{
  "abstract": "The set \\(P_{n}\\) consists of the characteristic vectors of pedigrees, which are combinatorial encodings in one-to-one correspondence with Hamiltonian cycles. For \\(n \\geq 4\\) and \\(τ_{n} = \\binom{n}{3}-1\\), the Membership Problem for the Pedigree Polytope (M3P) asks whether a given vector \\(X \\in \\mathbb{Q}^{τ_{n}}\\) belongs to \\(\\mathrm{conv}(P_n)\\). This paper gives a complete account, with the full proof statements and machine-verified Lean~4 certificates, of the necessary and sufficient (\\NS) condition established in Chapter~5 of \\emph{Pedigree Polytopes: New Insights on Computational Complexity of Combinatorial Optimisation Problems} (Arthanari, 2023): membership of $X$ in $\\mathrm{conv}(P_n)$ is characterised by the feasibility, at every stage $k=4,\\dots,n-1$, of a recursively constructed layered network, together with, when needed, a multicommodity flow problem $\\mathrm{MCF}(k-1)$ achieving its maximum possible flow $z_{\\max}$. We present the construction, the \\NS{} condition itself, and the proofs of this characterisation, together with the resulting polynomial-time algorithm for M3P. Both directions of the \\NS{} condition --- sufficiency and necessity (the multicommodity-flow feasibility argument of the book's Chapter~5) --- are machine-verified in Lean~4/Mathlib4 with zero unresolved \\texttt{sorry}s in either dependency chain. The \\NS{} characterisation therefore follows directly from the two machine-checked implications. The complete Lean~4 project is publicly available. A companion introductory article (arXiv:2507.09069) presents the same material for a broader readership, without the Lean~4 verification; the present paper adds full statements, proof sketches, and the complete machine-checked certificates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15914",
  "author_total": 1,
  "authors": [
    {
      "name": "T. S. Arthanari"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fzzky3czw36hq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15914",
  "fulltext_url": null,
  "id": "fzzky3czw36hq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15914",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7221799576",
  "pdf_url": "https://arxiv.org/pdf/2610.15914",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fzzky3czw36hq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15914",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15914"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "An Efficient, Lean 4 Verified Necessary and Sufficient Condition for Membership in Pedigree Polytopes",
  "title_text": "An Efficient, Lean 4 Verified Necessary and Sufficient Condition for Membership in Pedigree Polytopes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/fzzky3czw36hq",
  "venue": "arXiv",
  "versions": []
}