Electromagnetic Helicity from an Induced Line Connection Exact Chern–Simons Identities and a Conditional MTT Encoding

Abstract

A smooth rank-one orthogonal projector on a Hermitian bundle defines a line subbundle and an induced unitary connection. This elementary construction provides a precise setting in which Berry curvature, Abelian Chern–Simons functionals, magnetic helicity, and Hopf invariants can be compared. We derive the induced curvature, including its projector term, and formulate helicity only after specifying a global trivialization or a relative-helicity protocol. The exact slice balance is the usual electric–magnetic pairing plus an explicit boundary flux. There is no additional projector remainder when the electric and magnetic fields are those of the full induced connection: projector variation is already part of those fields. A quantitative comparison theorem bounds the error made when the Berry curvature is omitted and only the ambient Abelian field is retained. We also give a correctly scaled Riesz-projector derivative estimate and state the Hopf normalization with all conventions visible. Modal Triplet Theory (MTT) can use this construction as a conditional encoding once a selected source emits the Hermitian bundle, connection, rank-one projector, physical field identification, and boundary data. Current MTT results do not yet establish that source theorem for the physical electromagnetic sector. The paper therefore proves an exact geometric dictionary and a controlled comparison result, not a universal electromagnetic prediction.