Para-Hermitian Metric Compatibility as an Obstruction to Lorentzian Gravity

This work examines whether a para-Hermitian metric with neutral signature can serve directly as the gravitational metric experienced by light and matter while ordinary Lorentzian spacetime emerges on a lower-dimensional physical section. A general dimensional obstruction is derived: when a physical section has dimension (D>n) inside a (2n)-dimensional para-Hermitian geometry, it necessarily inherits null directions fixed by the para-complex structure and the section, independently of the dynamical metric. For the physically relevant (6\rightarrow4) case, this leads to fixed null directions, a reduced background symmetry, and strong restrictions on static gravitational solutions. The work also identifies conditions under which the obstruction is avoided and explains why double-field and Born-geometric formulations, where the neutral pairing and physical generalized metric are distinct structures, are not subject to the same problem. Consequences for multi-time and timelike Kaluza–Klein constructions are discussed.