{"paper":{"title":"Self-Referential $K$-SAT and the Finite Analogue of G\\\"odel's Incompleteness Theorem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DS","cs.IT","cs.LO","math.IT"],"primary_cat":"cs.CC","authors_text":"Jie Luo, Jun Liu, Ke Xu, Wen Fang, Xianxian Li, Yongxin Tong","submitted_at":"2026-07-02T03:52:23Z","abstract_excerpt":"Self-reference and solution independence are core properties underlying intractability. This paper establishes a finite combinatorial analogue of G\\\"odel's incompleteness theorems within Boolean $K$-SAT. While standard random $K$-SAT has assignment correlations that disrupt solution independence, we resolve this via a logarithmic-width ensemble ($K = O(\\log N)$). Here, satisfying assignments converge to a Poisson distribution, letting unsatisfiable and uniquely satisfiable formulas coexist. By executing a single-clause substitution conditioned on the unique solution, we construct structurally "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01671","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.01671/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}