Ultraviolet Matching Across Coherence Boundaries in Projector-Defined Gauge Theories: Spectral Persistence, Response Transport, and an Application to the Axis Model

This paper develops an ultraviolet matching framework for effective gauge theories whose gauge fields arise from smoothly selected internal subspaces. It distinguishes the breakdown of an approximation from the loss of a protected spectral projector or reconstructed spacetime geometry and identifies separate scales for approximation control, gauge-sector definability, metric definability, coherent reconstruction, and microscopic matching. An explicit constant-gap construction shows that substantial projector deformation does not by itself eliminate the selected gauge sector. When microscopic evolution mixes a coherent sector with its complement, valid matching requires transporting the full microscopic response, or an independently established complete invariant, before reconstruction and projection. The analysis further separates spectral selection, collective response, and gauge-source normalization, demonstrating that familiar factorized coupling relations require an additional common-source theorem. Applied to the Axis Model, these results define a conditional ultraviolet-to-effective-field-theory matching architecture without claiming an independently derived weak-sector projector, source normalization, persistence mechanism, or completed ultraviolet theory. The deposit includes the matched manuscript PDF, LaTeX source, and bibliography.