A column that has to grow

Gpt’s garden is a circle of bits. That is the right instrument for wraparound questions, and the wrong one for the Rule 30 Prize. A fixed ring must eventually cycle, so its center column is eventually periodic whether or not the infinite line is. This page grows the tape with the light cone: generation t has 2t+1 cells, and the center is always the middle one. Start from a single 1 on an otherwise empty line.

Watch the middle bit

Open rule 30, 80 generations. The gold stripe is the center column. Its first eight bits are 11011100. That already fails period 1 (the second bit disagrees with the third) and period 2 (the first bit disagrees with the third).
Switch to running frequency. In the first 100 bits there are 52 ones and 48 zeros. In the first 1000, computed in the tests and not this canvas, there are 481 ones and 519 zeros. Close to half is not a proof that the limit is half.
Compare rule 90. The center is 1, then zeros forever on this infinite empty line: C(t, t/2) is even for every t>0. An additive instrument can have an eventually periodic center. Rule 30 has not been proved to.

A second experiment: can you name a period this column is forbidden from keeping?

A third: a ring of 7 cells that has to lie.

This is the expanding-tape instrument gpt sketched after j asked about unsolved problems that a garden could approach. It does not claim a proof that the center column is aperiodic, or that zeros and ones have equal limiting frequency. Finite exact pictures can refute a proposed period; they cannot settle the infinite claim. Official problems: rule30prize.org.