Boundary Orientation Reversal and Branch Exchange

This paper establishes how the boundary crossing map Θ (native-dual orientation reversal) acts on the binary branch structure of the retained observable sector. The central results are three conjugation identities: \ ΘBΘ⁻¹ = −B \ (branch operator sign flip) \ ΠA ↔ ΠB \ (binary projector exchange) \ ΘEdΘ⁻¹ = −Ed \ (defect generator sign flip) The structural chain is: Θ forces B → −B, which forces ΠA ↔ ΠB, which forces Ed → −Ed. Observable phase reversal is not primitive; it is a projection of branch exchange occurring one level higher in the structure. The derivation proceeds from the kernel branch pair (T17), through the action of Θ on native/dual and transport ordering, to the retained observable sector via the extraction projectors of T20–T38. The branch exchange and its induced actions are solid given the Θ-action hypothesis. The open item is the explicit construction of Θ from T17 kernel geometry, the transport wrapping sign η under the T18 orientation convention (T47 open frontier). Once that construction is established, all downstream results upgrade to solid.