Research Note: From the Hermitian Seam Hypothesis to the Inversion Fibration — Two Null Results, the Newton Pullback Conjecture, and Branch-Indexed Organization in u-Space

This research note records a March 12–13, 2026 investigation into the Newton dynamics of the cover equation Cα(z) = z − exp(−α/z). Two initial hypotheses are tested and rejected: a Hermitian seam interpretation based on monodromy transport, and a local potential-landscape explanation of the visible z-space vesica. A more promising structure appears after inversion to u = α/z. In this chart, the basin geometry becomes more legible: Lambert branch territories separate coherently, an axial-relay organization emerges, and the visible z-space seam is reinterpreted as a compressed projection of a more native u-space structure. This note belongs to The Geometry of the Critical Line research programme and is published as a working research note rather than a final publication. Its open problems, limitations, and next steps are stated explicitly.