The D₄ Bridge as Perpendicular Span: Three Theorems on Orbit-Stabilizer Incidence

We prove three theorems concerning the relationship between the 24 Hurwitz units (binary tetrahedral group 2T) and the 24 grade-1 roots of E7. Theorem A (Orbit-Stabilizer): the 24 grade-1 roots form a single orbit under the Levi Weyl group WL = W(A2 × A1 × A3) of order 288, with stabilizer D6 ≅ S3 × C2 of order 12, giving 288/12 = 24. Theorem B (No-Go): there exists no injective group homomorphism 2T → WL, by Goursat’s lemma applied to the abelianization obstruction Ab(2T) ≅ ℤ3 vs. the ℤ2 factor of WL. Theorem C (Span): the D4 bridge is a span—two quotient maps from 2T into separate factors of WL—not an embedding. The common 6×4 block-fiber decomposition is proved to arise from distinct group-theoretic sources on each side.