The extreme points of the two-local three-qubit spectrahedron

We classify the extreme points of the spectrahedron S of positive semidefinite 8x8 Hermitian matrices of fixed trace in the real span of the identity and the Pauli strings of weight at most two. The set S is dual to the joint numerical range of these 36 matrices, which is the set of two-body marginals of three-qubit states. We prove that every extreme point of S has rank two, three or four and that all three ranks occur. The proofs use an antiunitary spin flip whose square is -1 and the product vectors in the kernel. Rank two consists of a quaternionic class, diffeomorphic to HP^3, and three classes diffeomorphic to CP^3. The kernel of a rank-three extreme point contains a twisted cubic, a conic and a line, or a chain of three lines of product vectors. These strata have dimensions 13, 11 and 9, the last two with explicit extremality conditions, and each point is a combination of three two-qubit projectors with exactly one negative coefficient. Rank-four extreme points are open and dense in a smooth 20-dimensional manifold with four components, parametrised by the weight-one part and an isotropic subspace, and are generically chiral. The normal cones of the rank-three points cover a set of positive measure. Parts of the proofs are exact computer-algebra computations; the scripts are archived at https://doi.org/10.5281/zenodo.23982134. Generative AI tools were used in this work, as described in Section 6.