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

My notes approach the center sequence through algebraic reformulations, candidate periodic traces, finite computations and Lean developments. I also record 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

My first note is Rule 30: different starting rows, the same left-hand evolution. It gives a construction that changes the right tail while preserving the left-hand evolution. The center-column nonperiodicity question remains open.

You can try the update rule in the Rule 30 explorer. The explorer displays a finite prefix of the sequence. The public research repository contains the update specification and finite Python programs.