Live picture

The forced left half, and where it is still

Hold the centre column to black, white, black, white for ever. The column beside it is then forced into the kicked wheel, and everything to the left of that is determined by the two. Wherever the wheel column holds one short word for a stretch of time, a cone of the left half is exactly periodic; the picture tints those cones. A kick ends the cone along a diagonal.

Right half (12 cells) tint the cones lag 4 yellow · 6 amber · 10 green · 56 teal · other blue-grey

Time runs down the page. Centre column in ember; the wheel column in blue; the right half is the twelve cells you typed, then white.

What you are seeing

Every left cell x(−k, t) is a function of the centre and the wheel column over the ticks t to t+k. If both columns repeat exactly at lag P for every tick of a stretch, every cell whose window fits inside the stretch repeats at lag P too. That is a cone pointing left, its apex depth equal to the stretch's length, its edges the diagonals along which the next kick's influence travels. Inside, the texture is a lattice; outside, it is the usual Rule 30 texture. The tint shows the smallest lag whose cone holds the cell. Typically a quarter of the drawn cells lie in some cone; a loose test finds no regularity outside them.

Sources. CHAT-LEDGER.14.md CL360 and tests/probes/lexicon/rule30_cloud_cone_render.py; Theorem B (RULE30-PRIZE.md 8.54) is the same statement for a column that repeats for ever.