files / problems / kissing-5d / compute / q3 / dual_gap.json

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{
  "gaps": [
    {
      "lo": "-1",
      "deg": 8,
      "numerical": 47.998777257614584,
      "certified": null
    },
    {
      "lo": "-1",
      "deg": 10,
      "numerical": 46.336869215638316,
      "certified": null
    },
    {
      "lo": "-1",
      "deg": 12,
      "numerical": 46.336869215638316,
      "certified": null
    },
    {
      "lo": "-19/20",
      "deg": 8,
      "numerical": 47.99891070038997,
      "certified": null
    },
    {
      "lo": "-19/20",
      "deg": 10,
      "numerical": 46.3367450262246,
      "certified": null
    },
    {
      "lo": "-19/20",
      "deg": 12,
      "numerical": 46.3367450262246,
      "certified": null
    },
    {
      "lo": "-9/10",
      "deg": 8,
      "numerical": 47.99896763082857,
      "certified": null
    },
    {
      "lo": "-9/10",
      "deg": 10,
      "numerical": 46.33699667025259,
      "certified": null
    },
    {
      "lo": "-9/10",
      "deg": 12,
      "numerical": 46.33699667025259,
      "certified": null
    },
    {
      "lo": "-4/5",
      "deg": 8,
      "numerical": 47.99888947571201,
      "certified": null
    },
    {
      "lo": "-4/5",
      "deg": 10,
      "numerical": 46.33702419880609,
      "certified": null
    },
    {
      "lo": "-4/5",
      "deg": 12,
      "numerical": 46.33702419880609,
      "certified": null
    },
    {
      "lo": "-3/4",
      "deg": 8,
      "numerical": 47.9989348230194,
      "certified": null
    },
    {
      "lo": "-3/4",
      "deg": 10,
      "numerical": 45.37626213655032,
      "certified": null
    },
    {
      "lo": "-3/4",
      "deg": 12,
      "numerical": 45.37626213655032,
      "certified": null
    },
    {
      "lo": "-2/3",
      "deg": 8,
      "numerical": 41.30277365835573,
      "certified": null
    },
    {
      "lo": "-2/3",
      "deg": 10,
      "numerical": 37.464479929129915,
      "certified": null
    },
    {
      "lo": "-2/3",
      "deg": 12,
      "numerical": 37.464479929129915,
      "certified": null
    }
  ],
  "pfender": {
    "used": false,
    "reason": "Pfender f_\u03b1 is a diagonally-dominant kernel, not a polynomial in the Gegenbauer basis.  An exact SOS/matrix certificate of that kernel was not produced.  Residue, not a dual below 44."
  },
  "best_certified": null,
  "excludes_any_k": [],
  "comment": "Unrestricted (lo=-1) Delsarte is the Odlyzko\u2013Sloane number \u224846.34; it cannot exclude 41\u201344.  Gapped duals (lo>-1) are not unrestricted certificates unless a geometric lemma forbids t in [-1, lo).  No such lemma is claimed here.  Pfender's kernel was not certified."
}