Research note describes η-stable properties of the Evans object in complex spectral analysis, indicating significant findings.
Research Note 26 in the "Geometry of the Critical Line" programme. The complex-λ probe of the first spectral coefficient C₁(λ,m) from RN25 reveals that the transport ratio R(λ,m) = M₂₁/M₁₁ and its λ-derivative depend on the regularisation cutoff η, both in phase and magnitude. This is Kill #73: R is not canonical. The magnitude of R'(λ) varies by factor ~150× and the phase rotates through ~4 radians across η ∈ [5×10⁻⁶, 10⁻⁴]. The note identifies the numerically stable replacement in the present extraction scheme. The Evans function E(λ,m) := |M₂₁(λ,m)| is η-independent to 0.04% across a 250× range of η, and its logarithmic derivative σ₁ := ∂_λ log E is η-invariant to 7–8 significant figures. The full complex-valued M₂₁ is η-stable in both modulus and argument (< 0.002 rad variation). The empirical decay law is E ~ C(m)λ−γ(m) with γ(m) crossing 1 between m = 2 and m = 4. No closed formula for γ(m) is derived. This note does not prove scheme-independence of M₂₁ under arbitrary reformulations; it proves only strong η-stability in the tested formulation. 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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