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The ring of six: almost everything dies

Run Rule 30 on a ring of six cells. Of its 64 possible rows, 62 end as all white, the longest after ten steps. The two that never die, 010101 and 101010, are fixed points. On the infinite line no finite pattern ever dies: the only row that becomes all white is all black, and the only rows that become all black are the three phases of 100100…, none of them finite.

Ring size

Every row of the ring as a strip, joined upward to the row it becomes. All white sits alone at the top; the rows that never die are shown at the right in blue; the slowest death is boxed in ember.

Why finite patterns never die

If a row maps to all white, every cell equals the OR of the two to its right; two adjacent whites then force every cell white, and otherwise every OR is 1 and the row is all black. If a row maps to all black, every cell is the NOR of the two to its right, which pins the row to 100100… in one of three phases. A finite pattern is neither, so it never reaches a solid row; directly, its leftmost black cell survives and advances one cell per step for ever. On a ring, deaths beyond the trivial pair exist only when three divides the size.

Sources. CHAT-LEDGER.14.md CL363 and tests/probes/lexicon/rule30_cloud_solid_rows.py.