The Geometric Origin of Feigenbaum Universality: Basin Membership, Renormalization Fixed-Point Uniqueness, and Poincaré's Classification Programme

In 1975, Mitchell Feigenbaum discovered that the ratio of successive bifurcation intervals in period-doubling cascades converges to a universal constant: δ = 4.669201609... The constant appears in every period-doubling system regardless of specific equations — logistic maps, sine maps, fluid convection, electronic circuits, population models. Feigenbaum demonstrated universality empirically and provided renormalization group arguments for how the constant operates, but the fundamental question — why this specific number, why universal — has remained incompletely answered for fifty years. This paper derives δ as a necessary consequence of the Universal Cascade Theorem (UCT, Randolph 2026). The UCT establishes three minimal conditions on any dynamical system — analytic dissipative boundedness (C₁), non-degenerate parametric fold (C₂), and transversal spectral crossing with infinite accumulating cascade (C₃) — that are necessary and sufficient for Feigenbaum cascade structure. The derivation proceeds through the UCT's Lemma SF (Schwarzian Freed), which maps C₁, C₂, C₃ onto the hypotheses of Lyubich (1999), establishing that the accumulation-point map φ_∞ lies in Basin(g*). Lanford's uniqueness theorem completes the derivation: g* is the unique hyperbolic fixed point of the renormalization operator, and δ is its sole unstable eigenvalue in the parameter direction. The constant is not computed from the Feigenbaum-Cvitanović functional equation. It is selected by uniqueness. Three simultaneous constraints — self-similarity (required by the UCT cascade structure), basin membership via Lemma SF (established from C₁, C₂, C₃ independently of Feigenbaum's own renormalization framework), and coupling topology (C₂, the quadratic maximum condition that selects the universality class) — intersect at one and only one point: δ = 4.669202 for quadratic-maximum systems. The Feigenbaum-Cvitanović functional equation is not invoked to identify this value. Uniqueness follows from Lanford (1982), confirmed by Campanino-Epstein (1981) and Eckmann-Wittwer (1987). Different coupling topologies correspond to different fixed points of the renormalization operator, each uniquely determining its own universal constant (cubic: 5.968, quartic: 7.285, sextic: 9.296), confirming that topology selects the constant through the same mechanism. The derivation is confirmed by Gaia DR3 stellar data, where 50,000 stars organize on a Feigenbaum sub-harmonic spectrum with active and passive populations separating into dual attractor basins at p < 10⁻³⁰⁰ (below machine precision, Randolph 2026). δ is not discovered. It is derived. From law to constant to stars. One chain. Unbroken.