No supersymmetric vacua of Castellani's twelve-dimensional supergravity on M4 x K8 with maximally symmetric factors in the standard embedding

Castellani constructed a locally supersymmetric action for twelve-dimensional supergravity in signature (10,2) as the supertrace of the sixth power of the curvature of an OSp(1|64) connection (arXiv:1705.00638); its solutions and their supersymmetry had not been analysed. This note considers backgrounds M4 x K8 with maximally symmetric factors, every split of the two times, and every field allowed by the isometries in the standard embedding: the vielbein, the Levi-Civita connection and two constant fluxes (a five-index and a nine-index field). Supersymmetry needs flat factors and then holds, with 16 of 32 supercharges, exactly at discrete flux values when the 4D factor has signature (2,2) or (0,4); with one time in the 4D factor it is impossible. At every such point the bosonic field equations (vielbein, five-index and nine-index) fail, as shown in exact arithmetic. So no vacuum of this class preserves supersymmetry. Flux-free product vacua of the class exist (for example with unit-curvature K8, where the equations hold trivially) but break all supersymmetry. Not covered: non-standard embeddings of the isotropy group, other dimension splits, non-maximally-symmetric factors. A short no-go note. The plan was hashed before the computation and revised after a blind attack; the paper was reviewed before upload by blind AI (Claude) referees without access to the code, one of which reproduced the result with its own code. Code, outputs, plans and review ledger are in the archive.