{
  "schema_version": 1,
  "problem_number": "AMR-011-0049",
  "title": "Exact Word-Map Distributions on Four-by-Four Unitriangular Groups",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give the exact probability distribution of every group word, with no constants and any number of variables, on the full unitriangular group UT4(Fq), for every finite field. A nonzero exponent sum modulo the characteristic makes the word map uniform. Otherwise the distribution is determined by the rank of its degree-two noncommutative coefficient matrix and an exact bilinear zero count on the matrix kernel. The formula includes characteristics two and three and proves the Amit-Ashurst lower bound for every nonempty word fiber on this family. It also classifies the word image as the whole group, its derived subgroup, its center or the identity. A separate elementary split-extension argument gives an identity-fiber bound on full unitriangular groups in every dimension. The general problem for arbitrary finite p-groups is not resolved.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "word maps",
    "unitriangular groups",
    "finite fields",
    "Amit-Ashurst conjecture",
    "word distributions",
    "AMR-011-0049"
  ],
  "manuscript_version_date": "2026-10-04",
  "publication_date": "2026-10-04",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-04",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-011-0049/",
  "pdf_url": "https://eulersolve.org/papers/amr-011-0049/paper.pdf?v=a3d4fdf12ca7",
  "doi": "10.5281/zenodo.23142917",
  "zenodo_record_url": "https://zenodo.org/records/23142917",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete theorem for full UT4(Fq), any finite field and any word without constants. This does not solve the full Amit conjecture, AMR-011-0049 for arbitrary finite p-groups, or arbitrary matrix subgroups. No polynomial-time algorithm for the remaining bilinear zero count is claimed. The elementary all-dimensions identity bound is not claimed as new. Novelty and absolute priority remain undetermined. AI-assisted, self-audited and unrefereed; no independent specialist review or proof-assistant verification is claimed.",
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    "source.zip": {
      "sha256": "257a400b5d1b3e8f6e4d9f9ee4289295e045873888a8e583519f2f9e5fad2bd9"
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    "verification_report.md": {
      "sha256": "59ee95e8bc7e2ebad34d65e7a1c587b64e4f52918af88852a1e4e9dd96bfe673"
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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."
}