[
  {
    "constant": "ramsey-3-4",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "9",
      "decimal_approx": 9,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "greenwood-gleason-1955"
  },
  {
    "constant": "ramsey-3-5",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "14",
      "decimal_approx": 14,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "greenwood-gleason-1955"
  },
  {
    "constant": "ramsey-3-6",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "18",
      "decimal_approx": 18,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "kery-1964"
  },
  {
    "constant": "ramsey-3-7",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "23",
      "decimal_approx": 23,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "graver-yackel-1968"
  },
  {
    "constant": "ramsey-3-8",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "28",
      "decimal_approx": 28,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "mckay-zhang-1992"
  },
  {
    "constant": "ramsey-3-9",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "36",
      "decimal_approx": 36,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "grinstead-roberts-1982"
  },
  {
    "constant": "ramsey-3-10",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "40",
      "decimal_approx": 40,
      "alt_display": null
    },
    "verified": {
      "latex": "39",
      "decimal_approx": 39,
      "alt_display": null
    },
    "verified_certificate": "ramsey-3-10-lower-39",
    "evidence": "witness",
    "source": "exoo-1989c"
  },
  {
    "constant": "ramsey-3-10",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "41",
      "decimal_approx": 41,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-2024"
  },
  {
    "constant": "ramsey-3-11",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "47",
      "decimal_approx": 47,
      "alt_display": null
    },
    "verified": {
      "latex": "46",
      "decimal_approx": 46,
      "alt_display": null
    },
    "verified_certificate": "ramsey-3-11-lower-46",
    "evidence": "witness",
    "source": "exoo-2013"
  },
  {
    "constant": "ramsey-3-11",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "50",
      "decimal_approx": 50,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "goedgebeur-radziszowski-2013"
  },
  {
    "constant": "ramsey-3-12",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "53",
      "decimal_approx": 53,
      "alt_display": null
    },
    "verified": {
      "latex": "49",
      "decimal_approx": 49,
      "alt_display": null
    },
    "verified_certificate": "ramsey-3-12-lower-49",
    "evidence": "witness",
    "source": "kolodyazhny-2015"
  },
  {
    "constant": "ramsey-3-12",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "59",
      "decimal_approx": 59,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "lesser-2001"
  },
  {
    "constant": "ramsey-3-13",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "61",
      "decimal_approx": 61,
      "alt_display": null
    },
    "verified": {
      "latex": "58",
      "decimal_approx": 58,
      "alt_display": null
    },
    "verified_certificate": "ramsey-3-13-lower-58",
    "evidence": "witness",
    "source": "nagda-raghavan-thakurta-2026"
  },
  {
    "constant": "ramsey-3-13",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "68",
      "decimal_approx": 68,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "goedgebeur-radziszowski-2013"
  },
  {
    "constant": "ramsey-3-14",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "67",
      "decimal_approx": 67,
      "alt_display": null
    },
    "verified": {
      "latex": "64",
      "decimal_approx": 64,
      "alt_display": null
    },
    "verified_certificate": "ramsey-3-14-lower-64",
    "evidence": "witness",
    "source": "kolodyazhny-2016"
  },
  {
    "constant": "ramsey-3-14",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "77",
      "decimal_approx": 77,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "goedgebeur-radziszowski-2013"
  },
  {
    "constant": "ramsey-3-15",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "74",
      "decimal_approx": 74,
      "alt_display": null
    },
    "verified": {
      "latex": "72",
      "decimal_approx": 72,
      "alt_display": null
    },
    "verified_certificate": "ramsey-3-15-lower-72",
    "evidence": "witness",
    "source": "kolodyazhny-2016"
  },
  {
    "constant": "ramsey-3-15",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "87",
      "decimal_approx": 87,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "goedgebeur-radziszowski-2013"
  },
  {
    "constant": "ramsey-4-4",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "18",
      "decimal_approx": 18,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "greenwood-gleason-1955"
  },
  {
    "constant": "ramsey-4-5",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "25",
      "decimal_approx": 25,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "mckay-radziszowski-1995"
  },
  {
    "constant": "ramsey-4-6",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "36",
      "decimal_approx": 36,
      "alt_display": null
    },
    "verified": {
      "latex": "34",
      "decimal_approx": 34,
      "alt_display": null
    },
    "verified_certificate": "ramsey-4-6-lower-34",
    "evidence": "witness",
    "source": "exoo-2012"
  },
  {
    "constant": "ramsey-4-6",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "40",
      "decimal_approx": 40,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-4-7",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "49",
      "decimal_approx": 49,
      "alt_display": null
    },
    "verified": {
      "latex": "47",
      "decimal_approx": 47,
      "alt_display": null
    },
    "verified_certificate": "ramsey-4-7-lower-47",
    "evidence": "witness",
    "source": "exoo-1989a"
  },
  {
    "constant": "ramsey-4-7",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "58",
      "decimal_approx": 58,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-4-8",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "59",
      "decimal_approx": 59,
      "alt_display": null
    },
    "verified": {
      "latex": "52",
      "decimal_approx": 52,
      "alt_display": null
    },
    "verified_certificate": "ramsey-4-8-lower-52",
    "evidence": "witness",
    "source": "exoo-tatarevic-2015"
  },
  {
    "constant": "ramsey-4-8",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "79",
      "decimal_approx": 79,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-4-9",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "73",
      "decimal_approx": 73,
      "alt_display": null
    },
    "verified": {
      "latex": "69",
      "decimal_approx": 69,
      "alt_display": null
    },
    "verified_certificate": "ramsey-4-9-lower-69",
    "evidence": "witness",
    "source": "exoo-2005"
  },
  {
    "constant": "ramsey-4-9",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "105",
      "decimal_approx": 105,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-4-10",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "92",
      "decimal_approx": 92,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "harborth-krause-2003"
  },
  {
    "constant": "ramsey-4-10",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "135",
      "decimal_approx": 135,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-4-11",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "102",
      "decimal_approx": 102,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "exoo-tatarevic-2015"
  },
  {
    "constant": "ramsey-4-11",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "170",
      "decimal_approx": 170,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-4-12",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "128",
      "decimal_approx": 128,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "su-luo-li-1998"
  },
  {
    "constant": "ramsey-4-12",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "210",
      "decimal_approx": 210,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-4-13",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "139",
      "decimal_approx": 139,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "nagda-raghavan-thakurta-2026"
  },
  {
    "constant": "ramsey-4-13",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "256",
      "decimal_approx": 256,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-4-14",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "148",
      "decimal_approx": 148,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "nagda-raghavan-thakurta-2026"
  },
  {
    "constant": "ramsey-4-14",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "307",
      "decimal_approx": 307,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-4-15",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "159",
      "decimal_approx": 159,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "nagda-raghavan-thakurta-2026"
  },
  {
    "constant": "ramsey-4-15",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "364",
      "decimal_approx": 364,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-5-5",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "43",
      "decimal_approx": 43,
      "alt_display": null
    },
    "verified": {
      "latex": "42",
      "decimal_approx": 42,
      "alt_display": null
    },
    "verified_certificate": "ramsey-5-5-lower-42",
    "evidence": "witness",
    "source": "exoo-1989b"
  },
  {
    "constant": "ramsey-5-5",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "46",
      "decimal_approx": 46,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2024"
  },
  {
    "constant": "ramsey-5-6",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "59",
      "decimal_approx": 59,
      "alt_display": null
    },
    "verified": {
      "latex": "54",
      "decimal_approx": 54,
      "alt_display": null
    },
    "verified_certificate": "ramsey-5-6-lower-54",
    "evidence": "witness",
    "source": "exoo-2023"
  },
  {
    "constant": "ramsey-5-6",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "85",
      "decimal_approx": 85,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-5-7",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "80",
      "decimal_approx": 80,
      "alt_display": null
    },
    "verified": {
      "latex": "71",
      "decimal_approx": 71,
      "alt_display": null
    },
    "verified_certificate": "ramsey-5-7-lower-71",
    "evidence": "witness",
    "source": "calkin-erdos-tovey-1997"
  },
  {
