{
  "table": "T623",
  "sage_version": "10.9",
  "skill_evidence": {
    "manifest": "agents/runs/20261030T103826Z-repair-skills/manifest.json",
    "local_sha256": "28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199",
    "live_sha256": "28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199"
  },
  "scope": "Repair verification only; no independent argument-review verdict.",
  "method": "Exact cyclotomic cotangent sum, live T622 eta differences, and unchanged generator comparison.",
  "precision": "Exact rationals; no approximate argument or rounding uncertainty.",
  "controls": [
    [
      2,
      1,
      1,
      "0"
    ],
    [
      3,
      1,
      1,
      "1/3"
    ],
    [
      3,
      2,
      1,
      "-1/3"
    ],
    [
      4,
      1,
      2,
      "1"
    ],
    [
      7,
      2,
      0,
      "0"
    ]
  ],
  "checked_rows": 550,
  "all_passed": true,
  "results": [
    {
      "p": 2,
      "q": 1,
      "u": 1,
      "rho": "0",
      "eta_difference": "0",
      "exact_cyclotomic_match": true
    },
    {
      "p": 3,
      "q": 1,
      "u": 1,
      "rho": "1/3",
      "eta_difference": "1/3",
      "exact_cyclotomic_match": true
    },
    {
      "p": 3,
      "q": 2,
      "u": 1,
      "rho": "-1/3",
      "eta_difference": "-1/3",
      "exact_cyclotomic_match": true
    },
    {
      "p": 4,
      "q": 1,
      "u": 1,
      "rho": "1/2",
      "eta_difference": "1/2",
      "exact_cyclotomic_match": true
    },
    {
      "p": 4,
      "q": 1,
      "u": 2,
      "rho": "1",
      "eta_difference": "1",
      "exact_cyclotomic_match": true
    },
    {
      "p": 4,
      "q": 3,
      "u": 1,
      "rho": "-1/2",
      "eta_difference": "-1/2",
      "exact_cyclotomic_match": true
    },
    {
      "p": 4,
      "q": 3,
      "u": 2,
      "rho": "-1",
      "eta_difference": "-1",
      "exact_cyclotomic_match": true
    },
    {
      "p": 5,
      "q": 1,
      "u": 1,
      "rho": "3/5",
      "eta_difference": "3/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 5,
      "q": 1,
      "u": 2,
      "rho": "7/5",
      "eta_difference": "7/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 5,
      "q": 2,
      "u": 1,
      "rho": "-1/5",
      "eta_difference": "-1/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 5,
      "q": 2,
      "u": 2,
      "rho": "1/5",
      "eta_difference": "1/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 5,
      "q": 4,
      "u": 1,
      "rho": "-3/5",
      "eta_difference": "-3/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 5,
      "q": 4,
      "u": 2,
      "rho": "-7/5",
      "eta_difference": "-7/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 6,
      "q": 1,
      "u": 1,
      "rho": "2/3",
      "eta_difference": "2/3",
      "exact_cyclotomic_match": true
    },
    {
      "p": 6,
      "q": 1,
      "u": 2,
      "rho": "5/3",
      "eta_difference": "5/3",
      "exact_cyclotomic_match": true
    },
    {
      "p": 6,
      "q": 1,
      "u": 3,
      "rho": "2",
      "eta_difference": "2",
      "exact_cyclotomic_match": true
    },
    {
      "p": 6,
      "q": 5,
      "u": 1,
      "rho": "-2/3",
      "eta_difference": "-2/3",
      "exact_cyclotomic_match": true
    },
    {
      "p": 6,
      "q": 5,
      "u": 2,
      "rho": "-5/3",
      "eta_difference": "-5/3",
      "exact_cyclotomic_match": true
    },
    {
      "p": 6,
      "q": 5,
      "u": 3,
      "rho": "-2",
      "eta_difference": "-2",
      "exact_cyclotomic_match": true
    },
    {
      "p": 7,
      "q": 1,
      "u": 1,
      "rho": "5/7",
      "eta_difference": "5/7",
      "exact_cyclotomic_match": true
    },
    {
      "p": 7,
      "q": 1,
      "u": 2,
      "rho": "13/7",
      "eta_difference": "13/7",
      "exact_cyclotomic_match": true
    },
    {
      "p": 7,
      "q": 1,
      "u": 3,
      "rho": "17/7",
      "eta_difference": "17/7",
      "exact_cyclotomic_match": true
    },
    {
      "p": 7,
      "q": 2,
      "u": 1,
      "rho": "-1/7",
      "eta_difference": "-1/7",
      "exact_cyclotomic_match": true
    },
    {
      "p": 7,
      "q": 2,
      "u": 2,
      "rho": "3/7",
      "eta_difference": "3/7",
      "exact_cyclotomic_match": true
    },
    {
      "p": 7,
      "q": 2,
      "u": 3,
      "rho": "5/7",
      "eta_difference": "5/7",
      "exact_cyclotomic_match": true
    },
    {
      "p": 7,
      "q": 3,
      "u": 1,
      "rho": "-3/7",
      "eta_difference": "-3/7",
      "exact_cyclotomic_match": true
    },
    {
      "p": 7,
      "q": 3,
      "u": 2,
      "rho": "-5/7",
      "eta_difference": "-5/7",
      "exact_cyclotomic_match": true
    },
    {
      "p": 7,
      "q": 3,
      "u": 3,
      "rho": "1/7",
      "eta_difference": "1/7",
      "exact_cyclotomic_match": true
    },
    {
      "p": 7,
      "q": 6,
      "u": 1,
      "rho": "-5/7",
      "eta_difference": "-5/7",
      "exact_cyclotomic_match": true
    },
    {
      "p": 7,
      "q": 6,
      "u": 2,
      "rho": "-13/7",
      "eta_difference": "-13/7",
      "exact_cyclotomic_match": true
    },
    {
      "p": 7,
      "q": 6,
      "u": 3,
      "rho": "-17/7",
