Regular Alena-Urbantke Geometries and Local Gauge-Sector Equivalence

A regular Alena-type visible tensor is shown to carry more geometry than is seen at the level of its relativistic image. In the four-dimensional self-dual setting, the visible tensor is represented by a Hermitian endomorphism of the complex self-dual bundle, and the same field determines the traceless part of the induced metric variable. On the regular locus, defined by positivity, full rank, and simple spectrum, this Hermitian field has three distinct positive eigenvalues and therefore determines a canonical ordered spectral flag. The associated self-dual eigenbasis defines a natural carrier geometry of Urbantke type, but this geometry does not in general coincide with the induced metric branch. A local equivalence is established between regular gauge-sector representations of the visible tensor and regular Alena-Urbantke geometries. In this form, the regular visible sector is organized by spectral data, flag geometry, and an induced tensor sector on the same self-dual bundle. Its natural reduced dynamics is written on the positive Hermitian cone, with kinetic term expressed in a logarithmic carrier variable. This separates common carrier scale, anisotropic spectral motion, and flag motion. On the first symmetry-enhancing boundary stratum, a distinguished local 2+1 reduction appears with residual U(2) x U(1) symmetry and a canonical carrier doublet. The same reduced cone sector also admits a natural amplitude lift: the positive Hermitian field arises as the Gram field of a carrier amplitude, the cone kinetic term is recovered from the Hermitian part of the corresponding multiplicative current, and the complementary anti-Hermitian part carries the orbital and gauge sector.