# Abstract This work develops the ZQCM geometric framework built on S³ spectral dimension analysis, Spin (9) Clifford algebra, Cartan depth hierarchy and chiral vacuum configuration, aiming to unify the origin of fermion mass hierarchies, CKM/PMNS mixing angles and Higgs mass renormalization group (RG) flow without extra free fitting parameters. First, we solve the spectral dimension fixed-point equation dₛ (t) =e via high-precision numerical summation of S³ scalar heat kernel series, extracting the intrinsic thermal scale ratio φ = tₑ/t*. Second, we construct the complete geometric Yukawa mass matrix including chiral interference term |σᵢDᵢ−σⱼDⱼ|·q^|i−j| from Spin (9) spinor projection, derive three-generation mass eigenvalues for up-type quarks, down-type quarks and charged leptons, and compute the 1–3 mixing angles of CKM and PMNS matrices via symmetric matrix diagonalization. Third, we establish the NLO+NNLO β-function for geometric gauge couplings, perform RG evolution from GUT geometric scale Mgeo=1. 1×10¹⁶ GeV down to electroweak scale MZ, and predict the renormalization-corrected Higgs pole mass consistent with experimental 125. 09 GeV measurement. All numerical calculations are fully reproducible via three independent mpmath/numpy Python scripts provided as supplementary files, covering spectral dimension solving, fermion mass matrix diagonalization and Higgs RG flow iteration. ## Version Note (V2 Update) This Version 2 revises the mass matrix construction formula: replacing the simplified product ansatz |DᵢDⱼ| with the full chiral difference quadratic term from complete B3-V2 derivation, eliminating artificial degeneracy between up and down quark matrices and correcting over-large mixing angle predictions from the V1 simplified approximation. All Python numerical codes are patched to adopt symmetric matrix dedicated diagonalization for stable physical observables output.
Qian Zhao (2026) studied this question.