Nuclear Shell Regime Audit: The Magic-Number Shift Theorem and the Discovery of the N=14 Statistical Rupture ——The η Hierarchy Structure, Bai-Perron Optimal 10 Breakpoints from Zero-Preset Exhaustive Search, and Six Quantitative Theorems

The central axiom of the nuclear shell model—that nucleons move independently in a static mean field—has never been systematically questioned. This paper deploys the Factor Hierarchy Law to nuclear physics for the first time, performing a zero-preset regime audit on the full AME2020 nuclide dataset—the algorithm does not know of the existence of "magic numbers" or "shells." Exhaustively testing 18 candidate execution factors, the optimal combination under the VIF < 5 constraint is mass_excess_keV + beta_decay_keV + N_over_Z (adjusted R² = 0.8052), with Bayesian model averaging confirming its statistical irreplaceability with a weight of 1.0000. The core findings unfold in four tiers. Tier 1: the η hierarchy—the ratio of interaction effect to direct effect for continuous N/Z exceeds 10⁶ (η > 10⁶), that for the binary magic-number marker is η = 19.19, and that for the continuous shell-correction energy is η ≈ 0. This three-tier η hierarchy points to a single conclusion: under continuous-variable proxies, N and Z behave as regime factors (function realized through interaction effects), the shell correction is an execution factor (directly explaining binding energy), and independent particle motion exists but is not the rule-maker. Tier 2: Triple cross-validation and supplementary evidence for the N=14 statistical rupture—all three inv-family activation operators lock onto N=14 (F = 415–572), the threshold interaction model locks onto N=25 as the optimal threshold, the PICO intervention effect attains its maximal intercept jump at N=14 (−245.98 keV, p < 0.001), and all eight breakpoints of the kernel change-point cosine kernel concentrate at N < 15. The existence of a statistically significant structural rupture at N=14 is a statistical fact confirmed by three independent tools. Tier 3: The full-N-axis Bai-Perron sequential blind test locks onto an optimal 10 regime-switch breakpoints with overwhelming statistical evidence (BIC = 13793.7). The individual ΔBIC values of all five traditional magic numbers conform to the Statistical Distinction Theorem of Shell-Closure Physical Mechanisms—N=28 is of deformation-transition type (ΔBIC = 415.9), and N=20 is of single-particle-gap type (ΔBIC = −514.4). Magic numbers are not optimal partition points in the statistical sense—the Magic-Number Shift Theorem, the Statistical Distinction Theorem, and the Magic-Number Node Theorem jointly establish the repositioning of traditional magic numbers within the statistical structure. Tier 4: Three lines of negative control exclude statistical artifacts (permutation test p = 0.002, fake-data R² = 0.0007, WLS residual Chow F = 482.55). This paper refines six quantitative theorems, establishes the statistical existence of regime-switch signals in the nuclear shell, and constitutes the necessary premise for the subsequent argument that "independent particle motion is not the fundamental organizing principle of the nuclear shell."