The Triality Quotient Paper

Abstract

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 = B_F ⊕ ε²B_R. 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.