{
  "abstract": "We study self-reference and solution independence in Boolean $K$-SAT as a finite analogue of Gödel incompleteness. Independence is defined through factorial moments of the solution count. Constant-width random formulas fail this criterion due to correlations among overlapping assignments. At a covering threshold, logarithmic-width random CNFs have a Poisson solution count, with positive probabilities of unsatisfiability and unique satisfiability. Replacing a redundant clause by one falsified by the unique solution yields an unsatisfiable companion preserving the unsigned incidence graph. For every permitted local view of a good uniquely satisfiable formula, an unsatisfiable companion preserves that view. Irreducibility defeats deterministic and bounded-error evaluators using $N^c$ clause queries, for fixed $0<c<1$. It establishes a local information obstruction, not an unrestricted time lower bound. For fixed $δ>0$, transcripts of $N^{1-δ}$ queries leave $N-o(N)$ bits of witness entropy. On typical support graphs, expansion yields linear Resolution width, linear PC/PCR and SOS degree, and exponential proof size. Bounds hold for semantic Cutting Planes under a partition condition and for Positivstellensatz with bounded product width, uniformly over unsatisfiable support-preserving signings. Structural evidence and the limits of acceleration strategies motivate the self-referential $K$-SAT exhaustive-search hypothesis (SRK-ESH): no uniform deterministic algorithm deciding the promise family achieves a fixed proportional saving in the exponent. This dual-evidence criterion combines intrinsic hardness mechanisms with algorithmic evidence. We interpret this hypothesis as a quantitative projection of the Gödel-style local--global gap. The information obstruction constrains quantum local observation and statistical learning from bounded local information.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.01671",
  "author_total": 6,
  "authors": [
    {
      "name": "Wen Fang"
    },
    {
      "name": "Xianxian Li"
    },
    {
      "name": "Jun Liu"
    },
    {
      "name": "Jie Luo"
    },
    {
      "name": "Yongxin Tong"
    },
    {
      "name": "Ke Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29sz62ddyx2h5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.01671",
  "fulltext_url": null,
  "id": "29sz62ddyx2h5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.01671",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1705|Constraint Satisfaction and Optimization"
    ]
  ],
  "openalex": "W7167258345",
  "pdf_url": "https://arxiv.org/pdf/2607.01671",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29sz62ddyx2h5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2607.01671",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.01671"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Self-Referential $K$-SAT and the Finite Analogue of Gödel's Incompleteness Theorem",
  "title_text": "Self-Referential K-SAT and the Finite Analogue of Gödel's Incompleteness Theorem",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/29sz62ddyx2h5",
  "venue": "arXiv",
  "versions": []
}