle physics. The possibility that one-to-many clamping arises from the sparsity of confirmed breakpoints cannot currently be excluded; this observation is positioned as a statistical feature awaiting clarification. This paper refines three new theorems (Theorems 7–9), deploys a methodological template of multi-level defense tests, and reports a total of fifteen new discoveries completely unknown to traditional nuclear physics. Together with Tang (2026v), this paper provides all five key pieces of evidence that "independent particle motion is not the fundamental organizing principle of the nuclear shell."