{
  "abstract": "This paper introduces and investigates a non-classical positional numeral system constructed on the normalized Mersenne basis $M_k = (2^k-1)/3$, structurally linking the classical Mersenne sequence (OEIS \\textbf{A000225}) with the integer sequence of alternating binary bitmasks (OEIS \\textbf{A002450}). The main objective of this study is to provide a rigorous algebraic proof of the system's completeness---specifically, theorems on the existence and absolute uniqueness of a canonical representation for any natural number. We show that the internal dynamics of the basis are deterministically governed by the splitting identity $3M_k = M_{k+1} + 2M_{k-1}$, which strictly restricts the valid alphabet of coefficients to the set $\\{0, 1, 2\\}$. Based on the lemma concerning the supremum of lower-order digits and a constructive greedy algorithm, it is proven that the proposed basis forms a closed and unambiguous algebraic space, entirely precluding the overlap of positional masses.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25091",
  "author_total": 1,
  "authors": [
    {
      "name": "E. Dyachenko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ce501fpjh9sm8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25091",
  "fulltext_url": null,
  "id": "ce501fpjh9sm8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25091",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Theories"
    ]
  ],
  "openalex": "W7214119453",
  "pdf_url": "https://arxiv.org/pdf/2609.25091",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ce501fpjh9sm8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25091",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25091"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2610|Advanced Mathematical Theories",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Algebraic Proof of Completeness of a Positional Numeral System in the Normalized Mersenne Basis",
  "title_text": "Algebraic Proof of Completeness of a Positional Numeral System in the Normalized Mersenne Basis",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/ce501fpjh9sm8",
  "venue": "arXiv",
  "versions": []
}