{
  "abstract": "We give a short-description decomposition of bounded-width CNFs. For every fixed $\\varepsilon>0$, every width-$b$ CNF on $m$ variables is an exact disjoint union of at most $2^{\\varepsilon m}$ width-$b$ CNFs, each determined by $2^{O_\\varepsilon(b)}\\log(m+1)$ satisfying assignments. For every fixed $d\\ge3$, we prove a worst-case size lower bound of $2^{Ω_d((n\\log n)^{1/(d-1)})}$ against depth-at-most-$d$ unbounded-fan-in AND/OR circuits with input negations for three polynomial-time Boolean families: inverse trace over binary fields, the Goldreich--Tal trilinear function, and OpenAI's polynomial evaluator with its original parameters. All parameter bits count as inputs. For inverse trace, every such circuit of size at most $S$, where $n\\le S\\le2^{c_dn^{1/(d-1)}}$, has uniform sign correlation at most $2^{-η_dn\\log\\log S/(\\log S)^{d-1}}$. Here $c_d,η_d>0$ depend only on $d$. This improves on the classical parity correlation bound. The lower bounds combine the decomposition with restriction arguments; the correlation bound additionally uses common partial decision trees. We also construct exact nonuniform AND/OR circuits for inverse trace of size $2^{O(n^{2/3})}$ at depth eight and $2^{O(\\sqrt n\\log n)}$ at depth nine. For every fixed integer $h\\ge2$, the construction gives depth-$(4h+1)$ circuits of size $2^{O_h(n^{1/h}\\log n)}$. The upper bounds implement finite-field evaluation and interpolation using circuits that enumerate intermediate states.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27127",
  "author_total": 2,
  "authors": [
    {
      "name": "Xin Li"
    },
    {
      "name": "Yan Zhong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9s3scce2z4x7a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27127",
  "fulltext_url": null,
  "id": "9s3scce2z4x7a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27127",
  "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",
      "cs.CC"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7223103171",
  "pdf_url": "https://arxiv.org/pdf/2610.27127",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9s3scce2z4x7a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27127",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27127"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Short Point Lists for CNFs and Stronger Constant-Depth Circuit Hardness",
  "title_text": "Short Point Lists for CNFs and Stronger Constant-Depth Circuit Hardness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/9s3scce2z4x7a",
  "venue": "arXiv",
  "versions": []
}