The Transverse Decoupling Correspondence: Real Coefficients, Seven Failed Coordinate Maps, and the Algebraic Backbone of Bridge 2

Boundary reference notice (22 September 2026). This record is linked as boundary context for Paper A12, Dynamics of the Lambert W Newton Iteration: Basin Geometry, Routing Transitions, and Finite-Rank Period-Three Resonance (DOI: 10.5281/zenodo.21845665). Its allocation to A12 is limited to the boundary note described in the publication plan. This link does not declare this record superseded and does not extend A12 beyond its stated Newton-dynamics and finite-matrix scope. The original description below and deposited files are retained. Paper 34 in the "Geometry of the Critical Line" programme. Paper 33 proved that the transport parity P_C,-(k+1) = P_C,k follows from conjugation equivariance of the Cover Function C_α(z) = z - e^(-α/z) for real α. Research Note 3 proved that the transverse velocity V_y = Im(Φ’_α(u)) at any vertical section in u-space is α-independent. Paper 5 proved that the derivative coupling coefficient A(σ) in the separated σ-equation on M^5_SCT vanishes if and only if σ = 1/2. This paper identifies a precise structural correspondence between the Newton-side and SCT-side decoupling results. Both are consistent with a shared algebraic backbone: the cover constraint Φ + e^(iπ - α/Φ) = 0 reduces via Euler’s identity e^(iπ) = -1 to the real equation Φ = e^(-α/Φ), giving C_α real coefficients for real α. The conjugation equivariance of Paper 33 (Bridge 1) and the transverse decoupling considered here (Bridge 2) are both consistent with that real-coefficient structure. However, the formal identification of the two decouplings across the arc boundary remains open. Seven coordinate-level attempts to construct such a map all fail. No simple coordinate-level bridge is found; any successful bridge is likely more indirect than a direct coordinate match. What this paper establishes is a strong structural correspondence, not a completed bridge theorem. Bridge 2 is not formally landed: the derived map sending the SCT decoupling to the Newton decoupling has not been constructed. A coordinate-level or operator-level bridge, if one exists, remains future work. Part of the Geometry of the Critical Line programme.