Supersymmetric product vacua of Castellani's twelve-dimensional supergravity with maximally symmetric factors in the standard embedding: none on M4 x K8 or M2 x K10, and on M6 x K6 only trivially solved ones

Version 2. 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). This note asks whether it has supersymmetric product vacua with maximally symmetric factors in the standard embedding, for every split of the two times. Version 1: on M4 x K8 supersymmetry needs flat factors and holds (16 of 32 supercharges) only at discrete flux values when the 4D factor has signature (2,2) or (0,4); at all of them the field equations fail (exact arithmetic). No supersymmetric vacuum. Version 2 adds the other even splits. On M2 x K10 no background with an invertible vielbein preserves supersymmetry. On flat M6 x K6 with one time in each factor, 16 supercharges survive at four flux points, where the field equations hold but trivially: the curvature satisfies R wedge R = 0, the field equations and their linearisation vanish identically, and the points lie on lines of trivially solved non-supersymmetric flux solutions (found by a blind referee of the draft); similar non-supersymmetric ones exist on M2 x K10 with signature (1,1) x (1,9). Whether such trivially solved backgrounds count as vacua is left open. Not covered: non-standard embeddings, odd splits, non-maximally-symmetric factors. A short no-go note with one trivially solved exception. Version 1 stays available. Plans were hashed before each computation and revised after blind attacks; each version was reviewed before upload by blind AI (Claude) referees without access to the code, one of which reproduced the results with its own code. Code, outputs, plans and review ledgers are in the archive.