{
  "abstract": "We give a self-contained pair-automaton proof of the exact compress-with-another threshold of the corrected Figure 3 family from a recent survey of open problems in synchronizing automata. For every $p \\geq 3$, the automaton has $n = 3p$ states and $μ(q_0) = 4p = 4n/3$. The word $(ba)^p(ab)^p$ attains this value. Two entrance potentials and an excluded region yield the lower bound, with the endpoint exception at $p = 3$ treated explicitly. A separate reset construction proves $\\operatorname{rt}(A_p) \\leq 3p^2 + 4p - 1$ for every parameter. Reproducible computations verify the transition and entrance identities; the all-parameter results follow from the explicit proofs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07044",
  "author_total": 1,
  "authors": [
    {
      "name": "Qichao Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/83n50669b5rh5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07044",
  "fulltext_url": null,
  "id": "83n50669b5rh5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07044",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.FL"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7220920499",
  "pdf_url": "https://arxiv.org/pdf/2610.07044",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/83n50669b5rh5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07044",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07044"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The compress-with-another threshold of Szykuła's Figure 3 family",
  "title_text": "The compress-with-another threshold of Szykuła's Figure 3 family",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/83n50669b5rh5",
  "venue": "arXiv",
  "versions": []
}