{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0153",
  "title": "An Exact Logarithmic-State Algorithm for Recursive Tight-Cycle Constructions",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "The classical recursive construction for tight-cycle-free three-uniform hypergraphs has optimal edge count R(n) = max_{0 <= m < n} {m binom(n-m,2) + R(m)}. We give an exact algorithm for a single input n that visits only O(log(n+2)) states. A square-completed cubic potential shows that every maximizing split is one of the two integers surrounding ((2n+1)sqrt(3)-(2n+3))/4. Their descendants occupy at most three consecutive integers per depth and halve in size. The same identity gives an explicit linear error bound and R(n) = (sqrt(3)/3-1/2)n^3 + (sqrt(3)-2)n^2/2 + O(n). Integer square roots suffice for exact evaluation and reconstruction of an optimal block sequence. Nearest-integer rounding alone fails at n=106. The construction, recurrence, leading density and finite directed-star interpretation are classical and credited. The single-forbidden-tight-five-cycle Turan density conjecture is not resolved. This is an unrefereed preprint; no absolute priority is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "cs.DS"
  ],
  "keywords": [
    "AIM-COMBINATORICS-0153",
    "tight cycles",
    "hypergraphs",
    "recursive constructions",
    "exact optimization",
    "integer rounding",
    "Turan density"
  ],
  "manuscript_version_date": "2026-10-16",
  "publication_date": "2026-10-16",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-16",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-combinatorics-0153/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0153/paper.pdf?v=e8aad00875fc",
  "doi": "10.5281/zenodo.23421447",
  "zenodo_record_url": "https://zenodo.org/records/23421447",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete exact-optimization theorem for the classical recursive construction, not a solution of the original AIM-COMBINATORICS-0153 single-forbidden-C5 Turan-density conjecture. The construction, recurrence, leading density and directed-star interpretation are credited to prior work. A corollary for forbidding both C4 and C5 uses the cited extremal theorem. O(log(n+2)) counts arithmetic and integer-square-root operations, not bit operations or full edge-list output. AI-assisted, self-audited and unrefereed; no certified historical priority, independent review or formal verification.",
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    "source.zip": {
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}