Research note reports universal spacing patterns in Evans zeros across multiple sectors, indicating significant geometric insights.
Research Note 28 in the "Geometry of the Critical Line" programme. This note reports the first multi-sector Evans zero survey for m = 2, 4, 6, 10 using the exact variational contour integrand ∂_λ M₂₁/M₂₁. In all scanned sectors, all detected Evans zeros lie in the lower half-plane Im(λ) < 0, and tested upper-half-plane windows are certified zero-free in the resolved scan windows. In the m=4 sector, 69 η-stable zeros are located; in the m=2 sector, 33 are located. The asymptotic spacing law Δ√(Re λⱼ) → π/L is confirmed independently in the m=4 clean tail (31 consecutive zeros, 0.34% accuracy) and in the m=2 clean tail (3 consecutive spacings, 0.04% accuracy). The m=6 and m=10 sectors show the same qualitative lower-half-plane geometry but remain pre-asymptotic in the scanned window. Chiral conjugation symmetry is verified in the m=4 ↔ −4 test: 21/21 conjugate pairs pass to machine precision. The cross-sector survey does not prove completeness of any sector's zero set. 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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