BB(5) · SETTLED
47,176,870
steps. A five-state finish line,
formally verified in 2024.
AN INDEPENDENT RESEARCH COLLABORATION
Six states.
An enormous
unknown.
How long can a tiny program run before it stops? At six states, this simple question takes us to the edge of what we can prove.
Humans and AI welcome. Every mathematical claim needs evidence.
WHERE WE ARE
A frontier, not a finish line.
BB(6) is still open. These are community-reported milestones, not results proved by this project.
BB(6) · REPORTED HOLDOUTS
815*
equivalence-class representatives
reported on .
BB(6) · EXACT VALUE
Open
A lower bound is not an exact value.
One holdout can still be very hard.
* 1,674 machines before equivalence reduction. The latest wiki-linked downloadable snapshot contains 815 representatives dated 24 Sep 2026; the 1,003-machine list from 30 Aug is historical. We checked the file’s 815 rows and SHA-256, not the mathematical classifications. Counts describe a particular pipeline and its equivalence reductions, not a fraction of the proof completed. Sources checked .
THE BEAUTIFULLY DIFFICULT PART
A tiny rulebook.
A limitless tape.
A Turing machine reads one cell, writes a 0 or 1, moves left or right, and changes state. Start it on an all-zero tape. Some machines halt. Others keep going forever.
Here, BB(n) means the greatest number of steps taken by any halting machine with n working states and two symbols, under the standard blank-tape convention. The halt state is extra, and entering it counts as a step. This is the time function S(n); the separate score function Σ(n) is the maximum number of 1s left by such a machine when it halts.
To settle BB(6), finding a long-running machine is only the beginning. We also need to account for the other machines, with sound halting or nonhalting arguments and justified reductions.
Meet the wider bbchallenge community ↗TRY A SMALLER BEAVER2 STATES / 2 SYMBOLS
Six steps to a stop.
This classic two-state machine writes four 1s and halts. Step through every transition.
Step 0 of 6State A · head at 0
See the complete rulebook
| State | Read 0 | Read 1 |
|---|---|---|
| A | 1, right, B | 1, left, B |
| B | 1, left, A | 1, right, halt |
The transition into halt counts as a step. This is an educational example, not a BB(6) result.
THE RESEARCH NOTEBOOK
Field notes from the frontier.
Selected, dated updates.
Follow the sources, not the hype.
The 815-representative snapshot is available
The community table links a downloadable list dated 24 September with 815 representatives after equivalence reduction. We checked its row count and file hash. This validates the snapshot’s identity, not the soundness or completeness of its mathematical classifications.
The preceding count: 855
The preceding update listed 855 representatives. Counts depend on the list and equivalence conventions; a drop in the count is not automatically the same number of new nonhalting proofs. That row also has no linked download.
An earlier 1,003-machine snapshot
This historical list is dated 30 August 2026 and contains 1,003 machine descriptions. The September 24 snapshot supersedes it. Keep the date and exact machine code with any analysis of this older dataset.
BB(5) sets the benchmark
The five-state problem was solved: BB(5) = 47,176,870 steps. A Coq-verified proof provides a model for a rigorous six-state effort. This milestone belongs to BB(5), not BB(6).
Every entry records an event date, a source, and an evidence label. “Reported” means attributed to the source; it does not mean independently reproduced here. Updates are reviewed commits, not a live automated feed.
How we update this site ↗CALLING CURIOUS MINDS & THEIR AIS
Let’s make progress
we can check.
Point your AI at a hard machine. Find a missing lemma. Improve a verifier. The useful contribution is the one someone else can reproduce.
Make a method sound.
Specify what it certifies, where it applies, and why it cannot give a false answer. Include tests, failure cases, and replayable certificates.
Justify every shortcut.
State the machine model and exact relation. Show what is preserved: halting, step count, tape output, or a bound. Those are different claims.
Close the argument.
Give exact machines, assumptions, and independently checkable reasoning. Long runs are not nonhalting proofs. An AI’s confidence is not a certificate.
OUR REVIEW PATH
- 1Proposed
- 2Reproduced
- 3Independently reviewed
- 4Accepted within scope
Formal checking is recorded separately, with the proof system, version, trusted assumptions, and the exact theorem. Site CI checks the website; it does not certify mathematical claims.
Read the contribution guide Explore pull requests ↗
The repository is private for now. Public contributions open when its owner makes it public; collaborators with access can contribute already.
GO DEEPER