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 stable manifold — they reflect off it in a damped oscillation we call the decay bounce. The parameter compression factor δ and the spatial contraction factor α enter through the derivative chain rule of the renormalization operator: δ from the unstable eigenvalue, 1/α from spatial rescaling. Their ratio δ/α governs the crossing steepness; their cancellation in σ_n leaves pure spatial decay at rate 1/α. We verify that this reflection geometry survives coupling (tested at five coupling strengths with broken synchronization), and demonstrate that it provides the mechanism for the self-grounding property: the three preconditions for Feigenbaum universality — boundedness, nonlinear fold, and parametric instability — each exhibit the Feigenbaum constant δ in their own parameter spaces, because every path to the cascade must pass through the same decay bounce geometry. A theorem and proof sketch are presented, assembling established results (Lanford 1982, Campanino–Epstein 1981, Sard 1942) with the new chain-rule argument.