{
  "schema_version": 1,
  "problem_number": "AIM-ANALYSIS-0056",
  "title": "Sharp Attraction Thresholds for Neutral Jordan Skew Products",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a class of holomorphic skew products tangent to the identity, we give necessary and sufficient conditions for an open attracting domain and for an open domain attracted along the base axis. The transverse leading coefficient is a fixed matrix with real spectrum and arbitrary Jordan blocks. If the base is g(z)=z-z^(2t+1)+a z^(3t+1)+O(z^(3t+2)) and the fiber multiplier is I-i z^t B-c z^(2t) I+R(z), with R(z)=O(z^(2t+1)) in the algebra generated by B, the attraction threshold is Re(c)>max_beta{beta^2/2+beta Im(a)+t(m_beta-1)}. Attachment to the base axis requires the same strict inequality with one added to its right-hand side. Here m_beta is the largest Jordan block size at beta. We prove exact rates at every Jordan height, including the boundary cases, and construct invariant open domains. The criteria distinguish equal-spectrum examples and exhibit the effect of a higher base coefficient. These are complete finite-jet criteria within the stated linear-fiber class, not a solution of the general AIM question about degenerate characteristic directions. This AI-assisted preprint is self-audited and unrefereed; no independent review, formal verification or absolute-priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "keywords": [
    "holomorphic dynamics",
    "tangent to the identity",
    "degenerate characteristic direction",
    "Jordan block",
    "skew product",
    "attracting domain"
  ],
  "manuscript_version_date": "2026-10-11",
  "publication_date": "2026-10-11",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-11",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-analysis-0056/",
  "pdf_url": "https://eulersolve.org/papers/aim-analysis-0056/paper.pdf?v=0819c4331fa1",
  "doi": "10.5281/zenodo.23304667",
  "zenodo_record_url": "https://zenodo.org/records/23304667",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete strict attraction and first-order attachment criteria for the specified commuting linear-fiber family with real spectrum, including all Jordan heights and a critical higher base coefficient. The general AIM degenerate-direction question remains partially resolved. Classical asymptotic and shearing methods are credited; no absolute-priority, independent-review or formal-verification certificate is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "0819c4331fa13ffdf22c9e373885f0130f110d46620ba644cbd4368e02e4c842"
    },
    "source.zip": {
      "sha256": "abc91a3ea2ab061feda86875875bca4db1eb3957d8338d2a37d717ad39f24003"
    },
    "verification_report.md": {
      "sha256": "134552f6f00ccdec2b2798bedc9bfb08a9ca46d73a29e91b1e5d179173ce48e2"
    }
  },
  "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."
}