ZQCM V1.4: First‑Principles Derivation of Standard Model Parameters from S³ Spectral Geometry and Spin(9) Root Systems

Abstract

# 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 M_geo=1.1×10¹⁶ GeV down to electroweak scale M_Z, 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.