Aurellion Function: Self-Referential Arrows Outrun Knuth and Reach ω-Level Growth — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: The Aurellion Function defines a fast-growing hierarchy via self-referential arrow counts, \(A_{n+1} = 10 \uparrow^{A_n} 10\), exceeding Knuth notation and primitive recursion. | MATH: \(A_1 = 10 \uparrow\uparrow\uparrow 10\); \(A_{n+1} = 10 \uparrow^{A_n} 10\). Growth rate: \(A_n \approx f_{\omega}(n)\) in the fast-growing hierarchy (since arrow count itself becomes the exponent of the next iteration — a diagonalization at the level of \(\omega\)). | CONNECTION: No direct geometric ratio (0.382, 0.618, 0.786, 1.618, 2.618) appears. However, the recursive self-embedding \(A_n \mapsto A_{n+1}\) mirrors the *golden spiral* self-similarity: each step scales the "complexity exponent" by its own magnitude — a discrete analogue of \(\varphi^n\) growth, but in ordinal space rather than Euclidean space. Also, the base 10 is arbitrary; in base 60 (sexagesimal), the same construction would yield \(B_1 = 60 \uparrow\uparrow\uparrow 60\), preserving the ordinal structure — showing base-in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212297992
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