{
  "schema_version": 1,
  "problem_number": "OWR-720-010",
  "title": "Sharp Kernel Penalties for Canonical Parseval Approximation",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a closed-range synthesis operator F, let P_F be the projection onto the orthogonal complement of its kernel. We study the metrics d_(p,kappa)(F,G) = (||F-G||^p + kappa^p ||P_F-P_G||^p)^(1/p), for finite p >= 1. The polar factor is universally a nearest Parseval synthesis operator among partial isometries with arbitrary ranges exactly at the penalty threshold kappa >= 1. Every minimizer then has the original kernel. We parametrize all minimizers and their ranges by a contraction, an orthogonal defect isometry and an operator inequality. Over complex Hilbert spaces we obtain an exact spectral uniqueness criterion and describe when distinct minimizing subspaces occur. A rank-collapse example excludes the same universal property for metrics inducing the operator-norm topology. The case p=2, kappa=1 is the existing kernel metric of Corach, Maestripieri and Mbekhta; the norm lower bound is a special case of Chiumiento's polar-factor theorem. This unrefereed preprint makes no absolute priority claim. Source dataset record: OWR-720-010 (ulamai/UnsolvedMath v1.6.0). The manuscript discloses OpenAI Codex assistance.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "keywords": [
    "OWR-720-010",
    "Parseval frame",
    "polar decomposition",
    "operator approximation",
    "kernel metric"
  ],
  "manuscript_version_date": "2026-10-24",
  "publication_date": "2026-10-24",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-24",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-720-010/",
  "pdf_url": "https://eulersolve.org/papers/owr-720-010/paper.pdf?v=3ca92ad37d9f",
  "doi": "10.5281/zenodo.23621570",
  "zenodo_record_url": "https://zenodo.org/records/23621570",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every finite p>=1, the kernel-gap metric d_(p,kappa) has the canonical Parseval synthesis as a global nearest partial isometry for all closed-range operators exactly at the universal penalty threshold kappa>=1. Every minimizer has the original kernel; all minimizers and their ranges are parametrized by a contraction and its orthogonal defect. Over complex spaces, uniqueness is equivalent to (A-I)^2=||A-I||^2 I. This constructively answers the frozen metric-existence and closest-subspace request, not a classification of every metric, every tight scale, or structured wavelet frames. The theorem constructs a metric and describes all its minimizers; it does not classify every possible metric or structured wavelet frames. Earlier CMM and Chiumiento results are credited. 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": "3ca92ad37d9fa864cfa528f8dc8eb6d49f71438d4e4d142821018e3f0eb237cd"
    },
    "source.zip": {
      "sha256": "beb9cda4f510e8236494ce1e36d7e114420cb1872f065adf1e91441975343cfa"
    },
    "verification_report.md": {
      "sha256": "d040475f4fd2c930eb88b113a554b4086b2e16e107fc4e76f46d21de79627d5c"
    }
  },
  "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."
}