The Geometric Necessity of Feigenbaum's Second Constant: Deriving α from the Universal Cascade Theorem

Mitchell Feigenbaum discovered two universal constants in 1975: δ = 4.669201609... and α = 2.502907875... Both govern every period-doubling system regardless of specific equations. δ — the parameter-space scaling ratio — has been derived from the Universal Cascade Theorem (UCT, Randolph 2026, Paper 06). α — the phase-space scaling ratio — has received no comparable derivation. The asymmetry is unwarranted. δ and α are two faces of the same cascade geometry: two properties of the same unique renormalization fixed point g*. This paper derives α from the UCT. The derivation runs parallel to Paper 06's derivation of δ. Conditions C₁ (analytic dissipative boundedness), C₂ (non-degenerate parametric fold), and C₃ (transversal spectral crossing) force the accumulation-point map φ∞ into Basin(g*) via Lemma SF and Lyubich (1999). Lanford's uniqueness theorem (1982) identifies the unique g* that the renormalization orbit converges to. Under normalization g*(0) = 1, the fixed-point equation R̃[g*] = g* forces g*(1) = α. Since g* is unique, α is unique. C₂ selects z = 2, pinning α = 2.502907875... The Feigenbaum-Cvitanović functional equation is not invoked. The constant is not computed. It is selected by uniqueness from the phase-space structure of the cascade. The derivation extends to the full α(z) family — one constant per universality class, indexed by the order z of the critical maximum. Each α(z) is geometrically necessary for systems with that coupling topology. The compound ratio δ/α = 1.86551..., identified in Paper 28 as a derived quantity, is here established as a necessary consequence of the UCT geometry. Two constants. Two faces. One cascade. α is not computed. It is derived.