This theorem constructs an exact intertwiner between the triadic carrier module CX = spane₇, e₆, e₂ established in Theorem 21 and the canonical spin-1 triplet factor T₃ = spanu₂, u₃, u₄ of the Theorem 22 decomposition H₁₂ ≅ V₄ ⊗ T₃, after complexification. Three propositions are established: (1) T₃ ⊗ℝ ℂ carries the standard spin-1 representation of su (2) via explicitly defined operators Jᵦ, Jₓ, Jᵧ satisfying the canonical commutation relations. (2) A Lie algebra isomorphism Ψ: CX ⊗ℝ ℂ → spanKₓ, Kᵧ, Kᵦ is constructed with signs fixed canonically by the T21 orientation convention σ_Γ = −1, via Ψ (e₇) = Kₓ, Ψ (e₆) = −Kᵧ, Ψ (e₂) = Kᵦ. (3) A module isomorphism Φ: CX ⊗ℝ ℂ → T₃ ⊗ℝ ℂ is constructed satisfying the intertwining identity Φ (adX Y) = Ψ (X) Φ (Y) for all X, Y in CX ⊗ℝ ℂ, via Φ (v₊) = u₂, Φ (v₀) = u₃, Φ (v₋) = −u₄. The main theorem combines these results: the T21 carrier adjoint module and the T22 spin-1 triplet module are canonically intertwined after complexification, yielding an explicit algebra and module equivalence. Both maps are canonical and free of external input; signs are fixed by σ_Γ, not chosen freely. All verifications are by direct matrix computation. This theorem closes the loop between the symmetry descent (T21) and the state decomposition (T22): loop structure ↔ symmetry ↔ representation is fully closed. Status: Solid. All results were established by direct matrix computation and weight-space matching. Dependencies: T18, T19, T20, T21, T22 (direct)
Craig Edwin Holdway (Sat,) studied this question.