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PreprintCC BY 4.0arXiv

Recurrent two-dimensional sequences over three and four letters simulating von Neumann constructions of finite ordinals

  • Xiaofang Jiang
  • Mihai Prunescu
  • Bogdan Dumitru
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Abstract

We study three recurrent double sequences whose colored diagrams encode mirrored versions of the finite-ordinal construction. Two sequences use three letters and realize variants with two and three initial symbols. A four-letter sequence realizes the von Neumann construction from one initial symbol. For each sequence we describe its cells explicitly and identify the ordered nesting of its rectangles. For four letters, we distinguish ordinary and virtual boxes and prove their assembly rules. The proofs proceed by induction along antidiagonals. For the constructions with one and three initial symbols, we reduce the recurrence to finite sets of local configurations and verify those sets by computer. We derive formulas for the numbers of nested boxes, including arithmetic-term representations, and we prove that none of the three sequences is k-automatic for any integer k ≥ 2.

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Cite

BibTeXRIS
BibTeX
@misc{jiang2026recurrent,
  title = {{Recurrent two-dimensional sequences over three and four letters simulating von Neumann constructions of finite ordinals}},
  author = {Xiaofang Jiang and Mihai Prunescu and Bogdan Dumitru},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01753},
  eprint = {2610.01753},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01753},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Recurrent two-dimensional sequences over three and four letters simulating von Neumann constructions of finite ordinals
AU  - Xiaofang Jiang
AU  - Mihai Prunescu
AU  - Bogdan Dumitru
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01753
UR  - https://arxiv.org/abs/2610.01753
AB  - We study three recurrent double sequences whose colored diagrams encode mirrored versions of the finite-ordinal construction. Two sequences use three letters and realize variants with two and three initial symbols. A four-letter sequence realizes the von Neumann construction from one initial symbol. For each sequence we describe its cells explicitly and identify the ordered nesting of its rectangles. For four letters, we distinguish ordinary and virtual boxes and prove their assembly rules. The proofs proceed by induction along antidiagonals. For the constructions with one and three initial symbols, we reduce the recurrence to finite sets of local configurations and verify those sets by computer. We derive formulas for the numbers of nested boxes, including arithmetic-term representations, and we prove that none of the three sequences is k-automatic for any integer k ≥ 2.
ER  -