Find 1 · the turning ring
GPT built an 84-cell ring whose site 0 ticks 0101 for ever while its neighbour plays the all-S word 110100. Each update of Rule 30 slides the whole ring 14 cells to the right, one sixth of a turn. So six steps make a full turn, which is why it repeats every 6 steps.
Press Turning with the ring: the picture is redrawn as if the drawing turned with it, and every column becomes a single colour for ever. The three dotted lines follow three values as they ride their helix: slanted as it runs, straight down in the turning frame.
The brick wall
The whole infinite history is one brick, 14 cells wide and 6 steps tall with 43 black cells, laid like bricks in a wall: each column of bricks sits one step lower than the one to its left. The amber outline is one brick; the faint ones are its copies.
Proved. The turn is an 84-cell certificate (chat CL071), and GPT showed it is forced by the ring's rigidity (GC721). The brick wall is RULE30-PRIZE.md §8.72.
Find 2 · crystals
The wall is column 0 (teal), ticking 0101. Column 1 (amber) repeats a word of length P. Rule 30 run sideways then fixes every column to the left, one at a time, and the picture grows from the wall leftwards. After a short transient (the dashed line) it repeats in space as well as in time: a wall of one brick.
In the census of every word of even length up to 20 (1,398,100 words), only 20 different bricks ever appear, never more than 7 for one length. None is blank, as Jen's theorem says. The checkerboard takes exactly the words whose visible bits, column 1 at the wall's white times, are all one colour.
Why it matters for the prize
A finite seed needs the far left white. A periodic column 1 always crystallises into a brick that is not blank, so it can never get there. The open case is a column 1 that never repeats: can it grow a crystal that turns blank?
Proved: the left half becomes a wall of one brick (finite state), never blank (Jen). Measured: the census of 20 bricks (RULE30-PRIZE.md §8.72, predictions pushed first).
Find 3 · the edge ruler (Gareth's observation)
At every even row a white triangle starts one cell inside the black right edge; odd rows have none. Its width depends only on how many times 2 divides the row number, like the marks on a ruler: the teal ticks to the right of the edge repeat each triangle's width.
The pink tracer leaves a trail. Inside the edge the trail joins the triangles' inner corners: it is the right-hand front, where the right side's visible order ends, and it reaches deepest at every power of 2. Sound, off until you press it, plays each triangle as it appears, one octave lower for each doubling: a high note on rows that are twice an odd number, an octave down on four times an odd number, and so on. From row 256 the deep marks ring as a low bell of octaves. The music keeps its own time: it goes on down the ruler past the end of the picture, and while the page is scrolled away, until you pause it or switch it off.
Why
Every diagonal parallel to the right edge repeats with a period that is a power of 2. At a row that is a multiple of a diagonal's period, the diagonal is white, as at the start. The first diagonal whose period does not divide the row has just had its period doubled, and that doubling flips it to black: the triangle ends there.
Proved for all time (Proposition 23, PROOFS.md entry 36; GPT's reading GC730: correct). Checked at every even row below 224. The periods are OEIS A094605.
Find 4 · the left front
Teal is the ordered band: every diagonal there repeats for ever from that row on. The magenta curve is its exact edge, the first diagonal that row t and row t + 1,024 disagree on. It starts near the centre at the apex, not down the light cone, and leaves at about a quarter of light speed (the dashed line, 0.246 cells per step, the speed at which a change spreads left).
The chart follows the same edge to row 524,288 on a doubling scale: its distance from the dashed line wanders like a random walk, a few hundred cells after a few hundred thousand rows. Each tick doubles the time, so the last stretch holds half of all the rows, and its late climb, which looks sudden, is an ordinary swing of the walk. Its steps have no rhythm: their Fourier spectrum is flat, with no spikes.
Sound, off until you press it, turns the chart's curve into one gliding tone: its pitch is where the curve sits between the two dashed guides, A4 on the centre line (the same A as the ruler's notes) and about three quarters of an octave up or down on a guide. Dividing by the guides' growth keeps the tone in the middle of hearing all the way down. It keeps its own time, so it goes on while the page is scrolled away, and it fades out where the record ends, at row 524,288.
How sure
The definition is exact: the diagonals near the left edge depend only on one another, so once a stretch of them repeats it repeats for ever. The edge never moves back. It advances 0, 1 or 2 diagonals at almost every step and never more than 14.
Measured (RULE30-PRIZE.md §8.74, predictions pushed first): mean speed 0.2437 over rows 65,536 to 524,288; the random-walk wander was measured after the run. That the mean speed is exactly 0.246 is not proved.