Relay Microstructure and the Kernel–Measure Handoff Across the Routing Transition
Pavel Kramarenko-Byrd · Zenodo (CERN European Organization for Nuclear Research) · 2026
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 31 in the "Geometry of the Critical Line" programme. Papers 25 and 29 characterised the routing transition through a 4-state traffic law and a measure-weighted overlap decomposition. This paper resolves the mechanism to a finer level through two complementary analyses. First, splitting the relay state C into core (Re(u) < 1.0) and fringe (1.0 ≤ Re(u) < 1.5) reveals that the routing floor at α ≈ 0.481 is a fringe-bottleneck regime: fringe-to-bounce permeability is minimised there for every tested split threshold (0.8–1.2), and the 4-state reduced-jump spectral step is not stable under the split. Second, a section-based routing kernel decomposes the bounce probability into a crossing measure μ_α and a conditional routing law T_α(y → bounce); the kernel is the stronger contributor across all tested reference choices (α* = 0.42, 0.481, 0.50, 0.60) and all tested bin counts (20, 40, 60, 80). A local contribution analysis reveals the dominant contribution alternates step-by-step across the lifecycle: some transitions are kernel-led (the routing rule itself evolves), others are measure-led (the orbit population redistributes), and others are mixed. Paper 29's finding that orbit redistribution is the stronger contributor is specific to the frozen-reference methodology; the local-contributions decomposition reveals a stage-dependent handoff. The routing transition is governed not by a single fixed dominant cause but by a coupled alternation of kernel and measure contributions across different α intervals.