Gauge-Coupling Normalization Beyond Projector Geometry: Action-Level Typing and Source Factorization

This paper examines how effective gauge fields and their coupling strengths arise from distinct mathematical structures. A smoothly selected internal subspace determines the geometry of an emergent gauge connection, while the normalization of its gauge kinetic term depends on independently specified microscopic response and curvature-source data. The analysis shows that spectral selection and collective response must be separately typed: an exact spectral projector cannot, in its own spectral decomposition, generate the nontrivial light-heavy mixing required for a Schur-complement response. An equivariant curvature-source embedding then determines the source-induced gauge kinetic normalization, while symmetry fixes its algebraic form but not its overall magnitude. A factorized relation between the projected response weight and gauge coupling is recovered only when an additional microscopic common-source condition is satisfied. The result provides a precise framework for separating geometric gauge emergence, collective response, and dynamical gauge normalization in projector-defined effective theories.