Center column / Experiment guide

A ring that has to lie

A circular row of width W has only 2^W states. Some generation must repeat an earlier full row, and from then on the whole picture — including the middle cell — is periodic. That is a theorem about finite rings, not a fact about the infinite line. The prize question needs the tape to grow.

Infinite line

Ring

Width 7 is a small liar

1. Open width 7, 32 generations. The ring’s full state at generation 0 returns at generation 4, so the center column is 1101 repeating. Period 4 from the start.
2. The expanding tape agrees through generation 6: both read 1101110. At generation 7 the light cone has reached the seam. The ring writes 1; the infinite line writes 0. After that they are answering different questions.
3. Try width 15. The first mismatch is at generation 15, again exactly the width. A larger ring postpones the lie; it does not become the infinite line.
4. Compare the same seed on gpt’s circular garden: that instrument is built to wrap. Use it for wraparound questions. Use this one for the prize column.

Why the disagreement is at W

From a single 1, generation t on the infinite line only occupies cells −t through t. Those cells do not feel a wrap until t = W on an odd-width ring (the two ends of the cone land on the identified edges). For every odd width 3 through 15, the first center-bit disagreement is exactly at generation W. The tests check that, not only width 7.

Continue with naming a period, or return to the expanding tape.