    "constant": "ramsey-5-7",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "133",
      "decimal_approx": 133,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-5-8",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "101",
      "decimal_approx": 101,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "harborth-krause-2003"
  },
  {
    "constant": "ramsey-5-8",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "193",
      "decimal_approx": 193,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-5-9",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "133",
      "decimal_approx": 133,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "kuznetsov-2016"
  },
  {
    "constant": "ramsey-5-9",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "282",
      "decimal_approx": 282,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-5-10",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "149",
      "decimal_approx": 149,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "exoo-tatarevic-2015"
  },
  {
    "constant": "ramsey-5-10",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "381",
      "decimal_approx": 381,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-5-11",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "183",
      "decimal_approx": 183,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "kuznetsov-2016"
  },
  {
    "constant": "ramsey-5-11",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "511",
      "decimal_approx": 511,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-5-12",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "203",
      "decimal_approx": 203,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "kuznetsov-2016"
  },
  {
    "constant": "ramsey-5-12",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "672",
      "decimal_approx": 672,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-5-13",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "233",
      "decimal_approx": 233,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "kuznetsov-2016"
  },
  {
    "constant": "ramsey-5-13",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "860",
      "decimal_approx": 860,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-5-14",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "267",
      "decimal_approx": 267,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "kuznetsov-2016"
  },
  {
    "constant": "ramsey-5-14",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "1081",
      "decimal_approx": 1081,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
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    "constant": "ramsey-5-15",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "275",
      "decimal_approx": 275,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "trivial"
  },
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    "constant": "ramsey-5-15",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "1341",
      "decimal_approx": 1341,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-6-6",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "160",
      "decimal_approx": 160,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-6-7",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "115",
      "decimal_approx": 115,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "exoo-tatarevic-2015"
  },
  {
    "constant": "ramsey-6-7",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "270",
      "decimal_approx": 270,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-6-8",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "134",
      "decimal_approx": 134,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "exoo-tatarevic-2015"
  },
  {
    "constant": "ramsey-6-8",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "423",
      "decimal_approx": 423,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-6-9",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "183",
      "decimal_approx": 183,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "kuznetsov-2016"
  },
  {
    "constant": "ramsey-6-9",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "651",
      "decimal_approx": 651,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-6-10",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "204",
      "decimal_approx": 204,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "kuznetsov-2016"
  },
  {
    "constant": "ramsey-6-10",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "944",
      "decimal_approx": 944,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-6-11",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "262",
      "decimal_approx": 262,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "tatarevic-2020"
  },
  {
    "constant": "ramsey-6-11",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "1346",
      "decimal_approx": 1346,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-6-12",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "294",
      "decimal_approx": 294,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "kuznetsov-2016"
  },
  {
    "constant": "ramsey-6-12",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "1855",
      "decimal_approx": 1855,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-6-13",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "347",
      "decimal_approx": 347,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "kuznetsov-2016"
  },
  {
    "constant": "ramsey-6-13",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "2499",
      "decimal_approx": 2499,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-6-14",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "3301",
      "decimal_approx": 3301,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-6-15",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "401",
      "decimal_approx": 401,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "trivial"
  },
  {
    "constant": "ramsey-6-15",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "4305",
      "decimal_approx": 4305,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-7-7",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "492",
      "decimal_approx": 492,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-7-8",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "219",
      "decimal_approx": 219,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "tatarevic-2020"
  },
  {
    "constant": "ramsey-7-8",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "832",
      "decimal_approx": 832,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-7-9",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "252",
      "decimal_approx": 252,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "kuznetsov-2016"
  },
  {
    "constant": "ramsey-7-9",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "1368",
      "decimal_approx": 1368,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-7-10",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "292",
      "decimal_approx": 292,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "kuznetsov-2016"
  },
  {
    "constant": "ramsey-7-10",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "2119",
      "decimal_approx": 2119,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-7-11",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "405",
      "decimal_approx": 405,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "xu-xie-e-r-2004"
  },
  {
    "constant": "ramsey-7-11",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "3197",
      "decimal_approx": 3197,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-7-12",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "417",
      "decimal_approx": 417,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "xiaodong-s-r-2011"
  },
  {
    "constant": "ramsey-7-12",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "4665",
      "decimal_approx": 4665,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-7-13",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "511",
      "decimal_approx": 511,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "xu-xie-radziszowski-2004"
  },
  {
    "constant": "ramsey-7-13",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "6653",
      "decimal_approx": 6653,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-7-14",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "9260",
      "decimal_approx": 9260,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-7-15",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "12635",
      "decimal_approx": 12635,
      "alt_display": "12,635"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-8-8",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "1518",
      "decimal_approx": 1518,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-8-9",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "329",
      "decimal_approx": 329,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "kuznetsov-2016"
  },
  {
    "constant": "ramsey-8-9",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "2662",
      "decimal_approx": 2662,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-8-10",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "343",
      "decimal_approx": 343,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "kuznetsov-2016"
  },
  {
    "constant": "ramsey-8-10",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "4402",
      "decimal_approx": 4402,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-8-11",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "457",
      "decimal_approx": 457,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "trivial"
  },
  {
    "constant": "ramsey-8-11",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "7040",
      "decimal_approx": 7040,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-8-12",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "10836",
      "decimal_approx": 10836,
      "alt_display": "10,836"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-8-13",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "817",
      "decimal_approx": 817,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "xu-xie-e-r-2004"
  },
  {
    "constant": "ramsey-8-13",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "27485",
      "decimal_approx": 27485,
      "alt_display": "27,485"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "huang-w-s-y-z-h-2007"
  },
  {
    "constant": "ramsey-8-14",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "41525",
      "decimal_approx": 41525,
      "alt_display": "41,525"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "huang-w-s-y-z-h-2007"
  },
  {
