{
  "abstract": "Dedekind's problem counts monotone Boolean functions, equivalently downsets of a Boolean lattice. We recast this enumeration as a finite layer-ratio reconstruction problem for the Whitney numbers of the ranked ideal lattice. An exact adjacent-layer double count expresses each layer ratio through local averages of the number of addable elements and the number of removable elements. Reversible fixed-layer Markov chains estimate these averages and hence estimate the Dedekind number $M(n)$. Backtests at $M(8)$ and $M(9)$ calibrate seed-level variability under the fixed protocol and measure the observed Monte Carlo budget scaling. The resulting estimate probes the Whitney-number sequence of the ideal lattice. Although these rows have previously been described empirically as unimodal, the high-precision $n=9$ estimate has a shallow two-shoulder feature around the central rank, contrary to that empirical description; $n=11$ and $n=13$ center-window estimates show a larger-contrast analogous pattern. The protocol estimate for $M(10)$ is \\[ \\widehat M(10)=(8.9360\\pm0.0010)\\times 10^{78}, \\] where the displayed uncertainty is the budget-based forecast scale from the cross-$n$ scaling law under the production budget.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Mathematics 14(18), 3300 (2026)",
  "arxiv": "2606.09795",
  "author_total": 5,
  "authors": [
    {
      "name": "Tian-Shun Chen"
    },
    {
      "name": "Hao Feng"
    },
    {
      "name": "Haozhe Wang"
    },
    {
      "name": "Chian-Shu Chen"
    },
    {
      "name": "Kilar Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/47xemt90k0s7b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.09795",
  "fulltext_url": null,
  "id": "47xemt90k0s7b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.09795",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "hep-th"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2606.09795",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/47xemt90k0s7b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:25Z",
      "native_id": "2606.09795",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.09795"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Finite-n Estimate of Dedekind Numbers by Layer-Ratio Monte Carlo",
  "title_text": "Finite-n Estimate of Dedekind Numbers by Layer-Ratio Monte Carlo",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/47xemt90k0s7b",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.3390/math14183300",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}