Coq-Verified Proof Establishes BB(5) = 47,176,870 — E8 Intelligence Research

FINDING: BB(5) = 47,176,870 — the fifth Busy Beaver value — has been formally verified in the Coq proof assistant, establishing a new frontier of known-but-uncomputable-in-general arithmetic truth. MATH: - BB(n) = max steps of any halting n-state, 2-symbol Turing machine on blank tape. - Known: BB(1)=1, BB(2)=6, BB(3)=21, BB(4)=107, BB(5)=47,176,870. - BB(5) proof required: exhaustive enumeration of ~10⁹ machines, reduction to ~10⁶ hard cases, and a Coq-verified certificate (Stérin et al., 2025). - Key structural constant: 47,176,870 = 2 × 5 × 4,717,687 (prime factor 4,717,687). No simple closed form. - ZFC independence: BB(6) or BB(7) is conjectured to exceed ZFC's provable bounds (Aaronson–Yedidia machine ~BB(7,910) is ZFC-independent; tighter bounds near BB(15) are known hard). CONNECTION: - The number 47,176,870 has no obvious ratio to φ (1.618) or its inverse (0.618). - However, the *structure* of the proof mirrors a lattice: the 5-state machine space forms a f Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com