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

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 found, numerically, supersymmetric solutions that are not trivially solved on flat M4 of signature (2,2) times the group manifold SU(3) with the adjoint embedding: 16 constant Killing spinors on a family with five free coefficients, and two solution branches with the B(9) coupling kappa9 at plus or minus 1/3 and 1/5. Version 4 gives these branches exactly, from computations modulo primes checked at random points (probabilistic certificates). At kappa9 = s/3 the field equation on the supersymmetric family collapses to a single polynomial equation F = 0 of degree 10 with rational coefficients, irreducible over Q (682 terms, deposited); F changes sign at rational points, so real solutions exist exactly, and a modular resultant test shows that every solution component of dimension at least 4 on the family has this value of kappa9. At kappa9 = -s/5 with mu1 = 0 the field equation collapses to two equations G1 = G2 = 0 (deposited), with G1 = (beta - gamma) H; the solution set consists of two two-dimensional pieces, one of which (beta = gamma) was missed by the numerical search of version 3. Why the rank drops at these values is not derived. Plans hashed before computation, a blind plan attack, blind review rounds and all code, outputs and polynomials are in the deposit. AI use: computation, text and reviews were carried out with Claude (Anthropic); the plan attack and the referees were separate AI instances without access to the code.