      "eta_difference": "-17/7",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 1,
      "u": 1,
      "rho": "3/4",
      "eta_difference": "3/4",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 1,
      "u": 2,
      "rho": "2",
      "eta_difference": "2",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 1,
      "u": 3,
      "rho": "11/4",
      "eta_difference": "11/4",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 1,
      "u": 4,
      "rho": "3",
      "eta_difference": "3",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 3,
      "u": 1,
      "rho": "1/4",
      "eta_difference": "1/4",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 3,
      "u": 2,
      "rho": "0",
      "eta_difference": "0",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 3,
      "u": 3,
      "rho": "1/4",
      "eta_difference": "1/4",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 3,
      "u": 4,
      "rho": "1",
      "eta_difference": "1",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 5,
      "u": 1,
      "rho": "-1/4",
      "eta_difference": "-1/4",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 5,
      "u": 2,
      "rho": "0",
      "eta_difference": "0",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 5,
      "u": 3,
      "rho": "-1/4",
      "eta_difference": "-1/4",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 5,
      "u": 4,
      "rho": "-1",
      "eta_difference": "-1",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 7,
      "u": 1,
      "rho": "-3/4",
      "eta_difference": "-3/4",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 7,
      "u": 2,
      "rho": "-2",
      "eta_difference": "-2",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 7,
      "u": 3,
      "rho": "-11/4",
      "eta_difference": "-11/4",
      "exact_cyclotomic_match": true
    },
    {
      "p": 8,
      "q": 7,
      "u": 4,
      "rho": "-3",
      "eta_difference": "-3",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
      "q": 1,
      "u": 1,
      "rho": "7/9",
      "eta_difference": "7/9",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
      "q": 1,
      "u": 2,
      "rho": "19/9",
      "eta_difference": "19/9",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
      "q": 1,
      "u": 3,
      "rho": "3",
      "eta_difference": "3",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
      "q": 1,
      "u": 4,
      "rho": "31/9",
      "eta_difference": "31/9",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
      "q": 2,
      "u": 1,
      "rho": "-1/9",
      "eta_difference": "-1/9",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
      "q": 2,
      "u": 2,
      "rho": "5/9",
      "eta_difference": "5/9",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
      "q": 2,
      "u": 3,
      "rho": "1",
      "eta_difference": "1",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
      "q": 2,
      "u": 4,
      "rho": "11/9",
      "eta_difference": "11/9",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
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      "u": 1,
      "rho": "-5/9",
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      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
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      "u": 2,
      "rho": "-11/9",
      "eta_difference": "-11/9",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
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      "u": 3,
      "rho": "-1",
      "eta_difference": "-1",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
      "q": 4,
      "u": 4,
      "rho": "1/9",
      "eta_difference": "1/9",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
      "q": 8,
      "u": 1,
      "rho": "-7/9",
      "eta_difference": "-7/9",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
      "q": 8,
      "u": 2,
      "rho": "-19/9",
      "eta_difference": "-19/9",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
      "q": 8,
      "u": 3,
      "rho": "-3",
      "eta_difference": "-3",
      "exact_cyclotomic_match": true
    },
    {
      "p": 9,
      "q": 8,
      "u": 4,
      "rho": "-31/9",
      "eta_difference": "-31/9",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 1,
      "u": 1,
      "rho": "4/5",
      "eta_difference": "4/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 1,
      "u": 2,
      "rho": "11/5",
      "eta_difference": "11/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 1,
      "u": 3,
      "rho": "16/5",
      "eta_difference": "16/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 1,
      "u": 4,
      "rho": "19/5",
      "eta_difference": "19/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 1,
      "u": 5,
      "rho": "4",
      "eta_difference": "4",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 3,
      "u": 1,
      "rho": "-2/5",
      "eta_difference": "-2/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 3,
      "u": 2,
      "rho": "-3/5",
      "eta_difference": "-3/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 3,
      "u": 3,
      "rho": "2/5",
      "eta_difference": "2/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 3,
      "u": 4,
      "rho": "3/5",
