Research Note 30 in the "Geometry of the Critical Line" programme. This note closes the first-order refinement cycle for the Evans depth law. Paper 46 derived the leading depth coefficient Dₘ^ (∞) = π|Im (r₁−r₂) |/L. RN29 identified and tested three first-order correction candidates. This note records the collective verdict: every isolated first-order refinement mechanism tested so far either cancels from the depth sector or is quantitatively insufficient. The leading depth law is protected against the three first-order correction mechanisms tested here. The visible finite-x gap in the m=4 sector (4% at current numerical scales) is a genuinely composite second-order matching problem, not a deficiency of the leading law. This note closes the first-order refinement cycle only. 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.
Pavel Kramarenko-Byrd (Sun,) studied this question.