The Feigenbaum constant δ ≈ 4.6692016, representing the universal scaling of period-doubling bifurcations, has traditionally been viewed as a purely numerical property of one-dimensional maps. We present a rigorous physical derivation of this constant within theAxiomatic Physical Homeostasis (APH) framework. We demonstrate that δ repre-sents the Dimensional Compression Ratio of the vacuum geometry as it undergoes aphase transition from the non-associative bulk (G2) to the stable associative cycle (S3).By accounting for the topological leakage of information mediated by the fine structureconstant α and the Euler characteristic of the manifold (χ = 24), we derive an exact closed-form expression for δ. This geometric prediction matches the standard numerical value toeight decimal places, suggesting that chaos is a signature of the topological decay of the G2vacuum.