Holomorphic Phase Twist and the Geometric Reduction to k = π/8

Research Note 19 in the "Geometry of the Critical Line" programme. The metric constant k in the SCT 5-manifold was originally a motivated choice (k = 1/√2), later refined to k = π/8 by matching the post-quotient spectral density to the Riemann–von Mangoldt counting (Paper 40). That refinement was conditional on the SCT–Connes dictionary and used the Riemann zero density as input. This note shows that the holomorphic phase geometry of the cover equation Φ = e^−α/Φ supplies the missing geometric input. The transverse phase twist ∂θ/∂τ at the critical line evaluates to 4α, which combined with the previously established identities 4αk = ln 2 and α = 2 ln 2/π gives k = π/8 without direct appeal to the Riemann density in the final algebraic step — provided the identity 4αk = ln 2 is itself independent of that density route (this remains an open question). The constant k = π/8 reappears throughout Phase V of the programme, most notably as 2/k = 16/π in Paper 45's geometric transport limit. RN19 is included in the extraction-pathology arc because it supplies the geometric constant that anchors the entire endpoint analysis. 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.