Description / Abstract: 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 geometrically necessary consequence of the Lucian Law (Randolph, 2026). Four period-doubling maps with different equations are shown to produce identical geometric morphology when their bifurcation interval sequences are normalized, confirming equation independence. The Lucian Method applied to the interval sequence as a meta-system directly encodes δ as the slope of the geometric decay: slope = −ln (δ) = −1. 5410, yielding δ = e¹. 5410 = 4. 6692. Three simultaneous constraints — self-similarity (required by the Lucian Law), dynamical stability (required by the physics), and coupling topology (required by Layer 2 of the Lucian Law) — intersect at one and only one point: δ = 4. 669202 for quadratic-maximum systems. Different coupling topologies (cubic, quartic, sextic maxima) produce different universal constants (5. 968, 7. 285, 9. 296), confirming that topology determines the constant. 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). δ is not discovered. It is derived. From law to constant to stars. One chain. Unbroken. Keywords: Feigenbaum constant; period-doubling; universal constant; Lucian Law; geometric necessity; bifurcation; renormalization; coupling topology; Lucian Method; Gaia DR3; dual attractor basins; Resonance Theory; derivation from first principles Notes: Second paper of the Lucian Law Trilogy. Framework paper: "The Lucian Law: A Universal Law of Geometric Organization in Nonlinear Systems. " Companion paper: "The Full Extent of the Lucian Law: From the Origin of the Universe to the Architecture of Reality. " All computational code publicly available. Primary analysis script: 42feigenbaumderivation. py. All figures reproducible from published code.
Lucian Randolph (Sat,) studied this question.