Local field-theoretic origin of the internal three-state holonomy

Identify a local field-theoretic origin for the internal three-state space and its loop phase in a $3+1$-dimensional embedding while preserving the chiral gauge structure of the Standard Model.

Background

The configuration-space winding mechanism used in the one-dimensional model does not directly survive as the same ordinary puncture winding in $3+1$ dimensions. The paper therefore proposes treating the cyclic structure as an internal three-state holonomy instead of as literal particle winding around a removed coincidence point.

A concrete unresolved issue is how to generate this internal three-state space and its rephasing-invariant loop phase from a local field theory, while retaining the chiral gauge structure required for a Standard Model embedding.