    "constant": "ramsey-8-15",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "873",
      "decimal_approx": 873,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "trivial"
  },
  {
    "constant": "ramsey-8-15",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "63609",
      "decimal_approx": 63609,
      "alt_display": "63,609"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "huang-w-s-y-z-h-2007"
  },
  {
    "constant": "ramsey-9-9",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "565",
      "decimal_approx": 565,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "mathon-1987"
  },
  {
    "constant": "ramsey-9-9",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "4956",
      "decimal_approx": 4956,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-9-10",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "581",
      "decimal_approx": 581,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "xiaodong-s-r-2011"
  },
  {
    "constant": "ramsey-9-10",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "8675",
      "decimal_approx": 8675,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-9-11",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "14631",
      "decimal_approx": 14631,
      "alt_display": "14,631"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-9-12",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "38832",
      "decimal_approx": 38832,
      "alt_display": "38,832"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "huang-w-s-y-z-h-2007"
  },
  {
    "constant": "ramsey-9-13",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "64864",
      "decimal_approx": 64864,
      "alt_display": "64,864"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "huang-w-s-y-z-h-2007"
  },
  {
    "constant": "ramsey-10-10",
    "group": "ramsey",
    "bound": "lower",
    "literature": {
      "latex": "798",
      "decimal_approx": 798,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "mathon-1987"
  },
  {
    "constant": "ramsey-10-10",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "16064",
      "decimal_approx": 16064,
      "alt_display": "16,064"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "angeltveit-mckay-2019-2024"
  },
  {
    "constant": "ramsey-10-11",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "45881",
      "decimal_approx": 45881,
      "alt_display": "45,881"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "huang-w-s-y-z-h-2007"
  },
  {
    "constant": "ramsey-10-12",
    "group": "ramsey",
    "bound": "upper",
    "literature": {
      "latex": "81123",
      "decimal_approx": 81123,
      "alt_display": "81,123"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "huang-w-s-y-z-h-2007"
  },
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    "constant": "ramsey-10-15",
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      "alt_display": "\u22481.732051"
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      "latex": "2 + \\frac{1}{\\sqrt{2}}",
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      "alt_display": "\u22482.707107"
    },
    "verified": {
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      "alt_display": "\u22482.236068"
    },
    "verified_certificate": "square-packing-in-square-5-lower-sqrt5",
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  },
  {
    "constant": "square-packing-in-square-5",
    "group": "square-packing",
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      "latex": "2 + \\frac{1}{\\sqrt{2}}",
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    "constant": "square-packing-in-square-10",
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    "verified": {
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    "source": "ejc-ds7-packing-unit-squares"
  },
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    "constant": "square-packing-in-square-37",
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      "alt_display": "\u22486.082763"
    },
    "verified_certificate": "square-packing-in-square-37-lower-sqrt37",
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  },
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    "constant": "square-packing-in-square-37",
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    },
    "verified": {
      "latex": "7",
      "decimal_approx": 7,
      "alt_display": null
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    "constant": "square-packing-in-square-38",
    "group": "square-packing",
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      "latex": "2\\sqrt{2}+\\frac{113+10\\sqrt{3}}{37}",
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      "alt_display": "\u22486.350603"
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    "verified": {
      "latex": "\\sqrt{38}",
      "decimal_approx": 6.164414,
      "alt_display": "\u22486.164414"
    },
    "verified_certificate": "square-packing-in-square-38-lower-sqrt38",
    "evidence": "lean",
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  },
  {
    "constant": "square-packing-in-square-38",
    "group": "square-packing",
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      "latex": "6+\\frac{1}{\\sqrt{2}}",
      "decimal_approx": 6.707107,
      "alt_display": "\u22486.707107"
    },
    "verified": {
      "latex": "7",
      "decimal_approx": 7,
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    },
    "verified_certificate": "square-packing-in-square-38-upper-7",
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  },
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    "constant": "square-packing-in-square-39",
    "group": "square-packing",
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      "latex": "2\\sqrt{2}+\\frac{113+10\\sqrt{3}}{37}",
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      "alt_display": "\u22486.350603"
    },
    "verified": {
      "latex": "\\sqrt{39}",
      "decimal_approx": 6.244998,
      "alt_display": "\u22486.244998"
    },
    "verified_certificate": "square-packing-in-square-39-lower-sqrt39",
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  },
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    "constant": "square-packing-in-square-39",
    "group": "square-packing",
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      "latex": "{\\approx}6.8107",
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    "verified": {
      "latex": "7",
      "decimal_approx": 7,
      "alt_display": null
    },
    "verified_certificate": "square-packing-in-square-39-upper-7",
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  },
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    "constant": "square-packing-in-square-40",
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      "alt_display": "\u22486.324555"
    },
    "verified_certificate": "square-packing-in-square-40-lower-sqrt40",
    "evidence": "lean",
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  },
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    "constant": "square-packing-in-square-40",
    "group": "square-packing",
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      "latex": "4+2\\sqrt{2}",
      "decimal_approx": 6.828427,
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    "verified": {
      "latex": "7",
      "decimal_approx": 7,
      "alt_display": null
    },
    "verified_certificate": "square-packing-in-square-40-upper-7",
    "evidence": "lean",
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  },
  {
    "constant": "square-packing-in-square-41",
    "group": "square-packing",
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    "literature": {
      "latex": "2\\sqrt{2}+\\frac{8}{\\sqrt{5}}",
      "decimal_approx": 6.406136,
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    },
    "verified": {
      "latex": "\\sqrt{41}",
      "decimal_approx": 6.403124,
      "alt_display": "\u22486.403124"
    },
    "verified_certificate": "square-packing-in-square-41-lower-sqrt41",
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  },
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    "constant": "square-packing-in-square-41",
    "group": "square-packing",
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      "latex": "{\\approx}6.9267",
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    "verified": {
      "latex": "7",
      "decimal_approx": 7,
      "alt_display": null
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  },
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    "constant": "square-packing-in-square-46",
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    },
    "verified": {
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      "decimal_approx": 6.78233,
      "alt_display": "\u22486.78233"
    },
    "verified_certificate": "square-packing-in-square-46-lower-sqrt46",
    "evidence": "lean",
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  },
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    "constant": "square-packing-in-square-47",
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      "latex": "7",
      "decimal_approx": 7,
      "alt_display": null
    },
    "verified": {
      "latex": "\\sqrt{47}",
      "decimal_approx": 6.855655,
      "alt_display": "\u22486.855655"
    },
    "verified_certificate": "square-packing-in-square-47-lower-sqrt47",
    "evidence": "lean",
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  },
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    "constant": "square-packing-in-square-48",
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      "alt_display": null
    },
    "verified": {
      "latex": "\\sqrt{48}",
      "decimal_approx": 6.928203,
      "alt_display": "\u22486.928203"
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    "verified_certificate": "square-packing-in-square-48-lower-sqrt48",
    "evidence": "lean",
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    "constant": "square-packing-in-square-50",
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      "alt_display": "\u22487.317426"
    },
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      "decimal_approx": 7.071068,
      "alt_display": "\u22487.071068"
    },
    "verified_certificate": "square-packing-in-square-50-lower-sqrt50",
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    "constant": "square-packing-in-square-50",
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    "verified": {
      "latex": "8",
      "decimal_approx": 8,
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    },
    "verified_certificate": "square-packing-in-square-50-upper-8",
