{
  "abstract": "We settle three conjectures concerning a doubly symmetric binary source $(X,Y)$ with crossover $p$. Consider Markov chains $U - X - Y - V$ with $U,V$ binary, and let $\\mathcal{A}$ be the set of rate triples $(I(U;V),I(U;X),I(Y;V))$ attainable with arbitrary binary channels $X\\to U$, $Y\\to V$, and $\\mathcal{B}$ the subset attainable with binary symmetric channels. The averaged BSC conjecture, Conjecture 5.2 of Pichler, Piantanida and Matz (2022), asserts $\\operatorname{conv}\\mathcal{A}=\\operatorname{conv}\\mathcal{B}$. We prove this for every $p\\in[0,1]$. Two conjectures of Dikshtein, Ordentlich and Shamai (2022) concern the double-sided information bottleneck at $p=0$, where $Y=X$ and the two channels see the same source: their Conjecture 1 identifies the exact maximum of $I(U;V)$ at prescribed rates $I(U;X)$ and $I(Y;V)$, and their Conjecture 2 the exact minimum. We prove both for binary $U,V$: the two extrema are attained by the same pair of Z/S-channels, in opposite orientation for the maximum and in the same orientation for the minimum. The proofs were found with substantial AI assistance, and all three theorems are formalised in Lean 4 with Mathlib, depending only on the standard axioms. The development is available at https://github.com/g-pichler/bsc-averaging . The proof of Conjecture 1 of Dikshtein, Ordentlich and Shamai (2022) contains three certified computations, a polynomial bound, an interval sweep and a polynomial positivity certificate, all of which are checked in Lean.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27991",
  "author_total": 1,
  "authors": [
    {
      "name": "Georg Pichler"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3nwqd5m1b3tmq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27991",
  "fulltext_url": null,
  "id": "3nwqd5m1b3tmq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27991",
  "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",
      "math.IT"
    ],
    [
      "openalex",
      "1705|Error Correcting Code Techniques"
    ]
  ],
  "openalex": "W7214151254",
  "pdf_url": "https://arxiv.org/pdf/2609.27991",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3nwqd5m1b3tmq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27991",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27991"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Three Conjectures on Binary Channels for the Doubly Symmetric Binary Source",
  "title_text": "Three Conjectures on Binary Channels for the Doubly Symmetric Binary Source",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/3nwqd5m1b3tmq",
  "venue": "arXiv",
  "versions": []
}