arXiv:2611.14747v1 Announce Type: new Abstract: We study how many polarized anti-invariant factors of the intermediate Jacobians of smooth Eckardt hyperplane sections determine a cubic fourfold with a symplectic involution. In rank three, the equation is determined by its fixed cubic surface and a conic in the dual projective space. The Torelli theorem of Casalaina-Martin--Zhang recovers the surface together with one plane section from each factor. When the surface has no automorphisms, these planes can be compared in a common dual space, and reconstruction reduces to classical conic interpolation. Five factors suffice without parameter data, and four suffice with their ordered parameter configuration; both bounds are sharp. In rank two, we construct two-dimensional families of pairwise nonisomorphic equivariant fourfolds with the same entire unparametrized Prym image. For a fixed fourfold, we also show that the closure of the anti-invariant factor family in an associated OG10 compactification is an integral projective Lagrangian subvariety.
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