Supersymmetric backgrounds of Castellani's twelve-dimensional supergravity: none on products of maximally symmetric spaces in the standard embedding, and numerically found solutions that are not trivially solved on M4 x SU(3) with the adjoint embeddi

Castellani constructed a locally supersymmetric action for twelve-dimensional supergravity in signature (10,2) (arXiv:1705.00638). Versions 1 and 2 of this note show that supersymmetric product vacua with maximally symmetric factors in the standard embedding are absent on M4 x K8 and M2 x K10 and only trivially solved (Y = 0) on M6 x K6. Version 3 tests a non-standard embedding: flat M4 of signature (2,2), which carries both times, times the group manifold SU(3) with the isotropy group acting by the adjoint representation. With every invariant constant field, 16 constant Killing spinors (16 of 32 real supercharges) exist on a family with five free coefficients for every radius. The field equations have solutions on it, found numerically, with one solution re-evaluated by separately written code: a branch with the B(9) coupling at plus or minus 1/3 (to fifteen digits), numerically four-dimensional near the solutions tested, with solutions at every radius tried, and a branch at plus or minus 1/5, numerically two-dimensional. These solutions are not trivially solved: the curvature R and Y are nonzero and the first-order symbol of the field equation does not vanish; R^6 and the on-shell Lagrangian vanish at the solutions checked. The solutions are for the metric sign of the latest arXiv version of the action; with the opposite sign of its earlier versions the same real coefficients fail, and that sign was not searched. Deviations from the hashed plans, the blind review rounds and all code and outputs are in the deposit. AI use: computation, text and reviews were carried out with Claude (Anthropic); the referees and the re-evaluation were separate AI instances without access to the code.