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      "held_target": 3899,
      "first_pivot": 3551,
      "required_second_pivot": {
        "numerator": 98377,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63923,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3552,
      "required_second_pivot": {
        "numerator": 98352,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63921,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3553,
      "required_second_pivot": {
        "numerator": 98327,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63919,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3554,
      "required_second_pivot": {
        "numerator": 4274,
        "denominator": 1
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 2779,
        "denominator": 1
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3555,
      "required_second_pivot": {
        "numerator": 98277,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63915,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3620,
      "required_second_pivot": {
        "numerator": 96652,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63785,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3621,
      "required_second_pivot": {
        "numerator": 96627,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63783,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3622,
      "required_second_pivot": {
        "numerator": 96602,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63781,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3623,
      "required_second_pivot": {
        "numerator": 4199,
        "denominator": 1
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 2773,
        "denominator": 1
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3624,
      "required_second_pivot": {
        "numerator": 96552,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63777,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3625,
      "required_second_pivot": {
        "numerator": 96527,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63775,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3680,
      "required_second_pivot": {
        "numerator": 95152,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63665,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3681,
      "required_second_pivot": {
        "numerator": 95127,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63663,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3682,
      "required_second_pivot": {
        "numerator": 95102,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63661,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3683,
      "required_second_pivot": {
        "numerator": 95077,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63659,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3684,
      "required_second_pivot": {
        "numerator": 95052,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63657,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3685,
      "required_second_pivot": {
        "numerator": 95027,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63655,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3705,
      "required_second_pivot": {
        "numerator": 94527,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63615,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3706,
      "required_second_pivot": {
        "numerator": 94502,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63613,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3707,
      "required_second_pivot": {
        "numerator": 94477,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63611,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3708,
      "required_second_pivot": {
        "numerator": 94452,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63609,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3709,
      "required_second_pivot": {
        "numerator": 94427,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63607,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3710,
      "required_second_pivot": {
        "numerator": 94402,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63605,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3765,
      "required_second_pivot": {
        "numerator": 93027,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63495,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3766,
      "required_second_pivot": {
        "numerator": 93002,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63493,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3767,
      "required_second_pivot": {
        "numerator": 92977,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63491,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3768,
      "required_second_pivot": {
        "numerator": 92952,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63489,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3769,
      "required_second_pivot": {
        "numerator": 92927,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63487,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3770,
      "required_second_pivot": {
        "numerator": 92902,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63485,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3835,
      "required_second_pivot": {
        "numerator": 91277,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63355,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3836,
      "required_second_pivot": {
        "numerator": 91252,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63353,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3837,
      "required_second_pivot": {
        "numerator": 91227,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63351,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3838,
      "required_second_pivot": {
        "numerator": 91202,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63349,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3839,
      "required_second_pivot": {
        "numerator": 91177,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63347,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3840,
      "required_second_pivot": {
        "numerator": 91152,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63345,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3895,
      "required_second_pivot": {
        "numerator": 89777,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63235,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3896,
      "required_second_pivot": {
        "numerator": 89752,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63233,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3897,
      "required_second_pivot": {
        "numerator": 89727,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63231,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3898,
      "required_second_pivot": {
        "numerator": 89702,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63229,
        "denominator": 23
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3899,
      "required_second_pivot": {
        "numerator": 3899,
        "denominator": 1
      },
      "second_pivot_sourced": true,
      "intermediate": {
        "numerator": 2749,
        "denominator": 1
      },
      "intermediate_sourced": false
    },
    {
      "held_target": 3899,
      "first_pivot": 3900,
      "required_second_pivot": {
        "numerator": 89652,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63225,
        "denominator": 23
      },
      "intermediate_sourced": false
    }
  ],
  "slope_valuation_certificate": {
    "E": [
      0,
      0,
      2
    ],
    "P": [
      1,
      0,
      1
    ],
    "J": [
      0,
      -1,
      2
    ],
    "target": [
      0,
      0,
      4
    ]
  },
  "integer_pivot_parameterization": "p=H+23t, q=H−25t; intermediate=H−1150−2t. Both pivots and the intermediate must already be sourced.",
  "counts": {
    "target_E2_maps": 13,
    "first_pivot_trials": 4849,
    "sourced_pivot_pair_solutions": 39,
    "distinct_pivot_pair_solutions": 26,
    "maps_with_sourced_path": 5,
    "endpoint_only_maps_repaired": 1,
    "complete_coordinate_paths": 5
  },
  "interpretation": "The source menu for possible pivots is deliberately generous: all373 existing regular coordinates. Coordinate source membership alone is tested, not a newly prescribed pivot grammar. Slope valuations force every positive E/P/J word equal to a target whole map to exactly two E factors. Failure in this larger menu excludes a sourced repair in the narrower existing role rules. Different slopes at one input, inverses and new sources are outside the claim."
}