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    "constant": "square-packing-in-square-51",
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      "latex": "2\\sqrt{2}+\\frac{101+3\\sqrt{14}}{25}",
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      "alt_display": "\u22487.317426"
    },
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      "alt_display": "\u22487.141428"
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    "verified_certificate": "square-packing-in-square-51-lower-sqrt51",
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  },
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    "constant": "square-packing-in-square-51",
    "group": "square-packing",
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    "verified": {
      "latex": "8",
      "decimal_approx": 8,
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    "verified_certificate": "square-packing-in-square-51-upper-8",
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    "constant": "square-packing-in-square-52",
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      "alt_display": "\u22487.211103"
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    "verified_certificate": "square-packing-in-square-52-lower-sqrt52",
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    "constant": "square-packing-in-square-52",
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    },
    "verified": {
      "latex": "8",
      "decimal_approx": 8,
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    },
    "verified_certificate": "square-packing-in-square-52-upper-8",
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    "constant": "square-packing-in-square-53",
    "group": "square-packing",
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      "latex": "2\\sqrt{2}+\\frac{101+3\\sqrt{14}}{25}",
      "decimal_approx": 7.317426,
      "alt_display": "\u22487.317426"
    },
    "verified": {
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      "alt_display": "\u22487.28011"
    },
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      "alt_display": "\u22487.822876"
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    "verified": {
      "latex": "8",
      "decimal_approx": 8,
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      "latex": "8",
      "decimal_approx": 8,
      "alt_display": null
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    "verified_certificate": "square-packing-in-square-55-upper-8",
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      "alt_display": "\u22487.874008"
    },
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    "constant": "square-packing-in-square-63",
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      "latex": "2\\sqrt{2}+\\frac{71}{13}",
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      "alt_display": "\u22488.185353"
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      "latex": "8+\\frac{1}{\\sqrt{2}}",
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    },
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      "latex": "9",
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    "constant": "square-packing-in-square-79",
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    },
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      "alt_display": "\u22488.888194"
    },
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    "constant": "square-packing-in-square-80",
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    },
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      "alt_display": "\u22488.944272"
    },
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    "constant": "square-packing-in-square-82",
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      "latex": "{\\approx}9.2667",
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    "source": "ejc-ds7-packing-unit-squares"
  },
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    "constant": "square-packing-in-square-82",
    "group": "square-packing",
    "bound": "upper",
    "literature": {
      "latex": "6+\\frac{5}{\\sqrt{2}}",
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    "verified": {
      "latex": "10",
      "decimal_approx": 10,
      "alt_display": null
    },
    "verified_certificate": "square-packing-in-square-82-upper-10",
    "evidence": "lean",
    "source": "ejc-ds7-packing-unit-squares"
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    "constant": "square-packing-in-square-83",
    "group": "square-packing",
    "bound": "lower",
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      "latex": "{\\approx}9.2667",
      "decimal_approx": 9.2667,
      "alt_display": null
    },
    "verified": {
      "latex": "\\sqrt{83}",
      "decimal_approx": 9.110434,
      "alt_display": "\u22489.110434"
    },
    "verified_certificate": "square-packing-in-square-83-lower-sqrt83",
    "evidence": "lean",
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    "constant": "square-packing-in-square-83",
    "group": "square-packing",
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      "latex": "{\\approx}9.6348",
      "decimal_approx": 9.6348,
      "alt_display": null
    },
    "verified": {
      "latex": "10",
      "decimal_approx": 10,
      "alt_display": null
    },
    "verified_certificate": "square-packing-in-square-83-upper-10",
    "evidence": "lean",
    "source": "chang-2026"
  },
  {
    "constant": "square-packing-in-square-84",
    "group": "square-packing",
    "bound": "lower",
    "literature": {
      "latex": "{\\approx}9.2667",
      "decimal_approx": 9.2667,
      "alt_display": null
    },
    "verified": {
      "latex": "\\sqrt{84}",
      "decimal_approx": 9.165151,
      "alt_display": "\u22489.165151"
    },
    "verified_certificate": "square-packing-in-square-84-lower-sqrt84",
    "evidence": "lean",
    "source": "ejc-ds7-packing-unit-squares"
  },
  {
    "constant": "square-packing-in-square-84",
    "group": "square-packing",
    "bound": "upper",
    "literature": {
      "latex": "9+\\frac{1}{\\sqrt{2}}",
      "decimal_approx": 9.707107,
      "alt_display": "\u22489.707107"
    },
    "verified": {
      "latex": "10",
      "decimal_approx": 10,
      "alt_display": null
    },
    "verified_certificate": "square-packing-in-square-84-upper-10",
    "evidence": "lean",
    "source": "gobel-1979"
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    "constant": "square-packing-in-square-85",
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      "latex": "{\\approx}9.2667",
      "decimal_approx": 9.2667,
      "alt_display": null
    },
    "verified": {
      "latex": "\\sqrt{85}",
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      "alt_display": "\u22489.219544"
    },
    "verified_certificate": "square-packing-in-square-85-lower-sqrt85",
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  },
  {
    "constant": "square-packing-in-square-85",
    "group": "square-packing",
    "bound": "upper",
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      "latex": "\\frac{11}{2}+3\\sqrt{2}",
      "decimal_approx": 9.742641,
      "alt_display": "\u22489.742641"
    },
    "verified": {
      "latex": "10",
      "decimal_approx": 10,
      "alt_display": null
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    "verified_certificate": "square-packing-in-square-85-upper-10",
    "evidence": "lean",
    "source": "ejc-ds7-packing-unit-squares"
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    "constant": "square-packing-in-square-86",
    "group": "square-packing",
    "bound": "upper",
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      "latex": "\\frac{17+\\sqrt{7}}{2}",
      "decimal_approx": 9.822876,
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    "verified": {
      "latex": "10",
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    "verified_certificate": "square-packing-in-square-86-upper-10",
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    "constant": "square-packing-in-square-87",
    "group": "square-packing",
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    "verified": {
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      "alt_display": null
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    "verified_certificate": "square-packing-in-square-87-upper-10",
    "evidence": "lean",
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      "alt_display": null
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    "verified_certificate": "square-packing-in-square-88-upper-10",
    "evidence": "lean",
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    "constant": "square-packing-in-square-89",
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    "constant": "square-packing-in-square-98",
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      "alt_display": null
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      "alt_display": "\u22489.899495"
    },
    "verified_certificate": "square-packing-in-square-98-lower-sqrt98",
    "evidence": "lean",
    "source": "nagamochi-2005"
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    "constant": "square-packing-in-square-99",
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      "alt_display": "\u22489.949874"
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    "verified_certificate": "square-packing-in-square-99-lower-sqrt99",
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    "constant": "kissing-number-1",
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      "alt_display": null
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    "evidence": "lean",
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    "constant": "kissing-number-2",
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      "decimal_approx": 6,
      "alt_display": null
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      "alt_display": null
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      "decimal_approx": 24,
      "alt_display": null
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      "alt_display": null
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    "constant": "kissing-number-6",
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    "constant": "kissing-number-7",
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      "alt_display": null
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    "constant": "kissing-number-8",
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      "alt_display": null
