The Triality Quotient Paper

Objective. We construct four-dimensional conformal spacetime from a marked split-real D4 datum by degeneration, radical quotient, spinor restriction, and projective null geometry. Why (4, 2;2). Let E = F ⊥ R, with F carrying signature (4, 2), R carrying signature (0, 2), and dim R = 2. Set Bε = BF ⊕ ε²BR. For ε > 0 this has signature (4, 4) ; at ε = 0, the radical of B₀ is exactly R, and the canonical quotient U = E/R has signature (4, 2). Thus Spin₀ (U) ≅ Spin₀ (4, 2) ≅ SU (2, 2), while the projective null cone of U compactifies Minkowski space. Two-plane geometry. Degenerating a fixed negative-definite two-plane is a different construction from varying isotropic two-planes in the orthogonal Grassmannian OG (2, 8): their Plücker lines land in disjoint parts of the adjoint variety, giving semisimple and minimal-nilpotent types respectively. The stabilizer-spinor incidence bridge. Restricting to the block subgroup Spin (F_ℂ) × Spin (R_ℂ) recovers the two chiral twistor modules T and T* as weight-multiplicity spaces inside the parent half-spin representations. On the rank-one locus X₁ = planes L: dim (L ∩ R_ℂ) = 1, quotienting gives a regular surjective map β: X₁ → ℙN (U_ℂ) that is not injective. With chosen enhanced spinor data, this null quadric is presented as the Grassmannian Gr (2, T) and, dually, as Gr (2, T*). Flatness and symmetry. A projective, flat, local-complete-intersection family specializes OG (2, 8) to a reduced scheme X₀ containing X₁. Separately, a free rank-28 Lie algebra family specializes inside the 31-dimensional stabilizer of the boundary form B₀, and its image acts on U through the conformal algebra so (4, 2). Scope. The construction is algebraic and kinematic, not dynamical. No nontrivial continuous homomorphism from Spin₀ (4, 4) to Spin₀ (4, 2) exists, and no map induced by the isotropic branch from OG (2, 8) to Gr (2, 4) is constructed - indeed, the exterior-square image of every plane in X₁ vanishes identically.