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This work addresses the open non-cancellation question of theorem T37 regarding remainder contributions to scalar extraction.

May 5, 2026Open Access

Mirror-Induced Cancellation of Remainder Contributions

Puntos clave

This work addresses the open non-cancellation question of theorem T37 regarding remainder contributions to scalar extraction.
Proved that the mirror involution preserves the remainder sector.
Demonstrated that the contribution to the scalar extraction functional L from orbits in the remainder sector sums to zero.
Identified the structural implications of the remainder sector's mapping into a non-rotational subspace.
The remainder sector contributes zero to the scalar extraction functional L, resulting in L(Ω_rem) = 0.
The effective embedded defect coefficient λ_eff(θ) is confirmed to equal 1 under the defined conditions.
The framework of commutator-driven extraction through the T29-T38 arc is solidified.

Resumen

This Theorem closes the open non-cancellation question of T37 (Remark 37. 8). Under explicit mirror-symmetry and projection-intertwining conditions, the remainder sector Ωᵣem of the terminal retained residue Ωₜerm contributes zero to the scalar extraction functional L, and the effective embedded defect coefficient satisfies λₑff (θ) = 1 exactly. The proof proceeds in three steps. First, the mirror involution J preserves Ωᵣem = ker (Pcomm), established by showing Pcomm·J = J·Pcomm implies J (Ωᵣem) = Ωᵣem. Second, L (J (X) ) = −L (X) for all X, so every J-orbit of size two in Ωᵣem contributes L (X) + L (J (X) ) = 0; fixed points satisfy L (X) = −L (X) = 0. Summing over orbits gives L (Ωᵣem) = 0. Third, the T37 correction term A − B (θI − D) ⁻¹C lies in the remainder image R_θ (Ωᵣem), so its contribution to λₑff (θ) vanishes, giving λₑff (θ) = 1. A structural upgrade is noted: the remainder sector maps into the non-rotational subspace Z (Ωᵣem) ⊆ spanI, σᵦ, σₓ, 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 = aₘax = 1 (T36) → defect holonomy Ed = iσᵧ (T37) → mirror cancellation λₑff (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 (Ωcomm) = Ωcomm is structurally motivated but not fully derived from T17 kernel generators. Conditional on Proposition 38 (commline): identification E_θ (Ωcomm) ⊆ spanEd 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.

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