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    "evidence": "lean",
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      "alt_display": null
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      "alt_display": null
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      "alt_display": null
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      "alt_display": null
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      "alt_display": null
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      "alt_display": null
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      "alt_display": null
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    "constant": "degree-diameter-3-2",
    "group": "degree-diameter",
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    "source": "wikipedia-degree-diameter"
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    "constant": "degree-diameter-3-3",
    "group": "degree-diameter",
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      "decimal_approx": 20,
      "alt_display": null
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    "source": "elspasgraphs"
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    "constant": "degree-diameter-3-4",
    "group": "degree-diameter",
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      "decimal_approx": 38,
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    "evidence": "lean",
    "source": "buset-2000"
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    "verified_certificate": "degree-diameter-3-5-lower-64",
    "evidence": "witness",
    "source": "afygraph"
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    "constant": "degree-diameter-3-6",
    "group": "degree-diameter",
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    "verified_certificate": "degree-diameter-3-6-lower-64",
    "evidence": "witness",
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    "verified_certificate": "degree-diameter-4-6-lower-513",
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    "evidence": "witness",
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    },
    "verified_certificate": "degree-diameter-5-4-lower-210",
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    "evidence": "lean",
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    "verified_certificate": "degree-diameter-6-3-lower-90",
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    },
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    },
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    "group": "degree-diameter",
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    "evidence": "lean",
    "source": "wikipedia-degree-diameter"
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    },
    "verified_certificate": "degree-diameter-7-3-lower-140",
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    },
    "verified_certificate": "degree-diameter-7-4-lower-602",
    "evidence": "witness",
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    "constant": "degree-diameter-7-5",
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    "verified_certificate": "degree-diameter-8-4-lower-1029",
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    "group": "cage",
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    "literature": {
      "latex": "620",
      "decimal_approx": 620,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-3-16",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "512",
      "decimal_approx": 512,
      "alt_display": null
    },
    "verified": {
      "latex": "510",
      "decimal_approx": 510,
      "alt_display": null
    },
    "verified_certificate": "cage-3-16-lower-510",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-3-16",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "936",
      "decimal_approx": 936,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "exoo-et-al-2026"
  },
  {
    "constant": "cage-4-5",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "19",
      "decimal_approx": 19,
      "alt_display": null
    },
    "verified": {
      "latex": "17",
      "decimal_approx": 17,
      "alt_display": null
    },
    "verified_certificate": "cage-4-5-lower-17",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-4-5",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "19",
      "decimal_approx": 19,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-4-7",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "67",
      "decimal_approx": 67,
      "alt_display": null
    },
    "verified": {
      "latex": "53",
      "decimal_approx": 53,
      "alt_display": null
    },
    "verified_certificate": "cage-4-7-lower-53",
    "evidence": "lean",
    "source": "wikipedia-cage"
  },
  {
    "constant": "cage-4-7",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "67",
      "decimal_approx": 67,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-4-9",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "270",
      "decimal_approx": 270,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "exoo-et-al-2026"
  },
  {
    "constant": "cage-4-10",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "320",
      "decimal_approx": 320,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "exoo-et-al-2026"
  },
  {
    "constant": "cage-4-12",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "728",
      "decimal_approx": 728,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-5-5",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "30",
      "decimal_approx": 30,
      "alt_display": null
    },
    "verified": {
      "latex": "26",
      "decimal_approx": 26,
      "alt_display": null
    },
    "verified_certificate": "cage-5-5-lower-26",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-5-5",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "30",
      "decimal_approx": 30,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-5-7",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "152",
      "decimal_approx": 152,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-5-10",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "1296",
      "decimal_approx": 1296,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "exoo-circa-2012"
  },
  {
    "constant": "cage-5-11",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "2688",
      "decimal_approx": 2688,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "araujopardo-2010"
  },
  {
    "constant": "cage-5-12",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "2730",
      "decimal_approx": 2730,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-6-5",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "40",
      "decimal_approx": 40,
      "alt_display": null
    },
    "verified": {
      "latex": "37",
      "decimal_approx": 37,
      "alt_display": null
    },
    "verified_certificate": "cage-6-5-lower-37",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-6-5",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "40",
      "decimal_approx": 40,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-6-7",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "294",
      "decimal_approx": 294,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-6-12",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "7812",
      "decimal_approx": 7812,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-7-6",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "90",
      "decimal_approx": 90,
      "alt_display": null
    },
    "verified": {
      "latex": "86",
      "decimal_approx": 86,
      "alt_display": null
    },
    "verified_certificate": "cage-7-6-lower-86",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-7-6",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "90",
      "decimal_approx": 90,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-7-8",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "672",
      "decimal_approx": 672,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-7-12",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "32928",
      "decimal_approx": 32928,
      "alt_display": "32,928"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-8-5",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "67",
      "decimal_approx": 67,
      "alt_display": null
    },
    "verified": {
      "latex": "65",
      "decimal_approx": 65,
      "alt_display": null
    },
    "verified_certificate": "cage-8-5-lower-65",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-8-5",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "80",
      "decimal_approx": 80,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-8-12",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "39216",
      "decimal_approx": 39216,
      "alt_display": "39,216"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-9-5",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "86",
      "decimal_approx": 86,
      "alt_display": null
    },
    "verified": {
      "latex": "82",
      "decimal_approx": 82,
      "alt_display": null
    },
    "verified_certificate": "cage-9-5-lower-82",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-9-5",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "96",
      "decimal_approx": 96,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-9-7",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "1152",
      "decimal_approx": 1152,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "araujopardo-2010"
  },
  {
    "constant": "cage-9-8",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "1170",
      "decimal_approx": 1170,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-9-11",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "74752",
      "decimal_approx": 74752,
      "alt_display": "74,752"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "araujopardo-2010"
  },
  {
    "constant": "cage-9-12",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "74898",
      "decimal_approx": 74898,
      "alt_display": "74,898"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-10-5",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "103",
      "decimal_approx": 103,
      "alt_display": null
    },
    "verified": {
      "latex": "101",
      "decimal_approx": 101,
      "alt_display": null
    },
    "verified_certificate": "cage-10-5-lower-101",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-10-5",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "124",
      "decimal_approx": 124,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-10-8",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "1640",
