{
  "schema_version": 1,
  "problem_number": "OWR-13346-004",
  "title": "Ambient Gaussian Landmarks and a Unit-Circle Obstruction to Nyström Approximation",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Ambient Gaussian Landmarks and a Unit-Circle Obstruction to Nyström Approximation Alper Ferudun, Mercury Software GmbH. English unrefereed preprint, version 1.0, 24 October 2026. For k(x,y)=1+x^T y and m ambient standard Gaussian landmarks in R^n, the manuscript derives an exact chi-square intercept law and a finite-sample lower bound on diagonal Nyström error. When m=o(n), the unit-query error tends to 2, even for data on a compact algebraic unit circle. More generally, if m/n tends to alpha, the approximant converges uniformly on that circle to alpha k. Three suitable on-circle landmarks instead reconstruct the rank-three restricted kernel exactly. This gives a full counterexample to the displayed global-constant tail bound on page 2501 of Oberwolfach Report 44/2014. It is a complete scoped obstruction result, not a claim to have resolved every suitably modified polynomial-kernel guarantee or the report's separate Gaussian-kernel problem. Classical feature-span and rotational-projection ingredients are credited. No absolute novelty or priority certification, formal verification or independent peer review is claimed. Source dataset record: OWR-13346-004 (ulamai/UnsolvedMath v1.6.0). The manuscript discloses OpenAI Codex assistance.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.NA",
    "stat.ML"
  ],
  "keywords": [
    "OWR-13346-004",
    "Nystrom approximation",
    "Gaussian landmarks",
    "kernel methods"
  ],
  "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-13346-004/",
  "pdf_url": "https://eulersolve.org/papers/owr-13346-004/paper.pdf?v=cbccec8a248a",
  "doi": "10.5281/zenodo.23621500",
  "zenodo_record_url": "https://zenodo.org/records/23621500",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For the kernel 1+x^T y with ambient standard Gaussian landmarks, a finite diagonal-error lower bound and an exact intercept law imply failure of the displayed global-constant tail bound on a compact unit circle. If m/n tends to alpha in [0,1], the approximant converges uniformly on that circle to alpha times the kernel. The broader request for modified approximation guarantees remains partially solved. The displayed global-constant bound is disproved. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
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      "sha256": "cbccec8a248a68cb1d7827ad6f8fc5136c2eb58279a1b0b59b53cd94eaf12026"
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    "source.zip": {
      "sha256": "4d02be0d1657d0967434e7788f2d52e61addb42a42a2eaa3a17d4816482d8be8"
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    "verification_report.md": {
      "sha256": "197aea5abb3c90438e8bbdc426ecaa7fa6ad53a1644681c4d320de6276a7a1aa"
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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."
}