{
  "schema_version": 1,
  "problem_number": "AMR-029-0066",
  "title": "Integer Decompositions for Independent Local Quotas on Forests",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let F be a finite forest. At each vertex v, choose a subset S_v of its neighbors and an integer interval [l_v,u_v]. We consider independent sets T for which l_v <= |T intersect S_v| <= u_v whenever v is not in T. Their convex hull is described by vertex bounds, edge inequalities and the perspective inequalities l_v(1-x_v) <= x(S_v) <= u_v(1-x_v). We give a constructive integer decomposition for every integer point in every positive integral dilation. The decomposition uses at most 3|V(F)|-1 distinct admissible sets, with integer multiplicities, and can be computed in time polynomial in the binary input size. The proof combines compressed circular star schedules with integer matching along tree separators. Exact Python diagnostics and source files accompany the manuscript. The ordinary forest-kernel description is a known specialization. This is a complete theorem for the stated forest quota class, not a characterization of kernel polytopes of arbitrary digraphs or closure of the entire AMR-029-0066 source programme. Unrefereed preprint; absolute novelty is not certified. Developed with AI assistance; the author is responsible for the claims.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.OC"
  ],
  "keywords": [
    "AMR-029-0066",
    "forest graph",
    "local quotas",
    "integer decomposition property",
    "polytope",
    "junction tree"
  ],
  "manuscript_version_date": "2026-10-24",
  "publication_date": "2026-10-24",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-24",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-029-0066/",
  "pdf_url": "https://eulersolve.org/papers/amr-029-0066/paper.pdf?v=ff5425fb0f2b",
  "doi": "10.5281/zenodo.23609821",
  "zenodo_record_url": "https://zenodo.org/records/23609821",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For finite forests with one conditional unweighted interval quota on a neighbor subset per vertex, the displayed inequalities describe the admissible independent-set hull. Every integer vector in qP has an integer weighted decomposition with support at most 3n-1, computable in polynomial binary input time. The general EGRES kernel-hull programme remains partially solved. The result concerns independent vertex sets with one unweighted interval quota per vertex on a forest. The ordinary forest kernel hull and general junction-tree gluing are known and credited; absolute novelty and priority are not certified. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "ff5425fb0f2bf17f28784dd10ba001d3ab09c0c27fcc7405ecfb51d8d1a31c5e"
    },
    "source.zip": {
      "sha256": "32cef079310ca4a77f9669aaf012ed5def3b2cacd1efe25e5b2b0f85269a1d7c"
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    "verification_report.md": {
      "sha256": "c3b970fa52bf6e8e00ded6ead9ba93706dd75f02f28e6a78ad1439e5b565d98f"
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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."
}