The Singular Locus of the Spin14–invariant Octic Is the Third Secant Variety of the Spinor Variety

Let S7 ⊂ P(Δ+) = P^63 be the spinor variety of Spin(14) and let F8 be the invariant octic on the half-spin representation Δ+. We prove that the singular locus of the octic hypersurface {F8 = 0} is, set-theoretically, the third secant variety σ3(S7), and that the scheme defined by the gradient of F8 is reduced at the general point of σ3(S7). Consequently, a pure even-parity state of seven fermionic modes has Gaussian border rank at most three if and only if the 64 partial derivatives of F8 vanish on it. We also determine the Gaussian rank and border rank of every pure even-parity state of seven modes, except for the states in one orbit, where the rank is 3 or 4; in particular, rank and border rank already differ for seven modes. Along the way we prove that σ3(S7) has dimension 58 (defect 5), confirming with a certified proof the value computed by Angelini; that its ideal I contains no forms of degree at most 6; that I7 is spanned by the 64 partial derivatives of F8; and that I8 = Δ+* · I7 (dimension 4005), so there are no new generators in degree 8. We conjecture that the 64 derivatives of F8 generate the ideal of σ3(S7). The proofs combine an extremal projector of Asherova–Smirnov–Tolstoy (used to count highest weight vectors in the coordinate ring of a secant variety), a stabilizer argument at a point of the dense orbit, Popov's classification of Spin(14)-orbits on the 64-dimensional half-spin representation, and computations modulo the prime 2^31 − 1 that are certified: modular ranks are used only as lower bounds for rational ranks. The supplementary material (Python/NumPy programs, a short C routine, a README with the exact commands and random seeds, and the complete output of all computations) reproduces every computational result in the paper.