Real-Axis Nonvanishing of M₁₁ for the Open Chiral SCT Connection Matrix

Research Note 13 in the "Geometry of the Critical Line" programme. Paper 41 constructed the 2×2 connection matrix M (λ, m) for the open chiral SCT operator and proved the involution M² = I, giving M₂₂ = −M₁₁. This note upgrades the numerical observation M₁₁ ≠ 0 to an analytic theorem: for the SCT family with k ∈ π/8, 1/√2 and |m| ≥ 1, M₁₁ (λ, m) ≠ 0 for all real λ. The proof uses the truncated flux-balance identity with a one-sided singular endpoint, exploiting the sign mismatch between the boundary and bulk divergence coefficients. Combined with M₂₁ ≠ 0 (Paper 40) and M₂₂ = −M₁₁ (Paper 41), this establishes a partial real-axis exclusion result for the connection matrix. No arithmetic interpretation is claimed. Part of a 46-paper open-access programme on the geometry of the Riemann zeta function's critical line, anchored by the SCT 5-Manifold and the cover equation Φ + e^iπ − 1/Φ = 0. The programme does not claim to prove the Riemann Hypothesis.