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 KA and KB, forces the branch operator B: = 2ΠA − I to flip sign, and reverses the defect generator Ed = iσᵧ. 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 (Mₙ ↔ Md^ι, Dₙ ↔ Dd^ι, X₍→₃ ↔ X₃→₍) and reversing operator ordering, giving Θ (UA − UB) = − (UA − UB) under the canonical T18 orientation convention. Three results follow: (1) ΘBΘ⁻¹ = −B; (2) ΘΠAΘ⁻¹ = ΠB and ΘΠBΘ⁻¹ = ΠA; (3) Θ² = I on the binary branch space of Λ₄. Conditional on Θ|₃₄₅₄₂ₓ ₏₋₀₍₄ being orientation-reversing on span|8⟩, |24⟩: ΘEdΘ⁻¹ = −Ed, since any orientation-reversing S ∈ O (2) satisfies SEdS⁻¹ = −Ed 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 (UA − UB) = − (UA − UB) 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. ΘEdΘ⁻¹ = −Ed 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.