{
  "schema_version": 1,
  "problem_number": "AMR-046-0017",
  "title": "A Sharp Three-Cycle Bound for Two-Zone Affine Systems with a Virtual Focus",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We prove that a planar two-zone piecewise-affine system with a straight switching line, two strict off-line foci, and at least one virtual focus has at most three isolated crossing limit cycles. The estimate includes nonhyperbolic cycles and every switching offset. Known three-cycle examples show sharpness.\n\nThe proof uses the Poincare half-map and extended Khovanskii framework of Carmona, Fernandez-Sanchez and Novaes. The additional argument rationally parametrizes contacts of their conic, rules out an exceptional denominator fiber, localizes the contacts by normalized equilibrium parameters, and establishes a trace-sign monotonicity certificate. Choosing between the full opposite-sign half-plane and an energy half-plane gives the intersection bound, with an explicit allowance for two components after clipping. Parameter perturbation treats degenerate conics and multiple isolated cycles.\n\nThis is a complete theorem for the stated focus family associated with AMR-046-0017 in UnsolvedMath v1.6.0, not a solution of the combined Gasull Problem 17 table or the general non-focus problem. The half-map methodology, endpoint-sum sign and three-cycle lower examples are credited prior work. The package includes an eight-page English manuscript and exact symbolic and rational regression checks. It is AI-assisted, originating-researcher self-audited and unrefereed. No independent review, proof-assistant verification, exhaustive novelty certification or absolute priority is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS"
  ],
  "keywords": [
    "piecewise-affine systems",
    "crossing limit cycles",
    "virtual focus",
    "Poincare half-map",
    "extended Khovanskii theory"
  ],
  "manuscript_version_date": "2026-10-10",
  "publication_date": "2026-10-10",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-10",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-046-0017/",
  "pdf_url": "https://eulersolve.org/papers/amr-046-0017/paper.pdf?v=b36181969da1",
  "doi": "10.5281/zenodo.23274752",
  "zenodo_record_url": "https://zenodo.org/records/23274752",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "At most three isolated crossing limit cycles in the stated canonical two-zone planar piecewise-affine system with two strict off-line foci and at least one virtual focus. Both signs and zero value of the switching offset, nongeneric contact conics and nonhyperbolic isolated cycles are included. Known examples give sharpness. No general non-focus, sliding-cycle or combined Gasull-table closure is claimed. The combined Gasull Problem 17 remains unresolved. This self-audited preprint is not independently peer-reviewed; absolute priority is not certified.",
  "files": {
    "paper.pdf": {
      "sha256": "b36181969da175edd76afe89111216ac64894a10cf1608682782268a3b01c914"
    },
    "source.zip": {
      "sha256": "f9bbf7de6e9177607f9c52d8fe504d6f478bcaff97d40b34a0f85cc106247503"
    },
    "verification_report.md": {
      "sha256": "5eea5f43c43aa7311bbcca13f5b52af1b17e0e748c844356c0e45134afe7d0cc"
    }
  },
  "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."
}