Regular Alena-Urbantke Geometries and Local Gauge-Sector Equivalence

A geometric normal form for the visible sector of Alena-type energy-momentum tensors in four-dimensional Lorentzian geometry is established. The traceless Ricci-type content of such tensors is encoded, via a curvature-type lift and self-dual reduction, by a Hermitian endomorphism of the complex self-dual bundle. A regularity condition (positivity, full rank, and simple spectrum of that endomorphism) selects a class of structures termed regular Alena-Urbantke geometries, in which a canonical ordered spectral flag and a natural chiral carrier geometry emerge. The main result is a local equivalence theorem: regular gauge-sector representations of the visible tensor correspond precisely to regular Alena-Urbantke geometries, and each determines the other locally. This formulation separates the visible self-dual block from the remaining hidden curvature data, clarifies the role of the Urbantke construction as a carrier geometry distinct from the induced metric variable, and identifies the degeneration loci at which the regular structure breaks down. Global existence of a regular visible sector is shown to require a splitting of the self-dual bundle with characteristic-class consequences. To support explicit computation, a Mathematica tool for spectral and regularity diagnostics of the visible Hermitian block is provided, together with an example report.