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 (K_A, K_B), canonical projector Π_A, and triadic carrier \[ C_X = span{e₇, e₆, e₂} \] , the branch-exchange involution J acts on C_X as an orientation-reversing involution with eigenvalues (+1, +1, −1), producing the decomposition: \[ E⁺ = span{e₇, e₆+e₂} \] (two-dimensional even sector) \[ E⁻ = span{e₆−e₂} \] (unique one-dimensional odd sector) The canonical retained axis \[ Im(Π_A) = span{e₇} \] 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 span{e₇} within the two-dimensional even sector E⁺, the inclusion \[ Im(Π_A) ⊂ Fix(J|C_X) \] is solid. Forcing the selection from first principles rather than canonical role assignment is the open item.