BB(5) Solved, BB(6) Unresolved: Busy Beaver Frontier Beyond Computability — E8 Intelligence Research

FINDING: BB(6) remains unresolved; BB(5) = 47,176,870 was proven in 2024 via the collaborative Busy Beaver Challenge, and the frontier has shifted to BB(2,5) and BB(6) where lower bounds already dwarf any computable function's practical reach. | MATH: BB(n) grows faster than any computable function; BB(5) = 47,176,870 (proven); BB(6) > 10↑↑15 (lower bound, Rado-style); BB(n) is Σ₁-complete, sitting at the Δ⁰₂ level of the arithmetical hierarchy — strictly above the halting problem's Δ⁰₁. Growth dominates Ackermann A(n,n) and every primitive recursive function. | CONNECTION: No direct geometric-harmony link in the evidence. The relevant structural analogy is ordinal-indexed fast-growing hierarchy: BB(n) ~ f_{ω}(n) in the Wainer hierarchy, whereas Ackermann ~ f_{ω} too but with vastly smaller constants; the true BB growth is conjectured to reach f_{ω₁^CK} territory (Church–Kleene ordinal), which is the "limit" of computable ordinals — a transfinite boundary, not a golden-ratio boundary. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com