Residual Symmetries and BRST Cohomology of Schwarzschild in the Kerr-Schild Double Copy

The Kerr-Schild double copy is celebrated for producing exact gravitational spacetimes from gauge fields, yet the preservation of symmetry content remains largely unexplored. We investigate the fate of residual symmetries in the Kerr-Schild double copy, focusing on the Schwarzschild solution. On the gauge theory side, we derive the residual transformations that preserve the Abelian and non-Abelian Kerr-Schild ansatzë, finding they both form an infinite-dimensional Lie algebra parameterized by arbitrary null functions. On the gravity side, we analyze the resulting residual diffeomorphisms of the Kerr-Schild Schwarzschild metric. Restricting our focus to the Killing vector class of solutions, we find that the only surviving diffeomorphisms are the finite-dimensional global isometries of Schwarzschild, reducing the residual gauge algebra to the Poincaré subalgebra containing exclusively time translations and spatial rotations. This finding confirms a fundamental structural mismatch: the infinite-dimensional algebra of the gauge side admits no analogous structure in this gravitational sector. We formalize this by showing that the BRST operator for the residual symmetry is trivialized under the Killing condition, establishing the consistency of this algebraic reduction within a quantum field theoretic framework. This paper is the first of a two-part series. In the second paper, we complete this analysis by examining the more complex proper conformal Killing vector class of solutions and formulating a unified BRST framework.