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PreprintCC BY 4.0arXiv

Finite Configurations Cannot Generate a Constant Trace in Rule 30

  • David L. Condrey
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Abstract

For every right half we determine the unique left half whose Rule 30 central trace is constant: an alternating tail selected by the leading one of the right half when the initial center is 0, and a single universal checkerboard when it is 1. Every nonzero member of either fiber carries infinitely many ones, so the zero row is the only finite configuration with a constant trace. For support radius w the sharp maximum constant-prefix length is 2*ceil(w/2)+1 when the initial center is 0 and 2*floor(w/2)+2 when it is 1, so the maximum over both is w+2, attained by exactly 2^w configurations for even w and 2^w - 1 for odd w. Hence no column of a nonzero finite Rule 30 orbit is eventually constant.

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Cite

BibTeXRIS
BibTeX
@misc{condrey2026finite,
  title = {{Finite Configurations Cannot Generate a Constant Trace in Rule 30}},
  author = {David L. Condrey},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09431},
  eprint = {2609.09431},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09431},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Finite Configurations Cannot Generate a Constant Trace in Rule 30
AU  - David L. Condrey
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09431
UR  - https://arxiv.org/abs/2609.09431
AB  - For every right half we determine the unique left half whose Rule 30 central trace is constant: an alternating tail selected by the leading one of the right half when the initial center is 0, and a single universal checkerboard when it is 1. Every nonzero member of either fiber carries infinitely many ones, so the zero row is the only finite configuration with a constant trace. For support radius w the sharp maximum constant-prefix length is 2*ceil(w/2)+1 when the initial center is 0 and 2*floor(w/2)+2 when it is 1, so the maximum over both is w+2, attained by exactly 2^w configurations for even w and 2^w - 1 for odd w. Hence no column of a nonzero finite Rule 30 orbit is eventually constant.
ER  -