Determine the form selection and vacuum realization of the modular operators

Construct a ultraviolet completion of the level-four modular framework that selects the required modular forms at each weight, fixes the Kähler split of the automorphy weights, supplies the partner of the metaplectic third-generation charged lepton, and generates the texture zeros required by the quark mixing structure.

Background

The operator charges and leading assignment determine the nome gradings only modulo eight. They do not determine which elements of the higher-weight modular-form spaces are selected, how total automorphy weights are divided between holomorphic forms and Kähler normalization, or how the metaplectic third-generation charged-lepton component is completed.

The same unresolved data must also account for the texture zeros underlying the Gatto–Sartori–Tonin relation and the CKM construction. The paper treats these issues as part of the vacuum-selection problem for a magnetized-torus or related ultraviolet completion.

References

What remains open is the form selection at each weight, the Kähler split of the automorphy weights, the partner of the metaplectic $ec_3$, and the texture zeros, which together constitute the vacuum-selection problem of Sec.~\ref{sec:uv}.

— Level-Four Modular Symmetry Selected by the Harmonic Pattern of Quark and Lepton Masses  (2609.09262 - Barger, 8 Sep 2026) in Appendix A, subsection “What charges do not fix”