Busy Beaver Function Has Turing Degree 0′ and Dominates All Computable Functions — E8 Intelligence Research

FINDING: The graph of the Busy Beaver function BB(n) is Σ₁⁰-complete, hence its Turing degree is exactly **0′** (the halting problem degree); no computable function dominates BB. MATH: - BB: ℕ → ℕ, BB(n) = max steps before halt over all halting n-state 2-symbol TMs (or max 1s written). - Undecidability: {⟨n,k⟩ : BB(n) = k} is Σ₁⁰-complete ⇒ Turing degree **0′** = deg(K) = deg(HALT). - Growth: BB is uncomputable and dominates every computable f; asymptotically BB(n) > f(n) for all total computable f, eventually. - Known values: BB(1)=1, BB(2)=6, BB(3)=21, BB(4)=107, BB(5)=47,176,870 (2024 coq-verified), BB(6) ≥ 10↑↑15 (lower bound, Antihydra 2024). - Σ₁⁰-completeness: A set A is Σ₁⁰-complete iff A ≡_m K; BB's graph is m-equivalent to the halting set. - Higher-order extension (arXiv 2507.20321): BB^(α) via oracle machines; degree climbs the arithmetical hierarchy — BB^(n) has degree **0^(n+1)**. - Post's theorem: Σ₁⁰-complete ⇔ degree 0′; Δ₂⁰ = computable in 0′. CONNECTION: No direct g Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com