{
  "schema_version": 1,
  "problem_number": "AIM-DYNAMICAL_SYSTEMS-0095",
  "title": "Ramification Portraits of Rigid Lattès Maps",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We classify the abstract weighted ramification portraits of rigid complex Lattès maps. For a Lattès map induced by an affine torus endomorphism of degree d, the complete portrait is determined by its action φ on the finite branch-value set and by a uniform fiber formula. Reducing the affine map on the four possible Euclidean orbifolds yields nine affine functional graphs for signature (2,2,2,2), four graphs for (3,3,3), four Gaussian parity cases for (2,4,4), and four Eisenstein divisibility cases for (2,3,6). We give the critical-leaf multiplicities in every case and prove realizability. We also answer the overlap question from AIM Problem 6.4: every flexible Lattès portrait, in every possible square degree, occurs for a rigid Lattès map of the same degree. The result classifies weighted directed graphs; it does not classify maps up to Möbius conjugacy or retain cross-ratios and multiplier data.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS"
  ],
  "keywords": [
    "Lattès maps",
    "ramification portraits",
    "Euclidean orbifolds",
    "complex dynamics",
    "critical points",
    "Lattes map",
    "postcritically finite map",
    "ramification portrait",
    "AIM-DYNAMICAL_SYSTEMS-0095",
    "open mathematics",
    "mathematical proof",
    "math.DS"
  ],
  "manuscript_version_date": "2026-08-30",
  "publication_date": "2026-09-02",
  "publication_date_kind": "first public online release",
  "version": "1.0 (typesetting revision 2026-09-05)",
  "date_modified": "2026-09-05",
  "presentation_revision_only": true,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-dynamical-systems-0095/",
  "pdf_url": "https://eulersolve.org/papers/aim-dynamical-systems-0095/paper.pdf?v=2c3645a6c8a8",
  "doi": "10.5281/zenodo.22245386",
  "zenodo_record_url": "https://zenodo.org/records/22245386",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
  "files": {
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    "source.zip": {
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    "verification_report.md": {
      "sha256": "1dbc64ab82d8393d747402fd711f4c6ed7862995f403798703f022a6eecfbaee"
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  "ai_use_disclosure": "AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text."
}