{
  "abstract": "Exactly four dimension vectors carry representations of the quotient algebra A = kQ/I that satisfy the convergence constraint dim(V_T) = 1 and are indecomposable in Rep(A): the full-cycle (1, 1, 1, 1), the contextualized (0, 1, 1, 1), the bound-oscillation (0, 0, 1, 1), and (1, 1, 1, 2). The involution relations are γ₁² = e_L, γ₂² = e_L, γ₃² = e_T. The dimension bound (combining the convergence constraint with the involution relations and indecomposability) restricts V_L and V_S to dimension at most 1 and V_C to dimension at most 2, giving twelve candidate dimension vectors with d_T = 1, d_L, d_S ∈ {0, 1}, and d_C ∈ {0, 1, 2}. The enumeration proceeds by one principle, stated here as Lemma 1.2: a cycle relation γ² = e_v is vacuous when V_v = 0 and binding when dim V_v = 1, because the identity on a zero space is the zero map. Applying it, six candidates are eliminated by involution failure, two are eliminated because every representation carrying them decomposes, and four survive: (1, 1, 1, 1), (0, 1, 1, 1), (0, 0, 1, 1), and (1, 1, 1, 2). The candidate (1, 0, 1, 1) does not survive, because V_S = 0 forces γ₁ to act as the zero map on a one-dimensional V_L, where the relation γ₁² = e_L is binding and fails. The candidate (1, 1, 1, 2) survives because the two maps leaving C, to L and to T, can be independent functionals on a two-dimensional V_C, and a representation carrying it has endomorphism ring k. The four are indecomposable in the sense that matters: decomposability is tested in the full category Rep(A) rather than in the constrained class, because the class of representations satisfying dim(V_T) = 1 is not closed under direct sums and therefore admits no internal notion of decomposition. **Keywords:** quiver representation, indecomposable module, dimension vector, path algebra, bound quiver algebra, involution, cycle relations, Krull-Schmidt",
  "abstract_shown": true,
  "affiliations": [
    "NeuraVeXium LLC"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Arthur Stewart",
      "orcid": "0009-0001-8139-2645"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dvzbt7sxz7y0y.bib",
  "cited_by": 5,
  "doi": "10.5281/zenodo.21644859",
  "fulltext_url": null,
  "id": "dvzbt7sxz7y0y",
  "issns": [],
  "landing_url": "https://doi.org/10.5281/zenodo.21644859",
  "language": "en",
  "licence": "CC-BY-NC-ND",
  "licence_label": "CC BY-NC-ND",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7171525445",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-28",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/dvzbt7sxz7y0y.ris",
  "sources": [
    {
      "first_seen": "2026-11-01T13:20:02Z",
      "native_id": "W7171525445",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.21644859"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Algebraic structures and combinatorial models",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Four Indecomposable Representations of a Four-Node Quiver Under a Convergence Constraint",
  "title_text": "The Four Indecomposable Representations of a Four-Node Quiver Under a Convergence Constraint",
  "type": "preprint",
  "updated_at": "2026-11-04T13:21:00Z",
  "url": "https://research.draconitic.com/paper/dvzbt7sxz7y0y",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}