{
  "abstract": "A set $A$ of integers is an $h$-basis for $n$ if every integer in $[0,n] = \\{0,1,2,\\ldots,n\\}$ belongs to the sumset $hA$, that is, can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. The classical postage stamp problem asks for the largest integer $n_h(k)$ such that $[0,n_h(k)] \\subseteq hA$ for some set $A$ of nonnegative integers of size $k+1$. We consider the effect of allowing negative elements. Let $n^{\\flat}_h(k)$ be the largest integer such that $[0,n^{\\flat}_h(k)] \\subseteq hA$ for some set $A$ of integers of size $k+1$ containing at least one negative integer. For $k=2$, we have $n^{\\flat}_h(2) < n_h(2)$ if $h$ is odd or $h=4$ and $n^{\\flat}_h(2) = n_h(2)$ if $h$ is even and $h \\neq 4$. Proofs of these comparison results were autonomously discovered and formally verified in Lean by https://aristotle.harmonic.fun/}{Aristotle}, a formal reasoning agent developed by https://www.harmonic.fun/}{Harmonic}. In this paper we give new proofs of these results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19470",
  "author_total": 3,
  "authors": [
    {
      "name": "Wouter van Doorn"
    },
    {
      "name": "Pietro Monticone"
    },
    {
      "name": "Melvyn B. Nathanson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cw2245w4eyjww.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19470",
  "fulltext_url": null,
  "id": "cw2245w4eyjww",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19470",
  "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",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7222238423",
  "pdf_url": "https://arxiv.org/pdf/2610.19470",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cw2245w4eyjww.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19470",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19470"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Positivity and negativity for $h$-bases for $n$",
  "title_text": "Positivity and negativity for h-bases for n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/cw2245w4eyjww",
  "venue": "arXiv",
  "versions": []
}