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 Wˢ (g*) of the Feigenbaum renormalization fixed point. At each period-doubling cascade level n, four observables are measured: the parameter interval Δrₙ, the spatial displacement dₙ, the transversality derivative dΦₙ/dr, and the normalized transversality σₙ. 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 σₙ 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 σₙ 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. Keywords: Feigenbaum universality, period-doubling cascade, renormalization group, stable manifold, transversality, decay bounce, reflection geometry, Feigenbaum constants, self-grounding property, nonlinear dynamics, universality class, bifurcation theory