Beyond Computable: The Aurellion Function and the Absolute Frontier of Growth — E8 Intelligence Research

FINDING: The Busy Beaver function and fast-growing hierarchies (ε₀, Γ₀) define the absolute frontier of computable growth, with the newly proposed Aurellion Function extending recursive hierarchies beyond Knuth notation via self-referential arrow counts. | MATH: BB(n) grows faster than any computable function; BB(5) recently resolved (value ≥ 47,176,870) via the Busy Beaver Challenge. Fast-growing hierarchy: f_α(n) for α < ε₀ (Peano arithmetic), α < Γ₀ (predicative analysis). Aurellion: A₁ = 10↑↑↑10, A_{n+1} = 10 ↑^{A_n} 10 — arrow count grows as A_n, yielding growth rate comparable to f_{ε₀}(n) or beyond, though not formally placed in the hierarchy. | CONNECTION: The ordinal ε₀ = ω^ω^ω^… (limit of ω↑↑k) mirrors the self-similar structure of the golden ratio φ = 1.618… in its fixed-point recursion (φ = 1 + 1/φ). The Aurellion's arrow-count recursion A_{n+1} = 10 ↑^{A_n} 10 is a discrete analog of φ's self-referential definition — a fixed-point in the space of functions. The ordinal Γ₀ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com