{
  "abstract": "We introduce and study \\emph{fixed degree addition chains} $d\\geq 2$, a natural generalization of classical (binary) addition chains in which each term may be formed as the sum of at most $d$ earlier terms. After giving a precise formal definition, we develop a four-class step partition (the $d$-dilate steps and three classes of non-$d$-dilate steps) that extends the usual doubling/large/medium/small decomposition from the binary case. For this partition, we define the notion of a \\emph{run} of step types (a maximal consecutive block of steps of a given type) and derive quantitative run-gap inequalities that relate run lengths and run composition to the absolute difference $s_j-s_{j-1}$ between consecutive chain elements; these estimates make the dependence on the degree $d$ and on a slowly varying parameter $\\delta(m)=1/\\log m$ explicit.\\\\ The main structural result, the \\emph{generalized run length-gap theorem}, gives simple combinatorial criteria which force unbounded gaps in any infinite sequence whose finite prefixes are fixed-degree chains. Concretely, under mild hypotheses asserting that, for all sufficiently large truncation indices $t$, the truncation contains runs whose lengths (or the number of large/medium steps they contain) grow at least on the order of $\\big(\\tfrac{\\log\\log T_t}{\\log d}\\big)^{1+\\epsilon}$ for some small $\\epsilon>0$, we prove $$ \\underset{n\\longrightarrow \\infty}{\\limsup}\\bigl(T_n-T_{n-1}\\bigr)=\\infty. $$ We also indicate that the growth hypothesis may be substantially weakened: it suffices that the relevant run length statistics tend to infinity with $t$.\\\\ The run length-gap framework clarifies how local patterns govern global gap behaviour, suggests natural density and typical-case questions for fixed-degree chains, and raises algorithmic problems about constructing short chains with a prescribed run structure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Theophilus Agama"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1nkp9e73cxwbb.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "1nkp9e73cxwbb",
  "issns": [],
  "landing_url": "https://hal.science/hal-05784814v1",
  "language": "en",
  "licence": "CC-BY-ND-4.0",
  "licence_label": "CC BY-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05784814/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1nkp9e73cxwbb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:50Z",
      "native_id": "hal-05784814",
      "source": "hal",
      "url": "https://hal.science/hal-05784814v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "ON THE GENERALIZED RUN LENGTH-GAP THEOREM",
  "title_text": "ON THE GENERALIZED RUN LENGTH-GAP THEOREM",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:50Z",
  "url": "https://research.draconitic.com/paper/1nkp9e73cxwbb",
  "venue": "HAL",
  "versions": []
}