The WKB Phase Transition: Waveguide, Barrier, and the λ = m² Threshold

Research Note 35 in the "Geometry of the Critical Line" programme. RN34 found numerically that the regular transport factor R(λ,m) has a power-law exponent α(m) that steepens rapidly with winding number m, with fit quality collapsing for m ≥ 8. This note identifies the mechanism: a WKB phase transition in the Liouville-transformed interior at the threshold λ = m². For λ > m², the interior is a pure transparent waveguide with no turning points and oscillatory transport. For λ < m², a central potential barrier forms and the transport transitions to evanescent tunnelling. The R² collapse in RN34 occurs precisely when data points cross this threshold. The waveguide optical length converges to L ≈ 2δ₀ as λ → ∞. This result explains quantitatively why the regular factor's behaviour changes character at λ = m² and identifies the waveguide/barrier transition as the controlling mechanism for interior transport in the SCT critical sector.