Carrier Axis Selection and Rotational Phase Emergence

This paper establishes the structural origin of the rotational generator R and the scalar phase µ from branch-exchange geometry. Given the terminal retained support Λ₄, orientation-sign invariant σ_Γ = −1, binary branch decomposition (KA, KB), canonical projector ΠA, and triadic carrier \ CX = spane₇, e₆, e₂ \, the branch-exchange involution J acts on CX as an orientation-reversing involution with eigenvalues (+1, +1, −1), producing the decomposition: \ E⁺ = spane₇, e₆+e₂ \ (two-dimensional even sector) \ E⁻ = spane₆−e₂ \ (unique one-dimensional odd sector) The canonical retained axis \ Im (ΠA) = spane₇ \ is the distinguished direction within E⁺, selected by its role in the commutator source structure. The rotational structure is confirmed by explicit computation: e₇ rotates P_⊥, exchanging even and odd directions via \ [e₇, e₆+e₂ = e₆−e₂ and e₇, e₆−e₂ = − (e₆+e₂) \]. The unique J-odd direction E⁻ is therefore the rotational generator R, and the scalar phase µ is its amplitude. The scalar phase is not fundamental; it is the projection of a unique rotation generator selected by branch-induced orientation reversal. The chain J ⇒ E⁺⊕E⁻ ⇒ unique odd direction ⇒ R ⇒ µ is solid. The remaining conditional is the canonical selection of spane₇ within the two-dimensional even sector E⁺, the inclusion \ Im (ΠA) ⊂ Fix (J|₂ₗ) \ is solid. Forcing the selection from first principles rather than canonical role assignment is the open item.