Quantization of Flavor Space and the Unified Algebraic Structure of Elementary Particle Masses: From Δ(27) Symmetry to Quantum ChromoElectrodynamics

Building upon three precise algebraic patterns in the quark mass spectrum uncovered in the author’s previous work 1, this paper constructs a unified dynamical framework—Quantum ChromoElectrodynamics (QChED). The core ideas are: (i) flavor space originates from color space; (ii) the weak interaction is a quantized relativistic mixing effect between the electromagnetic and strong interactions; (iii) the third-order differential equation replaces the Dirac equation as the fundamental equation of flavor dynamics. By introducing the flavor quantization hypothesis, defining the Liu constant ℏ𝐿 and the commutation relation 𝑥, 𝑝 = 𝑖ℏ𝐿, we construct a left-invariant third-order differential operator 𝐷3 = 𝑑𝑎𝑏𝑐𝑋𝑎𝑋𝑏𝑋𝑐 on the 𝑈(1)em ×𝑆𝑈(3)𝐶 group manifold, and prove that it reduces to a onedimensional third-order ordinary differential equation on the Δ(27) ×𝑈(1)-invariant subspace.The eigenvalues of this equation directly yield the up-type quark mass ratio 𝑚𝑡: 𝑚𝑐 : 𝑚𝑢 = 1 :√︃(3 √7)/2 : √︃(√7 − 2)/7. Extending the framework to down-type quarks, leptons, and electroweak gauge bosons, we derive the Wolfenstein parameters of the CKM matrix, the logarithmic lepton mass ratio 9 : 17 : 26, and the electroweak boson mass-squared ratio 𝑀2𝑊 : 𝑀2𝑍: 𝑀2𝐻= 7 : 9 : 17. These results are in excellent agreement with experimental data, revealing a common algebraic origin of fermion and boson mass spectra under Δ(27) flavor symmetry.The paper further establishes that the weak interaction is essentially a quantized relativistic mixing effect between the electromagnetic (𝑈(1)em) and strong (𝑆𝑈(3)𝐶) interactions under Δ(27) symmetry, with the weak mixing angle 𝜃𝑊 satisfying cos(2𝜃𝑊 (𝑀𝑍)) = 2𝜋√︁𝛼(𝑀𝑍) to within 0.2% accuracy. Weak bosons correspond to eigenstates on the adjoint representation 8 of Δ(27), and their mass-squared ratio 7 : 9 : 17 is isomorphically related to the meson mass-squared ratio in hadronic physics, suggesting a composite nature for weak bosons. The integral form of the third-order differential equation automatically avoids ultraviolet and infrared divergences, eliminating the need for renormalization. Extending the framework to the dark sector, we predict that dark quark mass ratios are identical to those of visible quarks, the dark weak mixing angle is 𝜋/4 at the ultraviolet fixed point, and the dark 𝑊𝐷 and 𝑍𝐷 mass-squared ratio is 1 : 2, providing testable inputs for dark matter research. The paper also demonstrates that the third-order differential equation can be naturally derived from the path integral of QCD and QED, establishing it as the fundamental equation describing flavor space dynamics.Finally, we prove that the fine-structure constant 𝛼 is the only independent constant in the theory; all other physical constants— including ΛQCD, 𝑚𝑒, 𝑚𝜇, 𝑚𝜏, 𝑚𝑡, 𝑚𝑐, 𝑚𝑢, 𝑀𝑊,𝑀𝑍, 𝑀𝐻 , and 𝜃𝑊—are uniquely determined by 𝛼 and the pure group-theoretic numbers of the Δ(27) symmetry (7, 9, 17, 9, 17, 26, 𝛼2, 𝛽). This represents the most significant reduction of fundamental parameters in the history of physics, from 19 in the Standard Model to 1 in Quantum ChromoElectrodynamics.