Branch Exchange Under Boundary Inversion and Orientation-Reversing Action on the Defect Generator

This Theorem establishes that the orientation-reversing boundary map Θ exchanges the two ordered kernel branches K_A and K_B, forces the branch operator B := 2Π_A − I to flip sign, and reverses the defect generator E_d = iσ_y. These results give the first theorem-level precision to the claim that the orientation-reversing boundary does not create a second independent architecture but exchanges the two observable orientations of one retained structure. From the T17 ordered branch structure, Θ acts by exchanging native and dual operators (M_n ↔ M_d^ι, D_n ↔ D_d^ι, Xn→d ↔ Xd→n) and reversing operator ordering, giving Θ(U_A − U_B) = −(U_A − U_B) under the canonical T18 orientation convention. Three results follow: (1) ΘBΘ⁻¹ = −B; (2) ΘΠ_AΘ⁻¹ = Π_B and ΘΠ_BΘ⁻¹ = Π_A; (3) Θ² = I on the binary branch space of Λ₄. Conditional on Θ|defect plane being orientation-reversing on span{|8⟩, |24⟩}: ΘE_dΘ⁻¹ = −E_d, since any orientation-reversing S ∈ O(2) satisfies SE_dS⁻¹ = −E_d for the rotation generator. The observable consequence is μ ∈ {+μ₀, −μ₀}, a discrete binary orientation branch, not a continuous family of independent structures. Status: Θ(U_A − U_B) = −(U_A − U_B) derived from T17 operator structure, conditional on transport wrapping sign η = +1 under T18 orientation convention, explicit verification requires full derivation of Θ from T17 kernel geometry (primary open target). ΘBΘ⁻¹ = −B and projector exchange derived given the branch exchange lemma. Θ² = I solid. ΘE_dΘ⁻¹ = −E_d additionally conditional on the orientation-reversing restriction of Θ to the defect plane. μ ∈ {+μ₀, −μ₀} strongly suggested, not yet proved at full kernel level. All results inherit T17, T18–T20, T36–T37 conditionality. Dependencies: T17, T20, T21, T29, T36, T37, T46.