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Theorem demonstrates feedback impacts observable classification in quantum systems, indicating novel spectral outcomes.
This Theorem derives the ambient Schur landing of complement-sector feedback and establishes its complete observable classification. The complement block of admissible form H_cc = aI + bJ (J² = I) is invertible for a² ≠ b² with H_cc⁻¹ = (aI − bJ)/(a² − b²). The Schur correction ΔH = H_rc·H_cc⁻¹·H_cr decomposes into: ΔH = (a/(a²−b²))·Δ₀ − (b/(a²−b²))·Δ_J where Δ₀ = H_rc·H_cr and Δ_J = H_rc·J·H_cr. Five lemmas establish the landing structure. The helicity-twisted term satisfies P_sc·Δ_J·P_sc = 0, J is odd under scalar projection, so the twisted feedback contributes nothing to the scalar channel. The transverse projection of Δ_J lands in span{I, σ_z, σ_x}, J anticommutes with R = iσ_y, so the twisted component has no R-component and lies entirely in the symmetric transverse sector. A critical bridge identifies Δ_J as the operator-level origin of the T30 spectral deformation: both the Schur denominator (a² − b²) and the T30 discriminant (p² − 4a²) arise as discriminants of two-level systems with identical admissible commutant structure. Four observable consequences: scalar channel protected (μ invariant); helicity-twisted correction annihilated by extraction functional L before readout; T30 spectral deformation identified as Schur correction; sole observable manifestation is spectral redistribution between channels at p and ω = √(p² − 4a²). Two independent protection mechanisms operate: local null-class annihilation (scalar observables) and global mirror-orbit cancellation via T38 (embedded observables). Status: Commutant inversion, Schur decomposition, scalar nullity given J odd under scalar projection, and transverse symmetric landing given anticommutation of J with R, all solid by direct computation. T30 spectral bridge structural, identification of denominator with p² − 4a² is a leading-order structural identification, not a derived equality. Scalar invariance and λ_eff = 1 solid given lemmas and T38. Orbit preservation solid via T34–T35. Admissible form H_cc = aI + bJ is a structural assumption; derivation from the full ambient T12/T16 block structure is the primary remaining open step. Dependencies: T12, T16, T26, T29, T30, T34, T35, T37, T38, T54.
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Craig Edwin Holdway (2026) studied this question.
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