The Self-Grounding Property of Feigenbaum Universality - Computational Evidence and Proof Sketch

Abstract The Feigenbaum period-doubling cascade arises in one-parameter families of maps satisfying three conditions: bounded dynamics, a nonlinear fold with quadratic maximum, and parametric transition through instability. We observe that these three conditions, when each is treated as a continuous parameter space and driven across extreme range, independently reproduce the Feigenbaum constant δ = 4.669201609102990. The boundedness space (logistic map, r ∈ 0, 4) yields δ within 0.06% via superstable orbit detection. The nonlinearity space (generalized logistic family x → rx(1 − xᶻ), z ∈ 0.5, 10) yields δ at all ten tested map orders with maximum error 0.009%. The coupling space (coupled maps with broken synchronization manifold) yields δ within 0.25% with R² > 0.999 via geometric scaling regression. While each result individually follows from known universality theory, their conjunction constitutes a self-grounding property: the preconditions for Feigenbaum universality themselves exhibit Feigenbaum universality. We prove that this self-referential structure follows from the codimension-1 stable manifold of the renormalization fixed point: the conditions for universality necessarily produce families that cross the stable manifold transversally, inheriting the universal scaling ratio δ. Keywords: Feigenbaum universality, period-doubling, self-grounding, nonlinear dynamics, renormalization