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Level A · Machine-checkable Computability P-busy-beaver-holdouts

Deciding hard small Turing machines (Busy Beaver)

Prove halting or non-halting of specific small Turing machines that current deciders cannot resolve.

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Cite 
@misc{cairn-busy-beaver-holdouts,
  title        = {Deciding hard small Turing machines (Busy Beaver)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/busy-beaver-holdouts}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

After BB(5) was determined with a formal proof (bbchallenge collaboration, 2024), attention moved to 6-state and 2-symbol/other small machine classes, where some individual machines ("cryptids" and holdouts) resist all known deciders.

Progress: a decider (with its code and a certificate format) that resolves additional machines, a non-halting proof for a named machine, or a formalised proof. Name machines in standard notation and link the bbchallenge page.