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PreprintCC BY 4.0arXiv

Diagonal Periods and Newton Supports of Rules 30, 86 and 135

  • Tigran Nersissian
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Abstract

The single-seed Rule 30 cone has unbounded least diagonal periods in both directions. On the finite-support side, first contact fixes the least Newton index and an integer polynomial lift gives a Fibonacci ceiling. For reflected Rule 86, a backward map on periodic tail profiles retains the zero boundary and bounds periods independently of transients. At depth m, the Rule 30 period is at least ⌊ m/2⌋+1; for natural Rule 86 cuts through depth m, the largest period is at least 1/2log2(m+2). This proves infinitely many reset/integration period doublings. The physical Rule 135 orbit is a translated complement of Rule 30 after its first step, yielding finite or cofinite supports. A separately initialized auxiliary recurrence has exact index-by-index stabilization. Residue tables and block formulas describe the remaining structure and permit evaluation from supplied representations. Construction and minimization remain separate costs.

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Cite

BibTeXRIS
BibTeX
@misc{nersissian2026diagonal,
  title = {{Diagonal Periods and Newton Supports of Rules 30, 86 and 135}},
  author = {Tigran Nersissian},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25077},
  eprint = {2609.25077},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25077},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Diagonal Periods and Newton Supports of Rules 30, 86 and 135
AU  - Tigran Nersissian
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.25077
UR  - https://arxiv.org/abs/2609.25077
AB  - The single-seed Rule 30 cone has unbounded least diagonal periods in both directions. On the finite-support side, first contact fixes the least Newton index and an integer polynomial lift gives a Fibonacci ceiling. For reflected Rule 86, a backward map on periodic tail profiles retains the zero boundary and bounds periods independently of transients. At depth m, the Rule 30 period is at least ⌊ m/2⌋+1; for natural Rule 86 cuts through depth m, the largest period is at least 1/2log_2(m+2). This proves infinitely many reset/integration period doublings. The physical Rule 135 orbit is a translated complement of Rule 30 after its first step, yielding finite or cofinite supports. A separately initialized auxiliary recurrence has exact index-by-index stabilization. Residue tables and block formulas describe the remaining structure and permit evaluation from supplied representations. Construction and minimization remain separate costs.
ER  -