{
  "abstract": "Let $V_i(n)$ denote the minimum worst-case number of comparisons needed to find the $i$-th smallest of $n$ distinct elements. We construct algorithms showing $V_7(16)\\le28$ and $V_8(16)\\le29$, improving the previously reported upper bounds of 30 and 31. Together with known lower bounds, these results establish $V_7(16)=28$ and leave $V_8(16)\\in\\{28,29\\}$. For 17 elements, we establish $V_5(17)=27$ and improve the upper bounds for ranks 6 through 9 to 29, 30, 31, and 31, respectively. Every new upper bound has an explicit decision-graph certificate verified by a static checker developed independently of the search. We also derive a lower bound for selection from a partially ordered state by combining the Fussenegger-Gabow weight with its order-reversed counterpart. All new upper bounds and this lower bound are formally verified in Lean 4. For $V_8(16)$, we show that a lower bound based solely on the total number of decision-tree leaves cannot establish $V_8(16)\\ge29$. The algorithms were found using fixed-budget search, information-guided comparison ordering, and proof-number search. Proof-number search was decisive in finding the 28-comparison algorithm; in our experiments, refuting a comparison budget remained substantially more expensive than finding an algorithm within it.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18740",
  "author_total": 1,
  "authors": [
    {
      "name": "Jonathan Cremer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/15t8vaq6gsnp2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18740",
  "fulltext_url": null,
  "id": "15t8vaq6gsnp2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18740",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7221756344",
  "pdf_url": "https://arxiv.org/pdf/2610.18740",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/15t8vaq6gsnp2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18740",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18740"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Selecting the 7th Smallest of 16 Elements Takes Exactly 28 Comparisons",
  "title_text": "Selecting the 7th Smallest of 16 Elements Takes Exactly 28 Comparisons",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/15t8vaq6gsnp2",
  "venue": "arXiv",
  "versions": []
}