{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1558",
  "content_hash": null,
  "slug": "gaussian-centered-ellipsoid-fitting-quarter-threshold-claim-literature-frontier",
  "type": "claim",
  "title": "Strongest checked neighboring result",
  "summary": "At the quarter threshold, approximate fits with bounded spectra are established; exact fitting is proved below some positive constant times \\(d^2\\), while the best general impossibility threshold remains near \\(d^2/2\\).",
  "relevance": "Locates the present research frontier immediately below The one-quarter threshold for fitting Gaussian points by a centered ellipsoid.",
  "relevance_source": "recorded",
  "body": "At the quarter threshold, approximate fits with bounded spectra are established; exact fitting is proved below some positive constant times \\(d^2\\), while the best general impossibility threshold remains near \\(d^2/2\\).\n\nThis leaves the following boundary unresolved: Upgrade approximate control to exact fits below one quarter and rule out ill-conditioned exact fits above one quarter. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.48550/arXiv.2504.20539",
      "locator": "Afonso S. Bandeira, Anastasia Kireeva, Antoine Maillard, and Almut Rödder, “Randomstrasse101: Open Problems of 2024,” arXiv:2504.20539, version dated 22 March 2026. Conjecture 6 and Theorems 3.1–3.2"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.48550/arXiv.2504.20539",
    "locator": "Afonso S. Bandeira, Anastasia Kireeva, Antoine Maillard, and Almut Rödder, “Randomstrasse101: Open Problems of 2024,” arXiv:2504.20539, version dated 22 March 2026. Conjecture 6 and Theorems 3.1–3.2"
  },
  "relations": [
    {
      "slug": "R1559",
      "title": "Current status and exact unresolved remainder",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "gaussian-centered-ellipsoid-fitting-quarter-threshold",
      "title": "gaussian centered ellipsoid fitting quarter threshold",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}