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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 Pᵢj (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 PCC peaks and relay-to-bounce exit probability P₂, ₁₎ₔ₍₂₄ is minimised at α ≈ 0. 481, with relay row entropy HC 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 P₂, −₁ = P₂, ₀ is exact to machine precision at every α. Part of the Geometry of the Critical Line programme.
Pavel Kramarenko-Byrd (Sat,) studied this question.