Oracle Busy Beavers: A Transfinite Hierarchy Encoding Decidability Boundaries — E8 Intelligence Research

FINDING: Higher-order Busy Beaver functions, defined via Turing oracle machines, create a transfinite hierarchy whose values encode the decidability of number-theoretic statements, revealing a sharp boundary between computable and non-computable truth. | MATH: Let BB(n) = max steps among halting n-state 2-symbol TMs. Define BB⁽ᵏ⁾(n) via k-th order oracle (Turing jump ∅⁽ᵏ⁾). Key result (arXiv:2507.20321): For each k, BB⁽ᵏ⁾(n) grows faster than any function computable with ∅⁽ᵏ⁾-oracle, and the value BB⁽ᵏ⁾(n) is undecidable in the corresponding arithmetic fragment (Σ_k vs Π_k). Specifically, BB⁽ᵏ⁾(n) is not Σ_k-definable, but its totality is Πₖ₊₁-provable. The hierarchy is strictly increasing: BB⁽ᵏ⁾ ≺ BB⁽ᵏ⁺¹⁾ in growth rate, and the diagonal limit BB^ω(n) = BB⁽ⁿ⁾(n) is not computable by any finite-order oracle. | CONNECTION: The growth rates of BB⁽ᵏ⁾(n) follow a discrete exponential tower: BB⁽ᵏ⁾(n) ~ fω^k(n) in the fast-growing hierarchy, where ω^k corresponds to ordinal exponents. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com