Mirror-Induced Cancellation of Remainder Contributions

This Theorem closes the open non-cancellation question of T37 (Remark 37.8). Under explicit mirror-symmetry and projection-intertwining conditions, the remainder sector Ω_rem of the terminal retained residue Ω_term contributes zero to the scalar extraction functional L, and the effective embedded defect coefficient satisfies λ_eff(θ) = 1 exactly. The proof proceeds in three steps. First, the mirror involution J preserves Ω_rem = ker(P_comm), established by showing P_comm·J = J·P_comm implies J(Ω_rem) = Ω_rem. Second, L(J(X)) = −L(X) for all X, so every J-orbit of size two in Ω_rem contributes L(X) + L(J(X)) = 0; fixed points satisfy L(X) = −L(X) = 0. Summing over orbits gives L(Ω_rem) = 0. Third, the T37 correction term A − B(θI − D)⁻¹C lies in the remainder image R_θ(Ω_rem), so its contribution to λ_eff(θ) vanishes, giving λ_eff(θ) = 1. A structural upgrade is noted: the remainder sector maps into the non-rotational subspace Z(Ω_rem) ⊆ span{I, σ_z, σ_x}, converting the scalar cancellation into an operator-level claim. The T29–T38 arc is now closed: commutator-driven extraction (T29) → spectral-propagator structure (T30–T35) → defect survival τ = a_max = 1 (T36) → defect holonomy E_d = iσ_y (T37) → mirror cancellation λ_eff(θ) = 1 (T38). The scalar observable layer is fully closed, and the embedded defect coefficient is exactly 1. Status: Subtraction anti-commutation, projection intertwining, remainder closure, phase-plane anti-symmetry L(J(X)) = −L(X), and orbit-pairing cancellation are all solid. Conditional on assumption (3), commutator-sector stability J(Ω_comm) = Ω_comm is structurally motivated but not fully derived from T17 kernel generators. Conditional on Proposition 38 (commline): identification E_θ(Ω_comm) ⊆ span{E_d} argued from T29/T36/T37, but complete derivation from explicit kernel structure is open. All results inherit T36 retained-pairing conditionality and T16/T17/T20 framework conditionality. Dependencies: T1, T7, T14, T15, T16, T17, T18, T19, T20, T27, T28, T29, T30, T36, T37.