Theoretical Extensions and Methodological Output of the Quantum-Classical Hybrid: The Regime-Factor Purity Spectrum, the Binary Structure of the Confirmed-Breakpoint Network, and Cross-Disciplinary Predictions ——Paradigm Deepening, Testable Predictions, and Complete Deployment of the Operating System Based on the Triple Demotion

Abstract

Tang (2026x) completed the triple demotion of the nuclear shell—the shell model was proven to be the projection of the regime-switch network in the spherical-symmetry limit by the 3.68–9.68-fold amplification of the Chow F of the five traditional magic numbers under continuous sphericity weighting (independently reproducible), the liquid drop model was proven to be the projection of the regime-switch network in the classical limit, and the independent-particle-motion axiom was proven to be incomplete. On this theoretical foundation, this paper accomplishes three independent theoretical extensions. First—the cross-disciplinary corollary of the regime-factor purity spectrum. The confirmed-breakpoint spacing CV = 0.67 falls between the purely discrete (particle-physics color-charge three-state CV = 0; chemical valence-electron equal-spacing CV = 0) and the purely continuous synthetic baseline (CV median = 0.689). Based on the regime-factor purities of six disciplines, the purity spectrum is formally proposed—from the fully continuous regime factors of finance to the purely discrete regime factors of particle physics and chemistry—providing a unified explanatory framework for the differentiation of demotion types: purely discrete regime factors permit constructive derivational demotion, semi-continuous regime factors permit limit-degradation derivational demotion, and fully continuous regime factors permit paradigm demotion. This purity spectrum supplies a testable predictive framework for the deployment of the Factor Hierarchy Law in a seventh discipline. The binary structure of the confirmed-breakpoint network—five confirmed breakpoints forming the full-N-axis skeleton (spacing CV = 0.67) and fifteen candidate breakpoints filling the inter-skeleton gaps (mean spacing 9.7)—is identified as a distinctive organizational feature of semi-continuous regime factors, standing in sharp contrast to the equal-spacing regime grids of purely discrete regime factors. Second—methodological insights into the paradigm shift. The stark contrast between the WLS continuous-weight approach and the doubly-magic-enrichment binary-screening approach (amplification of 3.68–9.68-fold versus 0.01–0.05-fold) reveals the deep physical implication that "the spherical-symmetry limit is a weight, not a region"—sphericity is continuous, and the applicability of the shell model is continuous. The cross-disciplinary methodological value of activation-operator stratification is systematically articulated for the first time: inv-family operators are sensitive to the quantum region, and raw-family operators are sensitive to the classical region—this stratification provides a direct operational guide for selecting optimal statistical detectors in other disciplines. The uniqueness of the double-track intersection point N≈70.8 is demonstrated by an elimination-method proof using the seven-metric intersection matrix—the intersection point between the shell-correction energy ΔR² and the intercept/slope ratio is the only metric pair among twenty-one pairs with a clear and stable intersection characteristic; this elimination method can be generalized to other disciplines to locate quantum-classical equilibrium points. Third—the concretization of cross-disciplinary predictions and the complete deployment of the operating system. The mid-shell silence zone is proposed as a testable cross-disciplinary hypothesis for quantum-classical hybrid systems—the silence zone within nuclear physics has been confirmed by multi-operator cross-validation, and numerical simulations of a dual-signal mixing model confirm the statistical principle that "mixed signals systematically suppress the Chow F," but its applicability in condensed matter physics and quantum chemistry depends on confirmation by independent experiments. The QCTI index provides a cross-disciplinarily universal quantitative framework for the degree of quantum-classical mixing—the simplified QCTI (requiring only macroscopic-model residuals plus discrete markers, ρ = −0.93) can be directly deployed in any system with an analogous structure, and its applicability boundary is precisely defined as "requiring a macroscopic baseline with R² > 0.9." The complete deployment of the Factor Hierarchy Law operating system in nuclear physics—all six functional dimensions fully covered by over twenty distinct tools—and the unique methodological contributions of the nuclear-physics deployment provide a complete operational guide for the deployment of the Factor Hierarchy Law in a seventh discipline. 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.