      "decimal_approx": 1640,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-10-12",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "132860",
      "decimal_approx": 132860,
      "alt_display": "132,860"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-11-5",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "124",
      "decimal_approx": 124,
      "alt_display": null
    },
    "verified": {
      "latex": "122",
      "decimal_approx": 122,
      "alt_display": null
    },
    "verified_certificate": "cage-11-5-lower-122",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-11-5",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "154",
      "decimal_approx": 154,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-11-6",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "224",
      "decimal_approx": 224,
      "alt_display": null
    },
    "verified": {
      "latex": "222",
      "decimal_approx": 222,
      "alt_display": null
    },
    "verified_certificate": "cage-11-6-lower-222",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-11-6",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "240",
      "decimal_approx": 240,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-11-8",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "2618",
      "decimal_approx": 2618,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-11-12",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "319440",
      "decimal_approx": 319440,
      "alt_display": "319,440"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-12-5",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "147",
      "decimal_approx": 147,
      "alt_display": null
    },
    "verified": {
      "latex": "145",
      "decimal_approx": 145,
      "alt_display": null
    },
    "verified_certificate": "cage-12-5-lower-145",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-12-5",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "203",
      "decimal_approx": 203,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-12-8",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "2928",
      "decimal_approx": 2928,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-12-12",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "354312",
      "decimal_approx": 354312,
      "alt_display": "354,312"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-13-5",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "174",
      "decimal_approx": 174,
      "alt_display": null
    },
    "verified": {
      "latex": "170",
      "decimal_approx": 170,
      "alt_display": null
    },
    "verified_certificate": "cage-13-5-lower-170",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-13-5",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "230",
      "decimal_approx": 230,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-13-6",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "336",
      "decimal_approx": 336,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-13-8",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "4342",
      "decimal_approx": 4342,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-13-12",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "738192",
      "decimal_approx": 738192,
      "alt_display": "738,192"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-14-5",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "199",
      "decimal_approx": 199,
      "alt_display": null
    },
    "verified": {
      "latex": "197",
      "decimal_approx": 197,
      "alt_display": null
    },
    "verified_certificate": "cage-14-5-lower-197",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-14-5",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "288",
      "decimal_approx": 288,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-14-8",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "4760",
      "decimal_approx": 4760,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-14-12",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "804468",
      "decimal_approx": 804468,
      "alt_display": "804,468"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-15-5",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "230",
      "decimal_approx": 230,
      "alt_display": null
    },
    "verified": {
      "latex": "226",
      "decimal_approx": 226,
      "alt_display": null
    },
    "verified_certificate": "cage-15-5-lower-226",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
  },
  {
    "constant": "cage-15-5",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "312",
      "decimal_approx": 312,
      "alt_display": null
    },
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    "source": "ejc-ds16-cages"
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    "constant": "cage-15-6",
    "group": "cage",
    "bound": "upper",
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      "latex": "462",
      "decimal_approx": 462,
      "alt_display": null
    },
    "verified": null,
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    "evidence": "witness",
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    "group": "cage",
    "bound": "upper",
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      "decimal_approx": 7648,
      "alt_display": null
    },
    "verified": null,
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    "evidence": "witness",
    "source": "ejc-ds16-cages"
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    "constant": "cage-15-12",
    "group": "cage",
    "bound": "upper",
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      "latex": "1957376",
      "decimal_approx": 1957376,
      "alt_display": "1,957,376"
    },
    "verified": null,
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    "evidence": "witness",
    "source": "ejc-ds16-cages"
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    "constant": "cage-16-5",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "259",
      "decimal_approx": 259,
      "alt_display": null
    },
    "verified": {
      "latex": "257",
      "decimal_approx": 257,
      "alt_display": null
    },
    "verified_certificate": "cage-16-5-lower-257",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
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    "constant": "cage-16-5",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "336",
      "decimal_approx": 336,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
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    "constant": "cage-16-6",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "504",
      "decimal_approx": 504,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
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    "constant": "cage-16-8",
    "group": "cage",
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      "latex": "8092",
      "decimal_approx": 8092,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
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    "constant": "cage-16-12",
    "group": "cage",
    "bound": "upper",
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      "latex": "2088960",
      "decimal_approx": 2088960,
      "alt_display": "2,088,960"
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    "verified": null,
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    "evidence": "witness",
    "source": "ejc-ds16-cages"
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      "latex": "294",
      "decimal_approx": 294,
      "alt_display": null
    },
    "verified": {
      "latex": "290",
      "decimal_approx": 290,
      "alt_display": null
    },
    "verified_certificate": "cage-17-5-lower-290",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
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    "constant": "cage-17-5",
    "group": "cage",
    "bound": "upper",
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      "latex": "448",
      "decimal_approx": 448,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
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      "latex": "8738",
      "decimal_approx": 8738,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
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    "constant": "cage-17-12",
    "group": "cage",
    "bound": "upper",
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      "latex": "2236962",
      "decimal_approx": 2236962,
      "alt_display": "2,236,962"
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    "verified": null,
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    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
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    "constant": "cage-18-5",
    "group": "cage",
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    "literature": {
      "latex": "327",
      "decimal_approx": 327,
      "alt_display": null
    },
    "verified": {
      "latex": "325",
      "decimal_approx": 325,
      "alt_display": null
    },
    "verified_certificate": "cage-18-5-lower-325",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
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    "constant": "cage-18-5",
    "group": "cage",
    "bound": "upper",
    "literature": {
      "latex": "480",
      "decimal_approx": 480,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
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    "constant": "cage-18-8",
    "group": "cage",
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    "literature": {
      "latex": "10440",
      "decimal_approx": 10440,
      "alt_display": "10,440"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
  },
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    "constant": "cage-18-12",
    "group": "cage",
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      "latex": "3017196",
      "decimal_approx": 3017196,
      "alt_display": "3,017,196"
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    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
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    "constant": "cage-19-5",
    "group": "cage",
    "bound": "lower",
    "literature": {
      "latex": "364",
      "decimal_approx": 364,
      "alt_display": null
    },
    "verified": {
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      "decimal_approx": 362,
      "alt_display": null
    },
    "verified_certificate": "cage-19-5-lower-362",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
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    "constant": "cage-19-5",
    "group": "cage",
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      "latex": "512",
