Busy Beaver Values Independent of ZFC for n ≥ 748 — E8 Intelligence Research

FINDING: Busy Beaver BB(n) is a well-defined, finitely-valued, uncomputable function whose values are provably independent of ZFC for sufficiently large n (Yedidia–Aaronson: n=7910 under ZFC, later reduced to ~748 states via improved encodings of a ZFC-consistency-checking Turing machine). | MATH: BB(n) = max steps before halt over all n-state 2-symbol halting TMs. Known values: BB(1)=1, BB(2)=6, BB(3)=21, BB(4)=107, BB(5)=47,176,870 (proven 2024 via Coq, "antihydra" resolved). BB(6) > 10↑↑15. Growth: BB(n) is non-computable, dominates every computable function; BB(n) > f(n) for any total computable f at some n. Aaronson–Yedidia construction: a TM of N states that halts iff ZFC is inconsistent ⇒ if ZFC consistent, BB(N) is ZFC-unprovable. Original N=7910; improved encodings (Riebel, O'Rear, et al.) reduced to N≈748. | CONNECTION: No direct geometric-harmony link. The relevant structural constants are combinatorial, not geometric: BB(5)=47,176,870 = 2·5·7·11·... (factorization: 2 × 5 × Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com