{
  "schema_version": 1,
  "problem_number": "AIM-ANALYSIS-0148",
  "title": "Eventual Scalar Recurrences for Constrained Orthogonal Polynomials",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let S be a finite-codimensional subspace of R[x] with a positive definite moment inner product, and let q_n be its degree-ordered monic orthogonal basis. For a fixed nonzero polynomial a of degree d, we prove that an eventual scalar band recurrence for a q_n exists if and only if multiplication by a preserves S plus its polynomial orthogonal complement. The width can then be chosen to be d. Some nonconstant multiplier works exactly when this sum contains a nonzero polynomial ideal. For the inner product on [0,1] and the single constraint integral_2^3 p(x) dx = 0, every nonconstant multiplier instead sends infinitely many q_n outside S. A mean-zero Legendre example shows why stabilizing S itself is not necessary for a tail recurrence. These elementary results refine an established stabilizer approach; they do not classify differential, coupled, or compressed-operator recurrences, or settle the broader AIM problem. The note is self-audited and unrefereed; absolute priority and independent review are not claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CA"
  ],
  "keywords": [
    "constrained orthogonal polynomials",
    "eventual recurrence",
    "polynomial stabilizer",
    "finite codimension",
    "moment inner product",
    "counterexample",
    "separated intervals"
  ],
  "manuscript_version_date": "2026-10-12",
  "publication_date": "2026-10-12",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-12",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-analysis-0148/",
  "pdf_url": "https://eulersolve.org/papers/aim-analysis-0148/paper.pdf?v=be01787faf90",
  "doi": "10.5281/zenodo.23325951",
  "zenodo_record_url": "https://zenodo.org/records/23325951",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete scoped real-moment theorem for one fixed polynomial multiplier and scalar band coefficients, with an explicit separated-interval counterexample. The broader AIM source and other recurrence notions remain unresolved here. Normand and the established stabilizer method are credited. Priority confidence is low; no independent review, formal verification or certified novelty is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "be01787faf902b8d24253b10790f1ab845e13f878a14bfe9497a3eccc929a459"
    },
    "source.zip": {
      "sha256": "fce7c7b1619f4b12a872fa4620b36b1449cc97579aeee6609385359cd7820e38"
    },
    "verification_report.md": {
      "sha256": "755ec2205a8f038ccf1dc3d288877c2284c21626db62582d60f0bc3c5f7a1650"
    }
  },
  "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."
}