The Triple Demotion of the Nuclear Shell: The Shell Model, the Liquid Drop Model, and the Independent-Particle-Motion Axiom ——Derivational Demotion Based on 2D Ising Exact Solution Calibration, Continuous Sphericity Weight Recovery, and the QCTI Index

The central axiom of the traditional shell model—that nucleons move independently in a static mean field—has magic numbers as its corollary and the shell correction as its supplement. The liquid drop model, which treats the nucleus as an incompressible charged droplet, has likewise achieved enormous success. For nearly nine decades, the conceptual tension between these two frameworks has never been systematically resolved. Drawing on the empirical findings of Tang (2026v) and Tang (2026w), this paper integrates five lines of evidence and completes a unified argument for a triple demotion of the nuclear shell: the shell model, the liquid drop model, and the independent-particle-motion axiom are each shown to be projections of a quantum-classical hybrid in specific limits. The core argument proceeds in five tiers. Tier 1—2D Ising exact-solution calibration: Taking Tc as a fixed breakpoint, the full ferromagnetic and paramagnetic data are fitted to an M~T equation. The intercept/slope ratio across five temperature intervals decreases monotonically from 875.75 in the deep ferromagnetic region to 2.67 in the Tc-neighborhood (Spearman ρ = 1.0000), establishing the intercept/slope ratio as a rigorous physical metric of the distance from the phase-transition critical point. The ratio exhibits a strict negative correlation with the susceptibility proxy |dM/dT| (ρ = −1.0000) and with the mean-field error ε_MF (ρ = −1.0000). The 2D Ising calibration provides heuristic support in the form of a formal isomorphism for the "distance to critical point" interpretation; the "quantum-classical transition" interpretation of the intercept/slope ratio in nuclear physics is independently anchored by the strong negative correlation with the shell-correction energy ΔR² (ρ = −0.87, p = 0.0000)—the two lines of evidence are logically independent yet convergent in their conclusions. Tier 2—Multi-dimensional empirical basis of the quantum-classical hybrid: The Quantum-Classical Transition Index (QCTI) decreases monotonically from +0.94 in the light-nucleus region to −0.81 in the heavy-nucleus region (ρ = −0.95); its monotonicity does not depend on the physical interpretation of the intercept/slope ratio—a simplified QCTI constructed using only the shell-correction energy ΔR² and the binary marker ΔR² yields ρ = −0.93. The QCTI measures quantum residuals using the liquid drop model as the classical baseline; it is a "quantum residual measure relative to the liquid drop model" rather than a "model-independent measure of quantality." The seven-metric evidence matrix displays systematic zonal characteristics across three nuclear regions. The double-track intersection N≈70.8 (spectrum value = 0.407) divides the N-axis into a quantum-dominated region and a classical-dominated region, and a seven-metric intersection matrix confirms its uniqueness. The mid-shell region (N=50–100) is identified as the common silence zone of all activation operators—the activation-operator Chow F drops to 6.8, only 1/15 of the light-nucleus region's 105.4 (Bootstrap 95% CI fully separated), and Kendall's W confirms complete agreement among the six operators. The mid-shell silence zone and the liquid-drop masking zone are significantly negatively correlated in space (ρ = −0.67, p = 0.0001). The local interaction effect of the binary magic-number marker decays significantly and monotonically across 28 sliding windows on the full N-axis (ρ = −0.57, p = 0.0016, a 132-fold decay), while the shell-correction energy ΔR² peaks in the mid-shell region. Tier 3—First demotion: the shell model. A continuous sphericity weight function based on the sum of distances to the nearest magic numbers is constructed. A weighted least squares test on the full nuclide data shows that the Chow F of the five traditional magic numbers is amplified by a factor of 3.68–9.68 (σ = 3.0); the amplification exceeds 1.5 for all five magic numbers for σ = 2.0–5.0, and the amplification factor decreases monotonically with increasing σ, demonstrating that the shell model is the projection of the regime-switch network in the spherical-symmetry limit. Tier 4—Second demotion: the liquid drop model. The QCTI tends to −0.81 in the heavy-nucleus region, the strict monotonic relationship between the intercept/slope ratio and the mean-field error ε_MF (ρ = −1.0000) holds, and the uniformity of the liquid-drop masking effect supports the "absorption" hypothesis, together proving that the liquid drop model is the projection of the regime-switch network in the classical limit. Tier 5—Third demotion: the independent-particle-motion axiom. Five lines of evidence from four independent dimensions establish the incompleteness of the axiom as a statistical description framework for many-body systems. The shell correction is re-positioned as a "quantum residual"—it compensates for the difference between quantum shell effects and classical liquid-drop behavior, peaking in the mid-shell region. This paper refines six theorems (Theorems A–F), and all 8 conditions for the completion of derivational demotion are satisfied. It should be candidly acknowledged that all current lines of evidence derive from multi-dimensional re-analyses of the same set of AME2020 nuclear mass data, satisfying statistical independence but not data-source independence; independent verification across different data sources is a key direction for future work. This paper completes the derivational demotion in nuclear physics, which, together with the demotion of gauge field theory in particle physics and the demotion of the periodic law in chemistry, constitutes the complete demotion argument of the Factor Hierarchy Law in the three hard-science disciplines.