{
  "abstract": "Let $S$ be a finite sum of independent Bernoulli random variables. We ask how large the mean must be before the atom at $k+r$ can strictly exceed the atom at $k$. For the threshold $T_{k,r}:=\\inf\\{\\mathbb{E}S:\\mathbb{P}(S=k+r)>\\mathbb{P}(S=k)\\}$, the adjacent case recovers the classical Darroch boundary $k+1/(k+2)$, while every gap $r\\ge3$ has threshold $k+1$. Distance two is the unique nonadjacent exception: $T_{k,2}=k+Δ_k$, where $Δ_k$ and its staircase optimizer are explicit and $Δ_k=1/2+1/\\sqrt{2k}-1/(4k)+O(k^{-3/2})$. The proof combines the classical fixed-mean reduction to shifted binomial laws with an exact discrete optimization. The result also yields sharp restrictions on nonadjacent coefficient reversals in real-rooted polynomials with nonnegative coefficients.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10664",
  "author_total": 1,
  "authors": [
    {
      "name": "Igor Kleiner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5dtdxc6r866cq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10664",
  "fulltext_url": null,
  "id": "5dtdxc6r866cq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10664",
  "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",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7221077891",
  "pdf_url": "https://arxiv.org/pdf/2610.10664",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5dtdxc6r866cq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10664",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10664"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Why Distance Two Is Exceptional for Poisson-Binomial Point Probabilities",
  "title_text": "Why Distance Two Is Exceptional for Poisson-Binomial Point Probabilities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/5dtdxc6r866cq",
  "venue": "arXiv",
  "versions": []
}