The Uncomputable Frontier: BB(5) Solved and Higher-Order Busy Beavers — E8 Intelligence Research

Abstract

FINDING: The Busy Beaver function BB(n) defines the maximum steps a halting n-state Turing machine can run; BB(5) was recently resolved via collaborative proof, and higher-order oracles extend the function into uncomputable hierarchies. | MATH: BB(5) = 47,176,870 (proven 2024, Scott et al.); BB(n) grows faster than any computable function — for any computable f(n), ∃N: BB(n) > f(n) ∀n>N. Higher-order BB: BB^α(n) where α is an ordinal (e.g., BB^1(n) = BB with oracle for halting of BB(0)-machines); BB^α(n) is uncomputable relative to lower-order oracles. Key relation: BB(n) is non-computable but Σ₁-definable; BB^α(n) corresponds to Σ_{α+1}-definability. | CONNECTION: No direct geometric ratio (0.382, 0.618, 1.618) appears. However, the *structure* of the proof for BB(5) used a decider tree with 27,000+ cases — a lattice-like combinatorial enumeration. The growth hierarchy of BB(n) mirrors ordinal hierarchies (ε₀, Γ₀) — a recursive "spiral" of complexity, analogous to self-similar scaling Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com