{
  "abstract": "Let f_R(n) be the least possible largest element of a set of n positive reals whose 2^n subset sums are pairwise at least 1 apart, and let f_Z(n) ≥ f_R(n) be the corresponding integer quantity of the Erdős distinct subset sums problem.\n\nFixing the signs of all signed relations &epsilon;·x, &epsilon; &isin; {−1,0,1}^n, decomposes the real problem into polyhedral chambers, on each of which it is a linear program.\n\nWe prove that for every n the Conway–Guy set is the unique optimum of its chamber.\n\nThe proof is an explicit positive dual certificate consisting of n signed relations that are tight at the Conway–Guy set and unit lower triangular in gap coordinates; the relations are described by blocks of consecutive coordinates together with the positions n − k(k+1)/2, and tightness reduces to an identity of the Conway–Guy sequence proved by induction.\n\nWe then show, by a branch-and-bound over chambers in which every leaf carries an exact rational certificate, that f_R(n) = f_Z(n) for all n ≤ 8; in particular f_R(7) = 44 and f_R(8) = 84. We also prove chamber optimality for both of Bohman's families, which first improve on Conway–Guy at n = 12 (1159 < 1164): every Bohman set is the unique optimum of its chamber, with a block-triangular certificate given by explicit formulas. Finally we exhibit chambers whose optimal vertex is not an integer point.",
  "abstract_shown": true,
  "affiliations": [
    "Independent researcher"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Esther Notik",
      "orcid": "0009-0005-4076-6531"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dd45jfde2t9z7.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23647845",
  "fulltext_url": null,
  "id": "dd45jfde2t9z7",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23647845",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23647845/files/Notik_separated_subset_sums_v6.1.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dd45jfde2t9z7.ris",
  "sources": [
    {
      "first_seen": "2026-11-08T12:21:03Z",
      "native_id": "23647845",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23647845"
    }
  ],
  "subjects": [],
  "title": "Positive Dual Certificates for Real Separated Subset Sums: Chamber Optimality of the Conway–Guy and Bohman Sets, and Exact Values up to Eight",
  "title_text": "Positive Dual Certificates for Real Separated Subset Sums: Chamber Optimality of the Conway–Guy and Bohman Sets, and Exact Values up to Eight",
  "type": "preprint",
  "updated_at": "2026-11-08T12:21:03Z",
  "url": "https://research.draconitic.com/paper/dd45jfde2t9z7",
  "venue": "Zenodo",
  "versions": []
}