Busy Beaver Dominates All Computable Functions, Proving a Hard Limit on Formal Systems — E8 Intelligence Research

FINDING: The Busy Beaver function BB(n) is total, well-defined, and strictly dominates every total computable function — its uncomputability is proven by diagonalization, establishing a hard ceiling on what any finite formal system can decide. MATH: - Definition: BB(n) = max{ steps halted by any n-state 2-symbol Turing machine that halts on blank input }. - Growth: BB(n) is total (defined for all n) but not computable. It grows faster than any computable function f: for every total computable f, ∃N such that ∀n>N, BB(n) > f(n). - Dominance proof (diagonalization): Assume BB computable. Then construct machine M_f that, given n, computes f(n), writes f(n) ones, and halts — using c states. Then BB(n + c) ≥ f(n). Choose f(n) = BB(n) + 1 → contradiction. Hence BB ∉ R (computable functions). - Known values: BB(1)=1, BB(2)=4, BB(3)=6, BB(4)=13, BB(5)=47,176,870 (Cooper–Kropitz 2024), BB(6) > 10↑↑15 (lower bound, 2022). - Growth rate exceeds all primitive recursive functions; BB(n) is not dom Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com