      "decimal_approx": 512,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
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    "literature": {
      "latex": "720",
      "decimal_approx": 720,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
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    "constant": "cage-19-8",
    "group": "cage",
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      "latex": "13642",
      "decimal_approx": 13642,
      "alt_display": "13,642"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
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    "constant": "cage-19-12",
    "group": "cage",
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      "latex": "4938480",
      "decimal_approx": 4938480,
      "alt_display": "4,938,480"
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    "verified": null,
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    "evidence": "witness",
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  },
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    "constant": "cage-20-5",
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    "literature": {
      "latex": "403",
      "decimal_approx": 403,
      "alt_display": null
    },
    "verified": {
      "latex": "401",
      "decimal_approx": 401,
      "alt_display": null
    },
    "verified_certificate": "cage-20-5-lower-401",
    "evidence": "lean",
    "source": "ejc-ds16-cages"
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    "constant": "cage-20-5",
    "group": "cage",
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      "latex": "576",
      "decimal_approx": 576,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
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    "constant": "cage-20-8",
    "group": "cage",
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      "decimal_approx": 14480,
      "alt_display": "14,480"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ejc-ds16-cages"
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    "constant": "cage-20-12",
    "group": "cage",
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      "latex": "5227320",
      "decimal_approx": 5227320,
      "alt_display": "5,227,320"
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    "verified": null,
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    "evidence": "witness",
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    "constant": "van-der-waerden-2-3",
    "group": "van-der-waerden",
    "bound": "upper",
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      "latex": "9",
      "decimal_approx": 9,
      "alt_display": null
    },
    "verified": null,
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    "evidence": "lean",
    "source": "chvatal-1970"
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    "constant": "van-der-waerden-2-4",
    "group": "van-der-waerden",
    "bound": "upper",
    "literature": {
      "latex": "35",
      "decimal_approx": 35,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "chvatal-1970"
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    "constant": "van-der-waerden-2-5",
    "group": "van-der-waerden",
    "bound": "upper",
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      "latex": "178",
      "decimal_approx": 178,
      "alt_display": null
    },
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    "evidence": "lean",
    "source": "stevens-shantaram-1978"
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    "constant": "van-der-waerden-2-6",
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      "latex": "1132",
      "decimal_approx": 1132,
      "alt_display": null
    },
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    "evidence": "lean",
    "source": "kouril-paul-2011"
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    "constant": "van-der-waerden-3-3",
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      "latex": "27",
      "decimal_approx": 27,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "chvatal-1970"
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    "constant": "van-der-waerden-3-4",
    "group": "van-der-waerden",
    "bound": "upper",
    "literature": {
      "latex": "293",
      "decimal_approx": 293,
      "alt_display": null
    },
    "verified": null,
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    "evidence": "lean",
    "source": "kouril-2012"
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    "constant": "van-der-waerden-3-5",
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      "latex": "2174",
      "decimal_approx": 2174,
      "alt_display": null
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    "verified": null,
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    "evidence": "witness",
    "source": "rabung-lotts-2012"
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    "constant": "van-der-waerden-3-6",
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      "alt_display": "11,192"
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    "verified": null,
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    "evidence": "witness",
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    "constant": "van-der-waerden-3-7",
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      "alt_display": "48,812"
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    "verified": null,
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    "evidence": "witness",
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    "constant": "van-der-waerden-3-9",
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      "alt_display": "932,746"
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    "evidence": "witness",
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    "constant": "van-der-waerden-3-10",
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      "alt_display": "4,173,725"
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    "verified": null,
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    "evidence": "witness",
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    "constant": "van-der-waerden-3-11",
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      "alt_display": "18,603,732"
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    "verified": null,
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      "latex": "76",
      "decimal_approx": 76,
      "alt_display": null
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    "evidence": "lean",
    "source": "beeler-oneil-1979"
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    "constant": "van-der-waerden-4-6",
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      "alt_display": "157,349"
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    "verified": null,
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    "evidence": "witness",
    "source": "monroe-2022"
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    "constant": "van-der-waerden-4-7",
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      "alt_display": "2,284,752"
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    "evidence": "witness",
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    "constant": "van-der-waerden-4-8",
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      "latex": "12288156",
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      "alt_display": "12,288,156"
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    "constant": "van-der-waerden-4-9",
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      "alt_display": "139,847,086"
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    "source": "monroe-2019"
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    "constant": "van-der-waerden-4-10",
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      "latex": "1189640579",
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      "alt_display": "1,189,640,579"
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    "evidence": "witness",
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    "constant": "van-der-waerden-4-11",
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      "alt_display": "3,464,368,084"
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    "constant": "van-der-waerden-5-3",
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      "latex": "171",
      "decimal_approx": 171,
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    "verified_certificate": null,
    "evidence": "witness",
    "source": "rabung-lotts-2012"
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  {
    "constant": "van-der-waerden-5-5",
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      "latex": "98742",
      "decimal_approx": 98742,
      "alt_display": "98,742"
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    "evidence": "witness",
    "source": "komkov-2016"
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    "constant": "van-der-waerden-5-6",
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    "literature": {
      "latex": "786741",
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      "alt_display": "786,741"
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    "evidence": "witness",
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  {
    "constant": "van-der-waerden-5-7",
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      "latex": "15993258",
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      "alt_display": "15,993,258"
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    "evidence": "witness",
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    "constant": "van-der-waerden-5-8",
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      "latex": "86017086",
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      "alt_display": "86,017,086"
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    "evidence": "witness",
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  {
    "constant": "van-der-waerden-5-9",
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      "latex": "978929596",
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      "alt_display": "978,929,596"
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    "evidence": "witness",
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    "constant": "van-der-waerden-5-10",
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      "latex": "8327484047",
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      "alt_display": "8,327,484,047"
