Horizon Residues, Matter-Trace Winding and Test-Scalar Modes in f(Q,T) Gravity

Abstract

We extend the analysis of local horizon remnants in quadratic non-metricity gravity to a specific matter trace theory with the functional form f(Q) = Q + alpha Q^2 + 16 pi lambda T. Here, the matter trace is defined by contracting the energy-momentum tensor with the metric from the original matter action. In the absence of explicit lambda-corrections, the theory inherits the vacuum logarithmic-connection hairy branch. To obtain a non-zero trace, we introduce polynomial non-linear electrodynamics and construct an exact magnetic branch in coincident Cartesian Kerr-Schild coordinates. The non-metricity scalar of this branch vanishes, while its matter trace is proportional to the inverse eighth power of the distance. Its metric component function is given by 1 - 2M/r + q^2/r^2 + C/r^6, where the correction term C equals xi(1 - 4 lambda)q^4 / 10. Consequently, the quadratic non-metricity coupling is sterile on this branch. We derive the metric and matter equations, the Euclidean temperature, the single-horizon remnant, the master scalar equation, and the first-order gauge shift. The logarithmic derivative of the matter trace imparts a winding number of -8 on a closed contour enclosing the central singularity, but a winding number of zero on a sufficiently small horizon contour. Nevertheless, the temperature and scalar propagation still vary continuously with the correction coefficient. Therefore, the trace winding number classifies divisor data rather than superseding the matter trace or determining its backreaction. This analysis also demonstrates a degeneracy between the trace coupling coefficient and the non-linear electromagnetic coefficient. These results are specific to this particular branch; this paper does not expound upon the full coupled gravity spectrum, nor does it claim priority for this magnetic metric.