Hierarchical Fractional Quantization in Superconducting and Superfluid Systems:A Unifying Framework from First Principles and Renormalization Group

The principle of symmetry serves as a cornerstone of modern physics, yet its role in condensed matter many-body systems has largely remained qualitative, acting primarily as a classificatory tool. In this work, we establish, for the first time, a universal framework of hierarchical fractional quantization (HFQ) for superconducting and superfluid systems with continuous symmetries and quantum fluctuations. We demonstrate that symmetry can impose rigorous, computable quantitative constraints on the numerical values of physical observables.We apply this framework to predict the response of the d-wave order parameter in high-temperature superconductors under uniaxial stress (predicting R = 1 and deriving new constraints on superfluid density anisotropy) and the excitation spectrum splitting of a Bose superfluid in an optical lattice under lattice distortion (predicting R = 1/2). Systematic quantitative validation using multiple published experimental datasets shows excellent agreement with theoretical predictions. This work elevates symmetry from a qualitative organizing principle to a quantitative predictive tool, offering a new paradigm for understanding competing orders and symmetry breaking in strongly correlated quantum materials.