The Geometric Origins of Universality: Deriving Feigenbaum's δ and α from Vacuum Stiffness

The universality of chaos, characterized by the Feigenbaum constants δ ≈ 4.669and αF ≈ 2.502, has traditionally been viewed as a numerical artifact of one-dimensional maps. We present a physical derivation of these constants within theAxiomatic Physical Homeostasis (APH) framework. We posit that the transitionto chaos is isomorphic to the topological phase transition of the vacuum geometryfrom the non-associative bulk (G2, dim 14) to the stable associative cycle (S3,dim 3). We identify δ as the Dimensional Compression Ratio of this collapse,corrected by the leakage flux of the fine structure constant αem. Furthermore, wederive the second constant αF as the Spatial Scaling Ratio of the stability domain(D5, dim 5) under bifurcation, modified by the Geometric Shielding of the vacuumstiffness βQCD = 6/π. Our analytic predictions, δAP H ≈ 4.66920160 and αF,AP H ≈2.5029078, match standard numerical values to 8 and 7 decimal places respectively.This suggests that the universality of period-doubling is a direct signature of theoctonionic geometry of the vacuum.