Binary Orientation Action on the Terminal Retained Support: Recovery of the Paired Phase Defect via Canonical Branch Projection

Theorem 19 established that the terminal ordered residue Ω_term has minimal complete support Λ₄ = (F₇, F₆, F₂). Theorem 20 establishes the internal structure of Ω_term|_Λ₄. The canonical binary orientation invariant σ_Γ = −1 from Theorem 18 induces a canonical binary projector Π_A, which selects the Gray-traversal-compatible (K_A-branch, defect-first) component of the retained residue. The two-sided projection R_A = Π_A(Ω_term|_Λ₄)Π_A is nonzero, has strictly lower orientation complexity than the full retained object, and collapses in two steps to the paired scalar phase defect of Theorem 15: R_A → C_A → Δφ(τ), where ρ_phase is the scalar phase readout of Theorem 15. The principal structural consequence is that Δφ(τ) is not a primitive object; it is the scalar readout of a binary-selected branch of a noncommutative retained residue on the minimal complete triadic support. The full derivation chain is: σ_Γ → Π_A → R_A → C_A → Δφ(τ). No symmetry group identification is made; that is the subject of Theorem 21.