Invisible Non-Abelian Holonomy: Two Parent Worlds, One Einstein–Maxwell Solution

We construct two normalized parent configurations that yield exactly the same gravitational and electromagnetic fields but have different internal holonomy. Here a parent configuration is a more detailed set of variables from which the classical fields and an internal connection are calculated. This separates consistency with the Einstein–Maxwell equations from uniqueness of the accompanying internal geometry. Within a specified classical field model, a smooth family above one local dyonic Reissner–Nordström solution preserves all electromagnetic, gravitational and auxiliary fields and an open neighbourhood of the exterior boundary. One endpoint has trivial restricted holonomy, whereas the other has the full non-Abelian group SU(2). Explicit curvature matrices certify that this distinction survives internal frame changes. A separate homogeneous local solution supplies a positive compatibility result: one normalized parent carries full SU(2) holonomy and a nonzero electromagnetic field, with a metric satisfying the gravitational constraints and short-time evolution equations. These constructions provide exact examples for testing whether reconstruction inputs distinguish different internal geometries, together with a coupled realization for checking the complete electromagnetic contribution to gravitational backreaction. They supply reproducible mathematical benchmarks for geometric model building and for evaluating the additional information needed to identify a specified parent structure.