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

Abstract

The central axiom of the traditional shell model—nucleons moving independently in a static mean field—and the liquid drop model have coexisted for nearly nine decades without resolution. Drawing on Tang (2026v) and Tang (2026w), this paper integrates five lines of evidence to complete a unified argument for the triple demotion of the nuclear shell: the shell model, the liquid drop model, and the independent-particle-motion axiom are projections of a quantum-classical hybrid in specific limits. The argument proceeds in five tiers. Tier 1—2D Ising calibration: With Tc as the fixed breakpoint, the intercept/slope ratio across five temperature intervals decreases monotonically from 875.75 in the deep ferromagnetic region to 2.67 near Tc (Spearman ρ = 1.0000), establishing it as a metric of distance from the phase-transition critical point. The ratio is strictly negatively correlated with susceptibility proxy |dM/dT| (ρ = −1.0000) and mean-field error ε_MF (ρ = −1.0000). The 2D Ising calibration heuristically supports the distance interpretation; the quantum-classical transition interpretation in nuclear physics is independently anchored by the strong negative correlation with shell-correction energy ΔR² (ρ = −0.87, p = 0.0000)—the two lines are independent yet convergent. Tier 2—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); a simplified QCTI using only shell-correction energy ΔR² and the binary marker yields ρ = −0.93. The seven-metric evidence matrix displays zonal characteristics across three nuclear regions. The double-track intersection N≈70.8 (spectrum value = 0.407) divides the N-axis into quantum-dominated and classical-dominated regions. The mid-shell region (N=50–100) is the common silence zone of all activation operators—Chow F drops to 6.8, only 1/15 of the light-nucleus region's 105.4 (Bootstrap 95% CI fully separated), with Kendall's W confirming complete agreement among the six operators. The mid-shell silence zone and the liquid-drop masking zone are negatively correlated (ρ = −0.67, p = 0.0001). The binary magic-number marker's local interaction effect decays significantly across 28 sliding windows (ρ = −0.57, p = 0.0016, a 132-fold decay), while shell-correction energy ΔR² peaks in the mid-shell region. Tier 3—First demotion: the shell model. A continuous sphericity weight function based on distances to the nearest magic numbers is constructed. Weighted least squares on the nuclide data shows that the Chow F of the five traditional magic numbers is amplified by 3.68–9.68 times (σ = 3.0); amplification exceeds 1.5 for all five magic numbers for σ = 2.0–5.0 and 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 monotonic relationship between the intercept/slope ratio and 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—compensating 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), with all 8 conditions for derivational demotion satisfied. All current evidence derives from multi-dimensional re-analyses of the same AME2020 nuclear mass data—statistically independent but not data-source independent; cross-data-source verification 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. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently directed by the author. DeepSeek assisted with code implementation, data presentation, and text drafting. The author takes full academic responsibility for the final content.