Uniqueness, Morse index, and nullity of Carlotto–Schulz minimal hypersurfaces

For the Carlotto–Schulz minimal embeddings of Sⁿ⁻¹ⁿ⁻¹¹ in S²ⁿ, we prove uniqueness and nondegeneracy of the normalized monotone closing profile. The first positive Laplace eigenvalue is $2n-1$, and its eigenspace consists of ambient coordinate functions. The full Morse index is $27$ for $n=2$ and (n³+9n²+11n+3)/3 for n≥3; the nullity is n²+2n, and every Jacobi field is induced by a rotation. The scalar argument also applies to graphical doubles with unequal spherical factors. Uniqueness for n≥3 follows from a monotone Jacobi-residual ratio and compact continuation. The kernel theorem gives local rigidity and, for nearby bumpy ambient metrics, minimal hypersurfaces of every index from the base index through that index plus n²+2n. In S⁴ there are at least twenty-one such three-tori.