Microscopic Theory of Double Rational Quantization in Mirror Nuclei: A Synthesis of Quantum Renormalization Group and Collective Quantization with Fundamental-Constant Connections

This paper establishes a novel microscopic theoretical framework by synthesizing quantum renormalization group (RG) theory with collective quantization models to explain the universal and precise mathematical regularity observed in the ratio (K) of magnetic moment differences (∆µ) to binding energy differences (∆B) for nuclear mirror pairs. The theory posits that nuclear many-body systems, under the RG flow towards low-energy infrared (IR) fixed points, develop emergent approximate collective symmetries (e.g., SU(3), U(5), SO(6)). Within the basis of these symmetries, the matrix elements of observable operators are dictated by group representation theory, naturally leading to rational ratios. A pivotal finding of this work is a striking numerical coincidence regarding the primary constant C. We demonstrate that the dimensionless combination 3C ≈ 0.8466 exhibits remarkable numerical identity with a specific algebraic construct derived from the quark mass spectrum: the arithmetic mean of the Koide-type ratios for intra-generational mass sums and mass splittings of quarks, calculated to be 0.8463 after consistent QCD scale evolution. This establishes a profound, yet currently phenomenological, quantitative link between the emergent nuclear-scale constant and the algebraic structure of the quark flavor sector, inviting further theoretical investigation. The primary and correction rational coefficients (p/q) and (m/n) are shown to be determined by the collective representation of each specific nucleus. The theory not only predicts the double rational structure but also classifies the nuclide chart into distinct universality classes based on their RG fixed points.A comprehensive analysis of existing experimental data provides striking confirmation of all theoretical predictions.