Transport Inversion and the Analytic Topography of the Relay Exit

Scoped supersession notice (22 September 2026). This record is part of the source research consolidated in Paper A12, Dynamics of the Lambert W Newton Iteration: Basin Geometry, Routing Transitions, and Finite-Rank Period-Three Resonance (DOI: 10.5281/zenodo.21845665). For the Newton-dynamics, routing, conjugation-parity, and finite-rank period-three results stated in that article, it supersedes the corresponding earlier programme presentation and is the current consolidated reference. Supersession is limited to that scope; other results, programme interpretations, and open problems in this record are not thereby replaced or resolved. In particular, the finite-matrix results do not assert a continuum resonance. The original description below and deposited files are retained for provenance. Paper 32 in the "Geometry of the Critical Line" programme. Paper 31 deferred the relay-to-section transfer kernel K_α(x,y) pending robustness testing. This paper subjects the transport inversion — the sign change in the fringe-vs-core bounce advantage — to partition, resolution, and recirculation stress tests. The inversion survives all three, indicating it is a robust feature of the relay-to-section architecture rather than a numerical artifact: it holds across 12 partition schemes, is stable to four decimal places across three spatial resolutions (800×600 to 1600×1200), and persists — indeed amplifies — under second-crossing recirculation dynamics. The existence of the inversion is robust; its location (the zero-crossing α) is split-dependent, shifting from α ≈ 0.485 at R_split = 1.0 to ≈ 0.527 at R_split = 0.75. Analytically, the transverse velocity at the relay exit is α-independent — a theorem following from the complex potential Φ_α(u) = αu + (u+1)e^{−u} — and the routing transition unfolds as variable forward transport over fixed transverse topography. The V_y theorem provides the analytic frame but does not yet formally derive the zero-crossing location. The SVD of the transfer kernel confirms a stable dominant mode (primary singular vector α-invariant) with the transition carried by secondary modes. The invariant lobe spacing (~0.032, Paper 30) is not explained by the exit-flow geometry and remains an open problem at a finer discrete scale.