The Aurellion Function: A Recursive Fast-Growing Hierarchy Beyond Knuth Notation

Abstract

We introduce the Aurellion Function, a novel recursively defined fast-growing hierarchy based on Knuth's up-arrow notation, defined by $A_1 = 10 \uparrow\uparrow\uparrow 10$, $A_{n+1} = 10 \uparrow^{A_n} 10$, where the number of arrows in the operation increases superexponentially with $n$. We analyze its growth rate relative to classical hierarchies such as the fast-growing hierarchy $(f_\alpha)_{\alpha < \varepsilon_0}$, and discuss its provability status in formal arithmetic. We provide formal bounds showing $A_n$ dominates all functions provably total in Peano Arithmetic, situating the Aurellion Function near the proof-theoretic ordinal $\Gamma_0$ due to its ability to majorize all functions $f_\alpha$ for $\alpha < \varepsilon_0$. We also outline possible transfinite extensions indexed by countable ordinals, thus bridging symbolic large-number constructions and ordinal analysis.