Nondegenerate product vacua of Castellani's twelve-dimensional supergravity action

Castellani proposed an action for supergravity in twelve dimensions of signature (10,2), built from the curvature of an OSp(1|64) connection, S = int STr(R^6 Gamma), and left the analysis of its field equations, their solutions and the counting of degrees of freedom for later work. We point out that its vielbein and spin connection sector is exactly twelve-dimensional Born-Infeld gravity, so that around its maximally symmetric vacuum the action has no terms below sixth order in the fluctuations and nothing propagates at linear order. On products of constant-curvature spaces, with all other fields set to zero, every bosonic field equation except the vielbein equation vanishes identically, and the remaining equations reduce to explicit polynomials. For M4 x K8 there is a degenerate branch (internal space at the vacuum curvature, four-dimensional curvature free, reduced four-dimensional action identically zero) and a single nondegenerate solution, on which the internal curvature is exactly twice the four-dimensional curvature, k4/k0 = 2 + [(2+sqrt3)^(2/3) + (2-sqrt3)^(2/3)]/2 = 3.4108, and the reduced action for the four-dimensional metric at fixed internal space contains a nonvanishing Einstein term. We prove that for any product of two constant-curvature factors, in any even dimension, every solution with neither factor at the vacuum curvature has equal curvature per dimension, k1/d1 = k2/d2, which reduces the vacuum problem to a single polynomial equation; for a 2+(D-2) split the solution is k1 = 2(D-1)k0/(D-2), k2 = (D-1)k0, and the rule fails for three factors. Numerically, such solutions exist exactly for even-even splits, one per split, for all d1+d2 <= 24. The geometric reading depends on the sign convention of the Clifford algebra, which differs between versions of Castellani's paper: the maximally symmetric vacuum is SO(10,3)/SO(10,2) (AdS-type) in one and SO(11,2)/SO(10,2) (dS-type) in the other, with identical numbers. In both, on this solution a four-dimensional spacetime with one time forces a noncompact internal space, while a compact Riemannian internal space forces a four-dimensional spacetime of signature (2,2). Every number is reproduced by deposited scripts. Version 2 adds the proof of the equal-curvature-per-dimension rule for two factors (observed but not proved in version 1), the closed-form 2+(D-2) solution, three-factor counterexamples, and two new scripts (kd_theorem.py, threeblock.py). It also corrects a stale equation reference in the README and a script comment. Version 1: 10.5281/zenodo.23001604. Source package: Python scripts (numpy only) reproducing every number in the paper, a README mapping each result to its script, and a withdrawn pre-registration for a follow-up spectrum computation with its hash.