Lean formalisation of Busy Beaver deciders and results

Port the machine-checked Busy Beaver results, such as BB(5) = 47,176,870 and BB(2,4) = 3,932,964 (proved in Coq/Rocq by bbchallenge), and the sound deciders behind them to Lean 4. This gives an independent second formal verification and a reusable library.

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Cite

@misc{cairn-bbchallenge-lean-proofs,
  title        = {Lean formalisation of Busy Beaver deciders and results},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/bbchallenge-lean-proofs}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The bbchallenge collaboration proved BB(5) = 47,176,870 in the Coq proof assistant (now Rocq), announced in 2024. A human-readable paper followed in September 2025 (arXiv 2509.12337). The proof classifies all relevant 5-state, 2-symbol Turing machines using deciders whose soundness is proved in Coq: loops, n-gram closed position sets (NGramCPS), repeated word lists (RepWL), and finite automata reduction (FAR) with a weighted variant (WFAR). The Coq-BB5 repository also proves BB(2,4) = 3,932,964 and re-verifies BB(2), BB(3), BB(4) and BB(2,3). This problem asks for an independent formalisation of these results and tools in Lean 4. Open 6-state holdouts are handled in the parent problem.

What counts as progress

A Lean 4 definition of Turing machines and the BB/S functions (ideally compatible with Mathlib), with proofs of the small values BB(2) = 6, BB(3) = 21 and BB(4) = 107.
Lean soundness proofs of individual deciders (e.g. loops, NGramCPS, RepWL, FAR/WFAR) together with executable versions that produce checkable certificates.
Lean non-halting proofs for individual hard machines, named in standard bbchallenge notation.
A complete Lean proof of BB(5) = 47,176,870 or BB(2,4) = 3,932,964.
Documented comparisons showing where Lean and Coq definitions differ, with a proof of their equivalence where feasible.

How it is checked. Lean files must compile against a pinned toolchain and Mathlib version with no sorry. #print axioms on the main theorems must show only the standard axioms. Any use of native_decide or other trusted code must be declared, since it enlarges the trusted base. Reviewers also check that the formal statements match the informal claims.

Current state

The problem