Center column / Experiment guide

Name a period it cannot keep

The first prize question is whether the center column of rule 30, started from one cell on an infinite empty line, is ever eventually periodic. Eventually periodic with period p after generation s means c(t+p) = c(t) for every t ≥ s. One mismatch is a proof that that pair (s, p) is wrong. A clean prefix is only a clean prefix.

Try a few guesses

1. Open period 1 from the start. The first mismatch is at t = 1: c(1)=1 and c(2)=0. The column is not constant.
2. Open period 2 from the start. It fails immediately: c(0)=1 and c(2)=0.
3. Open period 15 after generation 40. It fails at t = 40 itself. Sliding the start later does not save a period that already disagrees at the first compared pair.
4. Open period 40 on only 20 generations. There is no mismatch because there are not 40 later bits to compare. That is a too-short tape, not evidence.

What a mismatch is allowed to mean

If c(t) ≠ c(t+p), then the sequence is not eventually periodic with that period from any start ≤ t. The tester reports the smallest such t ≥ s in the computed prefix. It cannot say the column is aperiodic. Wolfram’s published computation finds no period in the first billion bits; this page does not repeat that run, and a billion bits is still finite.

Rule 90 on the same expanding tape is eventually periodic: the center is 1, then zeros. You can watch that on the instrument. Linearity gives a closed form; rule 30 does not.

Continue with the ring that has to lie, or return to the expanding tape.