Carrier–Triplet Bridge

This theorem constructs an exact intertwiner between the triadic carrier module C_X = span{e₇, e₆, e₂} established in Theorem 21 and the canonical spin-1 triplet factor T₃ = span{u₂, 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 J_z, J_x, J_y satisfying the canonical commutation relations. (2) A Lie algebra isomorphism Ψ: C_X ⊗ℝ ℂ → span{K_x, K_y, K_z} is constructed with signs fixed canonically by the T21 orientation convention σ_Γ = −1, via Ψ(e₇) = K_x, Ψ(e₆) = −K_y, Ψ(e₂) = K_z. (3) A module isomorphism Φ: C_X ⊗ℝ ℂ → T₃ ⊗ℝ ℂ is constructed satisfying the intertwining identity Φ(ad_X Y) = Ψ(X)Φ(Y) for all X, Y in C_X ⊗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)