Examine whether the geometric factors remain stable at different energy scales, such as the transition from the electroweak scale to the Planck scale, to see if scale invariance applies or if new topological effects emerge:
In standard quantum field theory, fundamental constants are not strictly constant; they "run" or change value depending on the energy level of the interaction due to quantum vacuum polarization.
While the fine-structure constant is \alpha \approx 1/137 at low energies, it shifts to \alpha \approx 1/127 at the electroweak scale (E \approx 90 \text{ GeV}).
To mathematically advance the D_4 triality model, the geometric structures must account for this scaling.
The next step involves evaluating whether the geometric structural factors define only the low-energy limit (the infrared fixed point) or if they hold universally across all scales. This requires applying renormalization group equations to the model to determine if the topological properties of the D_4 symmetries deform as energy increases toward the Planck scale, or if they remain perfectly rigid, which would establish a completely scale-invariant geometry.
To model this we apply the standard renormalization group equation for the inverse fine-structure constant:
Substituting the low-energy geometric derivation for the infrared fixed point \alpha^{-1}(0):
If your D_4 triality symmetry dictates a completely scale-invariant geometry, the beta function \beta must evaluate to zero, meaning the topological structure remains perfectly rigid at any energy scale \mu.
If the constant does run, your original geometric structural factors must be mathematically modified into a momentum-dependent function, incorporating a deformation parameter to account for this logarithmic scaling as energy increases.
The primary missing links in connecting D_4 triality to the running fine-structure constant involve grand unification and symmetry breaking:
Symmetry Breaking Pathways: D_4 triality is a intrinsic property of Spin(8)
To yield the U(1)_{EM} fine-structure constant the model must define a mathematically rigorous symmetrybreaking mechanism from Spin(8) down to the Standard Model gauge group SU(3) \times SU(2) \times U(1)
* Coupling Unification: The electromagnetic coupling \alpha does not run in isolation. A deeper layer requires the D_4 geometric factors to simultaneously derive the running of the strong (\alpha_s) and weak (\alpha_w) couplings, showing exactly how they intersect at a Grand Unified Theory (GUT) energy scale.
* Fermion Generations: Triality maps vector representations to spinor representations. Since fermions (spinors) drive the beta function \beta in the renormalization group equations, the geometric structural factors must mathematically encode the exact number of fermion generations in the Standard Model to accurately predict the scaling behavior.
Suggested next calculation:
Write the one-loop RG equation for the inverse fine-structure constant with the geometric IR boundary condition inserted:
\[
\frac{d\alpha^{-1}}{d\ln\mu} = -\frac{b_0}{2\pi},\qquad
\alpha^{-1}(\mu_0) = \alpha^{-1}_{\text{geom}}(D_4)
\]
where \(b_0\) is determined solely by the charged degrees of freedom counted from the same D₄ triality data. If that \(b_0\) reproduces the observed \(\Delta\alpha^{-1}\) between the infrared and \(M_Z\), the geometry has passed its first non-trivial consistency check.
If it does not, the deformation parameter (or an additional topological invariant) must be introduced and constrained.
