The α-Independence of Transverse Velocity at the Relay Exit

Research Note 3 in the "Geometry of the Critical Line" programme. The complex potential of the Newton map for C_α(z) = z − exp(−α/z) is Φ_α(u) = αu + (u+1)e^{−u} in the coordinate u = α/z. Its derivative Φ'_α(u) = α − ue^{−u} decomposes at any vertical section Re(u) = x₀ into a forward velocity V_x(y, α) that depends on α and a transverse velocity V_y(y) that does not. This note derives the decomposition, verifies it to machine precision, and records its structural consequences: the transverse exit geometry is a frozen function of the α-independent term (u+1)e^{−u}, the flow angle modulates from 72° to 14° as α increases from 0.42 to 1.0, the conjugate symmetry follows from V_y being an odd function of y, and…