      "eta_difference": "3/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 3,
      "u": 5,
      "rho": "0",
      "eta_difference": "0",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 9,
      "u": 1,
      "rho": "-4/5",
      "eta_difference": "-4/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 9,
      "u": 2,
      "rho": "-11/5",
      "eta_difference": "-11/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 9,
      "u": 3,
      "rho": "-16/5",
      "eta_difference": "-16/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 9,
      "u": 4,
      "rho": "-19/5",
      "eta_difference": "-19/5",
      "exact_cyclotomic_match": true
    },
    {
      "p": 10,
      "q": 9,
      "u": 5,
      "rho": "-4",
      "eta_difference": "-4",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 1,
      "u": 1,
      "rho": "9/11",
      "eta_difference": "9/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 1,
      "u": 2,
      "rho": "25/11",
      "eta_difference": "25/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 1,
      "u": 3,
      "rho": "37/11",
      "eta_difference": "37/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 1,
      "u": 4,
      "rho": "45/11",
      "eta_difference": "45/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 1,
      "u": 5,
      "rho": "49/11",
      "eta_difference": "49/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 2,
      "u": 1,
      "rho": "-1/11",
      "eta_difference": "-1/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 2,
      "u": 2,
      "rho": "7/11",
      "eta_difference": "7/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 2,
      "u": 3,
      "rho": "13/11",
      "eta_difference": "13/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 2,
      "u": 4,
      "rho": "17/11",
      "eta_difference": "17/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 2,
      "u": 5,
      "rho": "19/11",
      "eta_difference": "19/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 3,
      "u": 1,
      "rho": "3/11",
      "eta_difference": "3/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 3,
      "u": 2,
      "rho": "1/11",
      "eta_difference": "1/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 3,
      "u": 3,
      "rho": "5/11",
      "eta_difference": "5/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 3,
      "u": 4,
      "rho": "15/11",
      "eta_difference": "15/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 3,
      "u": 5,
      "rho": "9/11",
      "eta_difference": "9/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 5,
      "u": 1,
      "rho": "-7/11",
      "eta_difference": "-7/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 5,
      "u": 2,
      "rho": "-17/11",
      "eta_difference": "-17/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 5,
      "u": 3,
      "rho": "-19/11",
      "eta_difference": "-19/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 5,
      "u": 4,
      "rho": "-13/11",
      "eta_difference": "-13/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 5,
      "u": 5,
      "rho": "1/11",
      "eta_difference": "1/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 7,
      "u": 1,
      "rho": "-5/11",
      "eta_difference": "-5/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 7,
      "u": 2,
      "rho": "-9/11",
      "eta_difference": "-9/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 7,
      "u": 3,
      "rho": "-1/11",
      "eta_difference": "-1/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 7,
      "u": 4,
      "rho": "-3/11",
      "eta_difference": "-3/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 7,
      "u": 5,
      "rho": "-15/11",
      "eta_difference": "-15/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 10,
      "u": 1,
      "rho": "-9/11",
      "eta_difference": "-9/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 10,
      "u": 2,
      "rho": "-25/11",
      "eta_difference": "-25/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 10,
      "u": 3,
      "rho": "-37/11",
      "eta_difference": "-37/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 10,
      "u": 4,
      "rho": "-45/11",
      "eta_difference": "-45/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 11,
      "q": 10,
      "u": 5,
      "rho": "-49/11",
      "eta_difference": "-49/11",
      "exact_cyclotomic_match": true
    },
    {
      "p": 12,
      "q": 1,
      "u": 1,
      "rho": "5/6",
      "eta_difference": "5/6",
      "exact_cyclotomic_match": true
    },
    {
      "p": 12,
      "q": 1,
      "u": 2,
      "rho": "7/3",
      "eta_difference": "7/3",
      "exact_cyclotomic_match": true
    },
    {
      "p": 12,
      "q": 1,
      "u": 3,
      "rho": "7/2",
      "eta_difference": "7/2",
      "exact_cyclotomic_match": true
    },
    {
      "p": 12,
      "q": 1,
      "u": 4,
      "rho": "13/3",
      "eta_difference": "13/3",
      "exact_cyclotomic_match": true
    },
    {
      "p": 12,
      "q": 1,
      "u": 5,
      "rho": "29/6",
      "eta_difference": "29/6",
      "exact_cyclotomic_match": true
    },
    {
      "p": 12,
      "q": 1,
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