Compute the remaining two Standard Model mixing angles via the discrete framework

Develop calculations, analogous to those already proposed for six or seven angles, to determine the two remaining Standard Model mixing angles using the finite geometries and discrete groups G27 × Q8 × Z3; assess whether incorporating Dirac matrices is necessary to complete these computations.

Background

The paper claims progress toward computing several mixing angles from discrete geometric constructions tied to the finite groups underlying U(1), SU(2), and SU(3).

The author states that finding similar computations for the remaining two angles is a priority and suggests that including Dirac matrices may be required.

References

A priority for further work is therefore to try to find similar calculations for the remaining two mixing angles.

— A discrete model for Gell-Mann matrices  (2401.13000 - Wilson, 2024) in Section 9.2

At 1-loop the quark phases can be chosen so that $A{(*)}$ and $A{\prime ()}$ do all vanish at all six fixed points, but this has not been checked at higher loops. We conjecture that, even at higher loops, with the phases chosen so that $A{()}$ and $A{\prime (*)}$ vanish at one fixed point then they will vanish at all six, due to $S_3$ symmetry, but this has not verified this explicitly.

— Fixed points of the CKM matrix renormalization group running to all orders in perturbation theory  (2608.28149 - Dolan, 28 Aug 2026) in Section 2, footnote immediately following the discussion of the six CKM fixed points in Eq. (S_3)

In particular, it is still not known whether $\theta_{23}$ lies in the first octant ($\theta_{23}<45\circ$) or the second octant ($\theta_{23}>45\circ$), a problem commonly referred to as the octant degeneracy.

— Deviation from $μ-τ$ reflection symmetry under radiative corrections in the minimal seesaw framework  (2609.01196 - Pegu et al., 1 Sep 2026) in Section 1, Introduction