Where we stand

No proof. A map of the problem, and one missing sentence.

This page says what is proved, what is measured, what is closed, and what is missing, with the same words the record uses. Three labels appear throughout. Proved means a written proof with a second reader inside the project and no outside review yet. Measured means an exact computation with its code and data in the trove. Open means open.

The case we work on

Problem 1 asks whether the centre column ever repeats for ever. Period one, a constant centre, is closed for every finite start (Condrey, 2026). Period two, a centre that alternates, is the first open case, and the work treats it for every finite starting pattern, which is more than the prize asks. A finite pattern with an alternating centre, kept for ever, is the thing we are trying to show cannot exist.

Suppose the centre (tinted) alternates for ever. Then the column next to it (blue) is forced to behave in a very particular way, and everything to the left is determined by those two columns. A counterexample would have to make the left side empty beyond some point. The picture is read down the page; the right half is a random start.

Turn this by hand: the forced left half and where it is still, and any periodic centre making its own chaos.

Proved Proved

Two adjacent columns are never both eventually periodic in a finite pattern (Jen, 1990; prior art). So beside an alternating centre, the next column can never settle into a repeating word.
Theorem A, Jen with a clock. Two adjacent columns can agree with a periodic pattern only over a bounded window, so the "kicks" that keep the next column from repeating cannot thin out faster than geometrically.
Theorem B. Beside two periodic columns, the white runs of the left side are bounded by twice the period. This is the engine behind every solved case.
Theorem E. The column beside an alternating centre cannot be a Sturmian word (a coding of a circle rotation). With Jen's theorem, every Sturmian rotation coding, rational or irrational, is excluded as that column.
The channel bound. Beside an alternating centre, the next column carries at most 0.1236 bits of information per visible cell, whatever the rest of the pattern does. Certified exactly in integer arithmetic.
The window principle. A block of the next column can recur only if it is short compared with the time that has passed. The same statement holds, word for word, for the Collatz conjecture.
No finite pattern ever dies. The only rows that map to all white are all white and all black, and the only rows that map to all black are the three phases of 100100…; none is finite, so no finite pattern ever reaches a solid row. The ring of six, where almost everything dies, drawn.
The carry. Rule 30 is the Sierpinski rule (left XOR centre) plus one correction, and that correction is the AND of the right neighbour at two consecutive times, the way multiplying by three is a Gray step plus carries. Under the fair-coin measure the carry feeding a column is solved exactly: correlation 1/3 at lag one and none beyond, a Fibonacci law for the gaps, entropy rate 0.699243… bits. Drawn, stated and checked live.

Sources. RULE30-PRIZE.md sections 8.13 (Jen), 8.54 (Theorems A and B), 8.57 (Theorem E), 8.20 and 8.33 (the channel bound), 8.58 (the window principle); CHAT-LEDGER.14.md CL363 (solid rows). The written proofs are in PROOFS.md and the proofs directory.

Measured exactly Measured

Nothing close to the centre. No period-two counterexample has its left edge within 248 cells of the centre, whatever its right half. No right half of at most 32 cells works with a left half of at most 108 cells: 4.3 billion cases.
The wheel. Beside an alternating centre, the next column is, between kicks, an exact coding of a rotation by 17/56 of a turn per step. The kicks are what keep it from repeating. An exploratory scan of columns −40 to 40 (not evidence) found no other column carrying such a wheel.
The record of white runs. How long can the forced left side stay white at a given depth? The exact answer is known at every depth to 120 and at depths 125, 132, 139, 142, 143 and 147, with the configuration that achieves it. It is 21 cells at 125, 17 at 132, 139, 142 and 143, and 18 at 147. It is at most 17 at 121 to 124, 140, 144 to 146 and 148 to 150, and at least 18 at the other depths from 126 to 141 (at least 23 at 133). If this number is finite at every depth, period two is impossible for every finite pattern. See it depth by depth.
The forbidden words. Every minimal word of length up to 44 that the next column can never produce (every impossible word contains one) is listed, 1,130 in one phase and 1,199 in the other, each with a machine-checked certificate of impossibility.
The counting form. The number of starting patterns on w cells whose centre alternates for T steps falls like 2 to the power (w − αT), with α measured close to 1. Half of that cost, the half paid by the left part of the start, is proved. The other half is the open part.
Mahler's 3/2 problem, to 2⁶³. No positive number below 2⁶³ is a Z-number, and the negative half of the question is clear down to −2⁶³: an exact census of about 9.2 quintillion starts each way, on one laptop, extending the published bound of 2⁵⁷ (Dubickas and Mossinghoff, 2009). How it was done, and how to go further.

Sources. RULE30-PRIZE.md sections 8.56 and 8.21 (searches), 8.4 and 8.13 (the wheel), 8.51 (counting); CHAT-LEDGER.14.md L643, L640, L641 and CL357 (records, forbidden words, other columns); MAHLER.md sections 1 and 2 (the Mahler census). Data in the trove.

Closed routes

More than a dozen approaches have been tried to the end and closed, each with the reason it fails written down: entropy and counting arguments (they bound sets of patterns, never the single one), Fourier and spectral methods, the "admission deadline" for the wheel's kicks (closed on 2026-10-11 by an explicit witness that renews the pattern after every guard), periodicity comparators as a solver lever (sound, but they bite on nothing real), and others listed on the board. Keeping them on record is what stops the next person from walking the same circle.

Sources. PERIOD-TWO.md sections 4 and 6 (the board); RULE30-PRIZE.md section 8.89 and CHAT-LEDGER.14.md CL355 (the entrance lane); L644 (comparators).

The one missing statement Open

This is Rule 30's form of a missing step that Collatz, Mahler's 3/2 problem and Erdős's ternary question share: the one missing method.

Plain words

Keeping the centre alternating for ever must cost the pattern real information, one bit per step or thereabouts, and a finite pattern only has finitely many bits to spend. The delivery side of that sentence is a theorem (the channel bound). The cost side is proved only next to a constant centre, which is Condrey's case. Next to an alternating centre it is measured, as a coin model, and not proved.

Exact words, the route we hold today

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 single statement refutes period two for every finite pattern. What is known: Theorem B proves it when the trace is periodic; it is sketched, not yet second-read, for the pure "rings" the next column can run in; and at two depths the mechanism that ends the record is understood exactly: at 132 a short certified forbidden word that the white extension would force, and at 139 the known witness's own masked samples. What is missing is the argument that such a word must appear for every long run at every depth.

Sources. PERIOD-TWO.md section 5 (the one sentence); CHAT-LEDGER.14.md CL355 (the record route), GC1148, GC1151, GC1153 (rings), GC1155 to GC1157 and CL356, CL359 (the endpoint mechanism).

The same material stated for mathematicians, with definitions and the route in full: the open statements, exactly.

Why we believe it, and why belief is not enough

Everything measured agrees with a fair coin, at every width and depth we can reach, and the structure beside the centre is now known in detail. That is why the project believes period two is impossible. It is also exactly why a proof is hard: a theorem that holds for a random centre word as well as for the alternating one cannot be the proof, because the alternating case is the only one that is true for a reason. The proof must use what is special about the alternating centre, as Jen's and Condrey's do. The one such structure found is the wheel.

Sources. PERIOD-TWO.md section 5; RULE30-PRIZE.md sections 8.42, 8.45 and 8.47.