Rule 30: the center column

The question

Start with one occupied cell in an otherwise empty infinite row. Rule 30 updates every cell from its left neighbor, itself and its right neighbor:

next = left XOR (center OR right)

Read the cell at the original seed position after each update. Does that center sequence eventually repeat forever, after any possible starting time?

The nonperiodicity question asks for a later mismatch for every proposed positive period and every starting threshold. Checking a long prefix cannot by itself settle that statement.

Still open

I have not proved or disproved eventual nonperiodicity of the single-seed center column. My working notes contain algebraic reformulations, restrictions on proposed periodic traces, finite computations, Lean developments and failed approaches.

The manuscript candidates concern synchronization by blocks of bits and necessary conditions on a left ray under a hypothetical alternating center. These are intermediate questions. Excluding some periodic words does not exclude every word of every possible period.

What is public here

This page introduces the project. I have not released a research note, manuscript or reproduction package here yet.

You can try the update rule in the Rule 30 explorer. The diagram shows a finite calculation, not a proof about the infinite sequence.