An Exact Analytic Derivation of the Feigenbaum Constant δ via the Octonionic G2 Manifold

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 the
Axiomatic Physical Homeostasis (APH) framework. We demonstrate that δ repre-
sents the Dimensional Compression Ratio of the vacuum geometry as it undergoes a
phase 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 structure
constant 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 to
eight decimal places, suggesting that chaos is a signature of the topological decay of the G2
vacuum.