Establish or test the quantized second-torus phase conjecture

Establish whether a two-torus completion can share the flavor base and place the phase-carrying second modulus at the level-four quarter shift, thereby quantizing the CKM phase as \(\delta=n\pi/8\) and selecting the phenomenologically relevant value \(n=3\).

Background

The flavor modulus is constrained to the imaginary axis, so its modular-form values are real and cannot supply the observed CKM phase. The paper therefore introduces a second torus whose plaquette contributes both a suppression on the same lattice and a phase. With an unconstrained second-torus shape, the phase is fitted rather than predicted.

The authors propose a stronger, explicitly conjectural condition: the two tori share the same base and the second modulus has real part $1/4$, the quarter shift singled out by level four. Under these assumptions the phase becomes quantized in eighths. The value $n=3$, giving $\delta=3\pi/8$, remains a conjectural alternative and is to be distinguished from the continuous closure prediction by improved measurements of the CKM angles.