{
  "schema_version": 1,
  "problem_number": "OWR-13674-019",
  "title": "Sharp Two-Sided Root Selection for Common-Interlacing Polynomials with Fixed Trace",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let a finite family of monic real-rooted polynomials have a common interlacer and the same sum of roots. If a positive weighted average has all its roots in an interval with center c and radius R, then one member has all its roots in the concentric interval of radius 2R/sqrt(3). The factor 2/sqrt(3) is optimal in every degree at least three, even for characteristic polynomials of zero-diagonal real symmetric matrices. The optimal factor in degrees one and two is one. The proof uses nonnegative residues, a probability measure with a prescribed mean, and an exact affine majorant. Weak interlacing and repeated roots are included. A corollary selects a principal submatrix of a constant-diagonal Hermitian matrix using the root interval of the derivative of its characteristic polynomial. This is a complete scoped theorem in the direction of the two-extreme-eigenvalue question in OWR-13674-019. It requires a global common interlacer; it does not assert the same factor for arbitrary interlacing trees, a new paving/signing bound, or closure of the separate quantitative real-rootedness question. AI-assisted, self-audited preprint; no independent review, formal verification, or absolute-priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CA",
    "math.CO",
    "math.SP"
  ],
  "keywords": [
    "OWR-13674-019",
    "common interlacing",
    "fixed trace",
    "two-sided root selection",
    "sharp constant",
    "spectral norm"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-13674-019/",
  "pdf_url": "https://eulersolve.org/papers/owr-13674-019/paper.pdf?v=caef1fb65c78",
  "doi": "10.5281/zenodo.23642092",
  "zenodo_record_url": "https://zenodo.org/records/23642092",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For a finite family of monic real-rooted polynomials with a global common interlacer and the same trace, an average root interval of center c and radius R contains all roots of some member after dilation by 2/sqrt(3). This factor is sharp in every degree at least three; degrees one and two have factor one. A principal-submatrix corollary follows for constant-diagonal Hermitian matrices. This is a complete scoped theorem responding to the two-extreme-root direction, not a closure of the entire multi-question source record. The sharp factor 2/sqrt(3) applies to a finite family with one global common interlacer and fixed trace; the factor is one in degrees one and two. The wider source record remains partially solved: no quantitative real-rootedness distance result, arbitrary-interlacing-tree guarantee, or new paving/signing bound is claimed. The principal-submatrix corollary requires constant diagonal. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "caef1fb65c7801a62e78a8cd9f2e27416e6496e2ccc4a52de815d8074d3e955b"
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    "source.zip": {
      "sha256": "6c35a6ff82455a18d550512afbf2d5c27e612aa40a82fed14653c4da31b43a5a"
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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."
}