This Theorem closes the second open assumption of the T39 bridge lemma by deriving the leg-matching identity a = √2‖vL‖ + O (δ²ₑff) from Q5 carrier structure. Two independent Q5-native mechanisms force the leading-order identity. The complement involution ι: x ↦ 1 − x maps the Hamming-weight-1 subspace isometrically onto the Hamming-weight-2 subspace, intertwines the T24 ambient su (2) action by sign-reversing Ĵᵦ and exchanging Ĵ_±, and satisfies ι (vL) = vR with ‖vR‖ = ‖vL‖. Within the spin-1 module V₁ carried by the Hamming-weight-1 sector, the raising operator Ĵ₊ acts with matrix element √2 between adjacent weight states, a standard spin-1 Clebsch-Gordan coefficient. Combining: at leading order a = √2‖vL‖, where the √2 is the canonical spin-1 matrix element, not a normalization convention. The Y-transport correction is quadratic not linear: since θ = π/2 + O (δₑff) from T43, the amplitude scales as sin θ = cos (O (δₑff) ) = 1 − O (δ²ₑff), giving a = √2‖vL‖ + O (δ²ₑff). The leading-order result is stable to first order in the off-support leakage. Together with T43, both open assumptions in the T39 bridge lemma are now derived results. Status: Complement isometry solid: direct computation on the residual carrier basis. Spin-1 matrix element solid: standard result. Leading-order leg-matching conditional on identification of transport leg amplitude with Ĵ₊ matrix element on V₁ is structurally natural within the T24–T25 framework but not yet derived from explicit kernel-level transport generators. Quadratic correction inherits T43 parametric control conditionality. All results inherit T22–T25 and T39 conditionality. Dependencies: T22, T24, T25, T39, T43.
Craig Edwin Holdway (Sun,) studied this question.