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 + e^iπ − 1/Φ = 0.
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Pavel Kramarenko-Byrd
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Pavel Kramarenko-Byrd (Sun,) studied this question.
www.synapsesocial.com/papers/69ccb75916edfba7beb894c0 — DOI: https://doi.org/10.5281/zenodo.19323379