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.

Current status: unknown

This project has not proved or disproved eventual nonperiodicity of the single-seed center column. The research working files contain algebraic reformulations, restrictions on proposed periodic traces, finite computations, proof-assistant developments and records of 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 is an introduction to the project. No research manuscript or reproducibility package has yet been released through this archive. A future release must select one coherent source version and supply its scope, assumptions, checks and files.

The Rule 30 explorer in the software section draws a finite single-seed evolution in your browser. It illustrates the update rule; it is not evidence for an infinite theorem.