{
  "schema_version": 1,
  "problem_number": "AIM-DYNAMICAL_SYSTEMS-0042",
  "title": "Geometric and Arithmetic Independence of Affine Chebyshev Towers",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We determine joint Galois images for finite families of normalized Chebyshev maps with nonconstant affine targets over function fields. Over C(t), endpoint graphs and degree parity determine the geometric image; least common multiples merge repeated endpoint pairs, while greatest common divisors and quadratic square classes determine subfamily intersections. These formulas give exact strict-independence and finite-index criteria for the full preimage towers. Over a number-field function field k(t), globally pairwise coprime finite degrees give an arithmetic image combining endpoint cut-space signs with the cyclotomic Galois image. We compute its constant field and show that the joint index divides [k:Q]. For full towers, finite arithmetic index is equivalent to global pairwise coprimality of the degrees; over Q(t), this is equivalent to strict independence even when the geometric fields are entangled. The results are complete for this stated special family, not for arbitrary rational maps or constant number-field targets; the general AIM independence question remains open. Portable exact-check programs accompany the written proofs. The manuscript is AI-assisted, self-audited and unrefereed; no independent peer review, formal verification or absolute priority is claimed.",
  "result_type": "COMPLETE_PROOF_OF_DISPLAYED_SPECIAL_CASE",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "keywords": [
    "Chebyshev polynomials",
    "arithmetic dynamics",
    "arboreal Galois representations",
    "function fields",
    "linear disjointness",
    "cyclotomic fields",
    "graph cut spaces",
    "AIM-DYNAMICAL_SYSTEMS-0042"
  ],
  "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/aim-dynamical-systems-0042/",
  "pdf_url": "https://eulersolve.org/papers/aim-dynamical-systems-0042/paper.pdf?v=112faed822e0",
  "doi": "10.5281/zenodo.23273431",
  "zenodo_record_url": "https://zenodo.org/records/23273431",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete geometric and arithmetic independence theorems for normalized Chebyshev maps with nonconstant affine targets over function fields. The finite arithmetic formula assumes globally pairwise coprime degrees; the tower criterion treats every degree list. The general AIM independence question remains open: no arbitrary-map or constant-number-field-target solution is claimed. Classical individual-cover and quadratic-entanglement methods are credited. AI-assisted, self-audited and unrefereed; no independent review, formal verification or certified priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "112faed822e09dcb6fec925e240e92282c797cc70b4f17b42f2dc6f5f8fc5a84"
    },
    "source.zip": {
      "sha256": "4944ff5a234c5a4aaaa426e4bfd842202723c3453f0022cd54d382f2ac8ed8bc"
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    "verification_report.md": {
      "sha256": "de5c13873ca21a201cda996054b99f10e77788483d9dd419a3438f9cdfc93fff"
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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."
}