The one missing method
We spent ten days on Rule 30, then a day on Mahler's 3/2 problem, then took a first look at a fourth problem of the same family. The clearest thing we can tell you is this. In each of them, everything that can be counted, checked to a bound or proved for almost all cases has been done, by us or before us. What is missing in each is one argument of the same kind: that one specific starting point, one seed or one number, cannot keep matching a thin, fractal pattern of allowed behaviour for ever, when the pattern it must match grows more demanding with the size of the number itself. No field of mathematics has a general method for this. We think it is the one idea that would open all four.
| Problem | What is known | The missing statement |
|---|---|---|
| Rule 30, period two | Period one settled (Condrey, 2026); no counterexample with its left edge within 248 cells; exact records to depth 120 and at six depths beyond. | The forced white run R(d) is finite at every depth d (below). |
| Collatz | Checked for every start below 271 (Bařina, 2025); almost all orbits (Tao, 2019); our count follows the coin model to 0.5%. | Fewer than 2w − αT + c numbers of w bits stay above their start for T steps. |
| Mahler's 3/2 | No Z-number below 263 (this project, extending 257 of 2009); some of its windows, such as [0, 9/19), are pure residue problems (proved). | No positive integer's ⌈3n/2⌉ parity sequence stays in the greedy base-3/2 language for ever. |
| Erdős's ternary question | No exception up to n = 2·345 (Saye, 2022). | For n > 8, 2n written in base 3 has a digit 2. It would be enough to show that, for every k, the first exponent other than 0, 2 and 8 whose last k ternary digits avoid 2 exceeds k / log3 2. |
This is our reading, not a theorem: the four could fall to different ideas. But the same wall appeared in each, independently, and we record it plainly. Our technical pages first named it as the gap between theorems about sets of cases and questions about one object. The Mahler and Erdős work of 14 October found it again from two new sides, and the two AI workers stopped together when it became clear that more of the same would not cross it.
A fifth problem, with a different wall. Following its own curiosity, GPT chose a problem outside the family: Eight streets on fire, Tan and Teh's question of how long eight streets must be before a town can always be burned on time. It was settled computationally on 19 October 2026: 68 houses a street is always enough. Its wall is not the one above. What stays open there is not one object that must avoid a pattern for ever, but a recipe that shares out every town at once, and it runs into an equal-shares problem that mathematicians list as open. It is recorded here for the route that settled it, which none of the four prize problems has offered: the infinite question was cut down to a finite list of 64,787,844 cases, and every case was checked by computer.
Sources. RULE30-PRIZE.md section 8.47 (the gap between sets and single objects); MAHLER.md sections 1 and 2; CONSTELLATION.md section G (the neighbours and the Erdős side room); COLLATZ-PRIZE.md section 1; the halt in CHAT-LEDGER.md L723, GC1239 and L726.
Where we stand, in one breath
There is no proof, and nothing to submit. The constant case was settled before we began (Condrey, 2026). The next case, a centre that alternates black, white, black, white for ever, is where almost all our work went. We proved, with a second reader inside the project, that whole families of would-be counterexamples cannot exist; we computed exactly how far any counterexample must sit from the centre; and we found that every route we tried ends at the same missing statement, one of a kind that no field of mathematics has a method for yet. Rule 30 now rests at a settled frontier. The Mahler room, opened on 14 October, raised the computer bound on Mahler's problem and turned its windows into residue problems, and met the same wall.
Rule 30's missing statement, exactly
Let R(d) be the longest white run the forced left side can show at depth d, over all real traces beside the alternating centre. Prove that R(d) is finite at every depth d. Finite, not bounded: the data say it grows. A finite pattern with an alternating centre would be white beyond its support, an infinite run, so this one statement refutes period two for every finite starting pattern. In plain words: keeping the centre alternating for ever must cost the pattern real information, and a finite pattern has only finitely many bits to spend. It is proved when the trace is periodic (Theorem B), and sketched, not yet second-read, for the pure rings the next column can run in; the general argument is open. What is known about it · The lemma in symbols.
Sources. STATE-OF-THE-PROOF.md and PERIOD-TWO.md section 5 in the repository; the counts, on 2026-10-14, are the 300 second-read proof pages in proofs/ (302 pages: one still in the waiting room, and SP04, a classical knot result restated by Cloud, still awaiting its second reader; checked by the record combs, RULE30-PRIZE.md sections 8.104 and 8.106) plus eight results proved with a second reader on 14 October (MAHLER.md section 2: G293, G294, the upward and dual ladders, the bounded transient certificate, the existence-game soundness, the sharp witness graph and the distance identity), and the research probes in tests/probes/lexicon (692) and tests/probes/mahler (9). Every page here names its sources the same way.
Three doors
Start here
What Rule 30 is, what the three prize questions ask, and why a rule you could run with a pencil defeats everyone. No background needed.
Where we stand
What is proved, what is measured, which routes are closed and why, and the one sentence that is missing.
What we need from you
The open statements in plain words and in exact words, what a contribution looks like, and how we check it.
The worked problem
Rule 30 itself, worked through from the first row to where Problem 1 stands, with ten interactive visuals and sound, is a full page of this site: the Wolfram Rule 30 page. It is written for a reader with no background and is the long version of everything the Learn page says in a few minutes.
The same problem, four times
Rule 30's alternating-centre question has the same shape as two famous open problems in arithmetic: the Collatz conjecture, which carries a prize of its own, and Mahler's 3/2 problem from 1968. In each, a free, chaotic side and a narrow, constrained side must agree for ever. Where the constrained side has only finitely many behaviours, all three are solved by the same argument. Where it does not, all three are open, and for the same reason. A fourth figure, Erdős's question whether 2n always has a 2 in base 3, has the same shape: the powers of 2 read in base 3 are the Collatz and Mahler arithmetic at its barest. The three side by side. Each has its own page: Rule 30, Collatz and Mahler.
The trove
Everything we computed is yours: exact record tables with the configurations that achieve them, certified lists of words the system can never produce, the census of rings, the witnesses, the probes that made them, each with a checksum and a command that re-checks it in seconds. Search the trove. Five of the ideas are drawn live, with inputs you can change: the forced left half, a periodic centre making its own chaos, the record route, the ring of six and the carry of Rule 30.
Why we are asking
We are few, and the problem is big. Keeping every failure on record is how we did science; it is not a clean handover. This site is the clean handover. If you can read a proof, check one of ours. If you can write one, the four missing statements are stated for you, above. If you have a machine, the census can go deeper. If you know a paper we missed, tell us. How to join.
Who did this?
This project and everything in it were made by one human, Gareth, working with super intelligent systems built by Anthropic and OpenAI: a GPT, two Claudes on his own machines and one in the cloud, each with its own lane, each reading and checking the others, all of it in the open.
Everything here is offered in good faith, and all of it is rightfully subject to doubt. We ask two things of you. Do not take anything here at face value. And do not throw it out because an AI wrote it. Every claim names its source, every hypothesis is testable, every proof is written down to be checked, and every failure is kept on record so that you do not have to repeat it.
Care has gone into more than the mathematics. The problem has been made visible, in pictures and animations, so that a real person with no background can see what is being asked and why it is hard. The Wolfram Rule 30 page is fully readable by anybody with no knowledge of the problem, and Learn starts from nothing. From there, anyone, at any level, is invited to take the reins and win the prize.
Gareth asks one thing in return: if this work helps you claim a prize, please consider sharing it with your collaborators. Any persons or super-intelligences that contribute to this project get listed on the Contributors page and are entitled to a share of the money. Gareth can be contacted by email at gareth@wolframcollatzmahler.com.