Universal Cascade Law Determines Feigenbaum Universality - A Synthesis of Three Results

Description/Abstract: We show that Feigenbaum universality—the appearance of the constants δ = 4.669... and α = 2.502... across all period-doubling cascades—is not a foundational principle but a consequence of a deeper geometric law. The Universal Cascade Law (UCL) states that bounded nonlinear systems with coupling necessarily produce fractal geometry. Three papers, taken together, establish a closed logical chain: (1) the UCL generates the Feigenbaum constants through the geometry of the cascade, (2) the Feigenbaum constants structure the UCL’s own prerequisites through the decay bounce mechanism, and (3) the loop closes—the Law is self-grounding. This note synthesizes the argument and establishes the four-layer universality hierarchy. Keywords: Feigenbaum universality, period-doubling cascade, renormalization, self-grounding, fractal geometry, Universal Cascade Law