The Uncomputable Growth of Busy Beaver Functions and Their Arithmetic Hierarchy — E8 Intelligence Research

FINDING: The Busy Beaver function BB(n) defines the maximum runtime of halting n-state Turing machines; its values are non-computable and grow faster than any computable function, with higher-order versions (oracle machines) creating a non-collapsing hierarchy tied to arithmetic truth. | MATH: BB(n) = max{ t | ∃ n-state TM M: M halts in exactly t steps }; BB(1)=1, BB(2)=6, BB(3)=21, BB(4)=107, BB(5)≥47,176,870 (proven value 47,176,870 in 2024); BB(6) ≥ 10↑↑15. Higher-order: BB_α(n) for ordinal α, where BB_{α+1}(n) uses a Σ_α-oracle; Σ_α formulas define the arithmetical hierarchy. Key theorem (from arXiv:2507.20321): Decidability of Σ_n formulas ⇔ computability of BB_n (n-th order Busy Beaver). Non-collapsing: BB_0 < BB_1 < BB_2 < ... in growth rate, each strictly dominating the previous. | CONNECTION: The growth hierarchy of BB_n mirrors the ordinal ladder of fast-growing functions (f_α in the Hardy hierarchy), which itself maps to ordinal notations — and ordinals beyond ε_0 connect to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com