Aurellion Function: Extending Recursive Hierarchies Beyond Knuth Notation — E8 Intelligence Research

FINDING: Fast-growing hierarchies and ordinal collapsing functions extend computability into transfinite realms, with the Aurellion Function offering a novel recursive hierarchy beyond Knuth notation. | MATH: Aurellion Function: \(A_1 = 10 \uparrow\uparrow\uparrow 10\), \(A_{n+1} = 10 \uparrow^{A_n} 10\); ordinal collapsing functions (OCFs) map large ordinals (e.g., \(\omega_1^{CK}\), the Church-Kleene ordinal) to countable notations; fast-growing hierarchy \(f_\alpha(n)\) for \(\alpha\) up to and beyond \(\omega_1^{CK}\) via \(\Pi^1_1\)-CA0. | CONNECTION: No direct geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) appear; however, the recursive arrow-count growth \(A_n\) mirrors exponential/logistic scaling patterns seen in natural growth, and the ordinal hierarchy itself forms a well-ordered lattice structure (a total order with no infinite descending chains), analogous to root system posets in crystallography (e.g., \(A_n\) series) — though this is structural, not numeric. | DEPT Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com