The Decay Bounce: Reflection Geometry of the Feigenbaum Stable Manifold

Description/Abstract:We characterize the transversal geometry of one-parameter families crossing the stable manifold W^s(g*) of the Feigenbaum renormalization fixed point. At each period-doubling cascade level n, four observables are measured: the parameter interval Δr_n, the spatial displacement d_n, the transversality derivative dΦ_n/dr, and the normalized transversality σ_n. We discover that the transversality derivative grows as δ/α = 1.8655..., the ratio of the two Feigenbaum constants, verified to 0.0005% across eight cascade levels. The normalized transversality σ_n decays as 1/α with alternating sign, verified to 0.0003%. The sign alternation reveals that one-parameter families do not simply cross the…