Research Derives Asymptotic Evans Zero Law for Chiral SCT Operator, Indicating Key Behavior in Complex Analysis.
Paper 46 in the "Geometry of the Critical Line" programme. This paper derives the leading asymptotic law for the Evans zeros of the connection-matrix entry M₂₁(λ,m) of the chiral SCT operator. The proof uses exact Liouville reduction (the identity B = −A'/2), endpoint Laurent analysis, and complex-order Bessel matching. The Evans numerator reduces to sinh(β + iα), where β = Im(ν)π and α = Φ_tot − Re(ν)π − π/2. The main results at leading asymptotic order: (i) the leading asymptotic Evans model has no real zeros for m ≠ 0; (ii) the zeros satisfy Δ√(Re λₙ) → π/L with L = 2/k; (iii) for m > 0 the zeros lie in Im(λ) < 0 with depth D_m = π|Im(r₁−r₂)|/L, where r₁, r₂ are the Frobenius exponents at the right endpoint; (iv) for m < 0 the zeros lie in Im(λ) > 0 by chiral conjugation. Numerical verification in the m = 2 and m = 4 sectors confirms the predicted depth to within 1% and 4% respectively, with the discrepancy consistent with finite-λ convergence. Subleading corrections are not computed here. The lower-half-plane location is a leading-order result, not yet a full non-perturbative theorem. 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 + eiπ − 1/Φ = 0.
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Pavel Kramarenko-Byrd (2026) studied this question.
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