Generalized Minkowski Theorem for Tetrahedra in ${\mathrm{d}\mathrm{S}}^3$ and ${\mathrm{A}\mathrm{d}\mathrm{S}}^3$

We formulate and prove a constant-curvature, holonomy-valued Lorentzian analogue of Minkowski theorem for generalized tetrahedra in the constant-curvature Lorentzian spaces ${\mathrm{d}\mathrm{S}}^3$ and ${\mathrm{A}\mathrm{d}\mathrm{S}}^3$. Four non-trivial based ${\mathrm{S}\mathrm{O}}^{+}(1,2)$ holonomies, or equivalently $\mathrm{S}\mathrm{L}(2,\mathbb{R})$ spin lifts, determine intrinsic face normals, a dihedral Gram matrix $G$, and oriented triple products of intrinsic face normals. Under closure, nondegeneracy, and the outward convex branch condition, these data reconstruct a unique strictly convex tetrahedron up to ambient isometry. The sign of $\det G$ selects the de Sitter or anti-de Sitter model, and the prescribed holonomies are exactly the based Levi-Civita face holonomies of the reconstructed tetrahedron. The extrinsic face normals also define a polar-dual projective tetrahedron. In particular, the all-null AdS sector gives ideal dual tetrahedra, and the all-timelike AdS sector gives hyperideal dual tetrahedra. In the all-spacelike sector, changing to $\mathrm{S}\mathrm{U}(2)$ real form recovers the reconstruction theorem for Euclidean spherical and hyperbolic tetrahedra.