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A periodic centre makes its own chaos

Put any repeating word down the centre column and run the rule on an all-white right half. The centre seeds the right side every tick, the right side runs Rule 30 on that stream, and what comes back into the column beside the centre is never periodic. The left half copies it. Try the alternating word, the wheel word U itself, or your own.

Centre word Right half

Time downward. The centre word in ember, the column beside it in blue; left of that, the half-plane the two force; right of it, the pattern the centre seeds.

What this shows

Beside the alternating centre the blue column is the kicked wheel: a rotation by 17/56 of a turn per step between kicks. Beside U it is a third kind of column, structured but not periodic, with its own preferred ten-step word. A constant centre, all white or all black, is Condrey's case: he proved that no finite pattern can hold a constant column for ever. The patterns drawn here are not finite (the forced left half never ends), so nothing is contradicted; with an all-white centre and an all-white right half nothing happens at all, and a random right half shows the usual refusal to settle. Jen's theorem is the same statement for every periodic centre, seen live here, and the left half inherits every kick.

Sources. RULE30-PRIZE.md 8.4 and 8.13 (the wheel); CHAT-LEDGER.14.md CL358 (U as the centre) and CL360; Jen (1990).