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    "evidence": "witness",
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      "latex": "38108048914",
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      "alt_display": "38,108,048,914"
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      "decimal_approx": 226,
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    "evidence": "witness",
    "source": "blankenship-c-t-2018"
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    "constant": "van-der-waerden-6-5",
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      "alt_display": "98,755"
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    "evidence": "witness",
    "source": "komkov-2016"
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  {
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      "alt_display": "1,555,550"
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    "constant": "van-der-waerden-6-7",
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      "latex": "111952800",
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      "alt_display": "111,952,800"
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    "constant": "van-der-waerden-6-8",
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      "alt_display": "602,119,596"
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    "constant": "golomb-ruler-15",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "151",
      "decimal_approx": 151,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "shearer-1990"
  },
  {
    "constant": "golomb-ruler-16",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "177",
      "decimal_approx": 177,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "shearer-1990"
  },
  {
    "constant": "golomb-ruler-17",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "199",
      "decimal_approx": 199,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "sibert-1993"
  },
  {
    "constant": "golomb-ruler-18",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "216",
      "decimal_approx": 216,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "sibert-1993"
  },
  {
    "constant": "golomb-ruler-19",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "246",
      "decimal_approx": 246,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "dollas-rankin-mccracken-1998"
  },
  {
    "constant": "golomb-ruler-20",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "283",
      "decimal_approx": 283,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "garry-vanderschel-1997"
  },
  {
    "constant": "golomb-ruler-21",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "333",
      "decimal_approx": 333,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "garry-vanderschel-1998"
  },
  {
    "constant": "golomb-ruler-22",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "356",
      "decimal_approx": 356,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "garry-vanderschel-1999a"
  },
  {
    "constant": "golomb-ruler-23",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "372",
      "decimal_approx": 372,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "garry-vanderschel-1999b"
  },
  {
    "constant": "golomb-ruler-24",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "425",
      "decimal_approx": 425,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "distributed-net-ogr-24"
  },
  {
    "constant": "golomb-ruler-25",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "480",
      "decimal_approx": 480,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "distributed-net-ogr-25"
  },
  {
    "constant": "golomb-ruler-26",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "492",
      "decimal_approx": 492,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "distributed-net-ogr-26"
  },
  {
    "constant": "golomb-ruler-27",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "553",
      "decimal_approx": 553,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "distributed-net-ogr-27"
  },
  {
    "constant": "golomb-ruler-28",
    "group": "golomb-ruler",
    "bound": "lower",
    "literature": {
      "latex": "585",
      "decimal_approx": 585,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "distributed-net-ogr-28"
  },
  {
    "constant": "schur-1",
    "group": "schur",
    "bound": "upper",
    "literature": {
      "latex": "1",
      "decimal_approx": 1,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "trivial"
  },
  {
    "constant": "schur-2",
    "group": "schur",
    "bound": "upper",
    "literature": {
      "latex": "4",
      "decimal_approx": 4,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "trivial"
  },
  {
    "constant": "schur-3",
    "group": "schur",
    "bound": "upper",
    "literature": {
      "latex": "13",
      "decimal_approx": 13,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "trivial"
  },
  {
    "constant": "schur-4",
    "group": "schur",
    "bound": "upper",
    "literature": {
      "latex": "44",
      "decimal_approx": 44,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "golomb-baumert-1965"
  },
  {
    "constant": "schur-5",
    "group": "schur",
    "bound": "upper",
    "literature": {
      "latex": "160",
      "decimal_approx": 160,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "heule-2017-schur"
  },
  {
    "constant": "schur-9",
    "group": "schur",
    "bound": "lower",
    "literature": {
      "latex": "17803",
      "decimal_approx": 17803,
      "alt_display": "17,803"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ageron-et-al-2021"
  },
  {
    "constant": "weak-schur-1",
    "group": "weak-schur",
    "bound": "upper",
    "literature": {
      "latex": "2",
      "decimal_approx": 2,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "trivial"
  },
  {
    "constant": "weak-schur-2",
    "group": "weak-schur",
    "bound": "upper",
    "literature": {
      "latex": "8",
      "decimal_approx": 8,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "trivial"
  },
  {
    "constant": "weak-schur-3",
    "group": "weak-schur",
    "bound": "upper",
    "literature": {
      "latex": "23",
      "decimal_approx": 23,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "blanchard-harary-reis-2006"
  },
  {
    "constant": "weak-schur-4",
    "group": "weak-schur",
    "bound": "upper",
    "literature": {
      "latex": "66",
      "decimal_approx": 66,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "blanchard-harary-reis-2006"
  },
  {
    "constant": "weak-schur-7",
    "group": "weak-schur",
    "bound": "lower",
    "literature": {
      "latex": "2146",
      "decimal_approx": 2146,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "rowley-2020"
  },
  {
    "constant": "weak-schur-8",
    "group": "weak-schur",
    "bound": "lower",
    "literature": {
      "latex": "6976",
      "decimal_approx": 6976,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "rowley-2020"
  },
  {
    "constant": "weak-schur-9",
    "group": "weak-schur",
    "bound": "lower",
    "literature": {
      "latex": "22536",
      "decimal_approx": 22536,
      "alt_display": "22,536"
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "witness",
    "source": "ageron-et-al-2021"
  },
  {
    "constant": "snake-in-the-box-1",
    "group": "snake-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "1",
      "decimal_approx": 1,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "trivial"
  },
  {
    "constant": "snake-in-the-box-2",
    "group": "snake-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "2",
      "decimal_approx": 2,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "davies-1965"
  },
  {
    "constant": "snake-in-the-box-3",
    "group": "snake-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "4",
      "decimal_approx": 4,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "davies-1965"
  },
  {
    "constant": "snake-in-the-box-4",
    "group": "snake-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "7",
      "decimal_approx": 7,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "davies-1965"
  },
  {
    "constant": "snake-in-the-box-5",
    "group": "snake-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "13",
      "decimal_approx": 13,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "davies-1965"
  },
  {
    "constant": "snake-in-the-box-6",
    "group": "snake-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "26",
      "decimal_approx": 26,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "davies-1965"
  },
  {
    "constant": "snake-in-the-box-7",
    "group": "snake-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "50",
      "decimal_approx": 50,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "kochut-1996"
  },
  {
    "constant": "snake-in-the-box-8",
    "group": "snake-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "98",
      "decimal_approx": 98,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "ostergard-pettersson-2014-snake"
  },
  {
    "constant": "coil-in-the-box-2",
    "group": "coil-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "4",
      "decimal_approx": 4,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "kautz-1958"
  },
  {
    "constant": "coil-in-the-box-3",
    "group": "coil-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "6",
      "decimal_approx": 6,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "kautz-1958"
  },
  {
    "constant": "coil-in-the-box-4",
    "group": "coil-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "8",
      "decimal_approx": 8,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "kautz-1958"
  },
  {
    "constant": "coil-in-the-box-5",
    "group": "coil-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "14",
      "decimal_approx": 14,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "kautz-1958"
  },
  {
    "constant": "coil-in-the-box-6",
    "group": "coil-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "26",
      "decimal_approx": 26,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "davies-1965"
  },
  {
    "constant": "coil-in-the-box-7",
    "group": "coil-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "48",
      "decimal_approx": 48,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "kochut-1996"
  },
  {
    "constant": "coil-in-the-box-8",
    "group": "coil-in-the-box",
    "bound": "upper",
    "literature": {
      "latex": "96",
      "decimal_approx": 96,
      "alt_display": null
    },
    "verified": null,
    "verified_certificate": null,
    "evidence": "lean",
    "source": "ostergard-pettersson-2014-coil"
  }
]