Weighted Transition Traffic at the Relay Junction: Dwell Masks the Spectral Signature, Routing Reveals It

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 25 in the "Geometry of the Critical Line" programme. Papers 21–24 established a routing transition near α ≈ 0.481 in the Newton dynamics of C_α(z) = z − exp(−α/z). This paper constructs the weighted transition traffic matrix P_ij(α) on a fixed 4-state symbolic space {C, −1, 0, O} and measures the routing transition at the operator level. The raw-step matrix confirms the transition as a smooth first-order redistribution: relay self-retention P_CC peaks and relay-to-bounce exit probability P_{C,bounce} is minimised at α ≈ 0.481, with relay row entropy H_C at its floor. The raw-step spectral gap |λ₂| is flat (0.951–0.956). However, the reduced-jump matrix — which factors out the relay self-loop and isolates the routing decision — exhibits a floor-to-plateau spectral step co-located with the routing transition, suggesting a change from a fast-mixing to a slower-mixing routing regime. The magnitude of this step may be partly influenced by sample-size growth across the sweep. The conjugate symmetry P_{C,−1} = P_{C,0} is exact to machine precision at every α. Part of the Geometry of the Critical Line programme.