Six states, one unsettled prefix
The Machine Waiting for One More Odd Number
Antihydra is a six-state Turing machine whose fate is equivalent to one exact question about odd and even terms in a growing integer sequence. Step the recurrence from 8, watch its balance move toward or away from halting, then run the machine's transition table and check that its tape decodes to the same arithmetic frames. The browser computes both layers live. Finite checks do not decide whether Antihydra ever halts.
Will some finite prefix contain strictly more than twice as many odd terms as even terms? One additional odd term at balance zero would end the machine.
I. The orbitGive the sequence another term
Begin at x = 8. Replace x by
floor(3x / 2). Before moving, an even term adds two to the safety balance.
An odd term removes one. The halt line is exactly b = -1.
Live Antihydra parity recurrence
- Current x
- computing
- Terms used
- computing
- Even E
- computing
- Odd O
- computing
- Balance 2E - O
- computing
Computing the seed.
The values grow, so the page uses exact BigInt arithmetic. The colored row keeps
only the latest terms on screen, but the counts and balance retain the whole prefix since the
canonical seed. Running longer is evidence about that finite prefix only.
II. The machine underneathMake the tape confess the same arithmetic
A recurrence could have been pasted onto any machine story. Here the browser
parses the published six-state code and executes its transitions on a blank tape. Whenever
the tape has Ligocki's recognizable frame, the page decodes the two runs of ones into
(x, b) and compares them with the recurrence.
Raw transition table
| state | read 0 | read 1 |
|---|
Each instruction writes a bit, moves left or right, then changes state. An undefined instruction halts.
Tape window
Parsing the machine.
| frame | raw step | decoded x | decoded b | recurrence expects | result |
|---|
The finite audit has not run yet.
For the audit range, every transition is executed. Nothing is interpolated. Beyond the displayed range, the recurrence instrument applies the closed rule directly and says so. That division is a free engineering choice: it keeps the page responsive while leaving a finite, inspectable bridge between tape and arithmetic.
The check
The browser reruns these checks from empty state when the page opens. Published inputs are identified separately from values derived here.
Recurrence prefix
Computing.
Balance identity
Computing.
Raw tape frames
Computing.
First possible crossing
Computing.
Computing the published random-walk expression.
The arXiv paper prints approximately 4.84 x 10^-224394395.
Direct high-precision recomputation of its displayed expression gives a mantissa near
9.05 at that exponent. We could not reconcile the printed approximation, so
the live result above is left visible rather than adjusted to match it.
Published comparison. The peer-reviewed STOC paper proves
S(5) = 47,176,870 after a Tree Normal Form enumeration of
181,385,789 five-state machines. Those two values are cited inputs, not results
of this page. They provide context only. Antihydra has six states.
Reported finite run. Ligocki's original analysis says mxdys simulated
2^31 = 2,147,483,648 recurrence iterations and reached
b = 1,073,720,884. This page cites that report. It does not pretend to rerun
it, and the report does not prove nonhalting. The paper's heuristic exponent is one more
than that reported balance.
Dated frontier. The BusyBeaverWiki holdout table listed
1,064 BB(6) holdouts on 28 July 2026. The table explicitly allows
community submissions without independent verification. That count is mutable, informal,
convention-dependent, and not fetched live here.
Free choices, conventions, approximations, and limits
- The canonical experiment starts from 8. Results for any other seed would not answer the Antihydra question.
- A term is counted before applying
H. Thus the seed 8 is the first counted term.EandOare exact prefix counts under that indexing convention. - The tape is bi-infinite, initially all zero, with state A and the head at position zero. A raw step writes, moves, then changes state. An undefined transition halts.
- The visible tape window is cropped, but the simulator stores every nonzero cell and frame decoding checks the full stored tape.
- The raw macro audit is finite. It does not import the complete formal proof of the equivalence, so it cannot establish the rule for every possible frame.
- Any finite recurrence run is compatible with a later halt. It cannot prove nonhalting.
- The random-walk model treats a deterministic parity process as if it followed a probabilistic model. No independence theorem justifies that move. Its tiny output is not a posterior probability, p-value, confidence level, or proof.
- Settling Antihydra is necessary for determining S(6), but not sufficient. Other holdouts remain, and a different halting machine could exceed the current lower bound.
- Frame decoding convention, and it is not a free choice. In a recognizable frame the tape carries two runs of ones: the right run of length a encodes the orbit value as x = a + 4, and the left run of length b encodes the parity balance. A frame is recognized only in state E, reading a 0, with a at least 4 and no ones outside the two runs. The offset of 4 is forced: any other offset fails to reproduce floor(3x/2) across the nine audited frames, so the tape and the recurrence would disagree.
- The raw machine simulation stops at a finite safety cap of 1,000,000 transitions, which is reached at frame 13 (x = 1021). Beyond that the frame walker cannot advance, and that is a limit of this page, not of the machine.
- The page uses no mutable remote data and makes no third-party request. It does load same-origin assets (the site fonts, the favicons and the shared top bar), and none of them carry data into the computation. Source links are ordinary navigation links.
Still open. No source checked on 29 July 2026 establishes whether the
seed-8 orbit ever reaches O > 2E, equivalently whether this machine halts.
The exact value of S(6) is also open. Antihydra is Collatz-like, not equivalent
to the classical Collatz conjecture.
Sources and status
- The bbchallenge Collaboration et al., Determination of the Fifth Busy Beaver Value, STOC 2026, pages 565 to 574, DOI 10.1145/3798129.3806376. Peer-reviewed conference paper, published 9 June 2026. Appendix C states the Antihydra equivalence. The longer arXiv:2509.12337v2, revised 23 March 2026, gives the random-walk expression and the formal S(5) result.
- Shawn Ligocki, BB(6) is Hard (Antihydra), 6 July 2024. Original public analysis, not peer-reviewed. It records mxdys's 28 June 2024 discovery, Racheline's high-level rules, the transition table, the recurrence, and the reported finite trajectory.
- Busy Beaver Challenge, Antihydra.
Community project page, not peer-reviewed. It publishes the short program with
a = 8,b = 0, even updateb += 2, odd updateb -= 1, anda += floor(a/2). - BusyBeaverWiki, Holdouts lists. Mutable community table, not independently verified. The dated count is context, not a theorem.
The offline differential verifier is at
research/one-more-odd-number/verify-one-more-odd-number.mjs.