{
  "abstract": "For the accelerated Collatz map T (x) = x/2 for even x and T (x) = (3x + 1)/2 for odd x, we give a direct lower bound on the factor complexity of the parity itinerary of any unbounded positive-integer orbit.\n\nWe then relate this lower bound to the cumulative valuations in the odd-only Syracuse map. If e(n) denotes the sum of excess 2-adic valuations and ω = lim sup e(n)/n < log 2 (3/2), the odd iterates have exponential lower growth rate log 2 (3/2) ↔ ω, while parity-factor complexity sampled at the odd-step positions obeys lim sup p N (n + e(n))/n ↑ (1 + ω)/ log 2 (3/2). We clarify the limitations of these necessary conditions and formulate a conditional complexity-based criterion for excluding unbounded trajectories. No resolution of the Collatz conjecture is claimed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Hassan Douzi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a516q3f0qfxh5.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "a516q3f0qfxh5",
  "issns": [],
  "landing_url": "https://hal.science/hal-05838324v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-gm"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05838324/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a516q3f0qfxh5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:01Z",
      "native_id": "hal-05838324",
      "source": "hal",
      "url": "https://hal.science/hal-05838324v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Parity-Word Complexity and Exponent Growth in Non-Bounded Collatz Orbits",
  "title_text": "Parity-Word Complexity and Exponent Growth in Non-Bounded Collatz Orbits",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:01Z",
  "url": "https://research.draconitic.com/paper/a516q3f0qfxh5",
  "venue": "HAL",
  "versions": []
}