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      "required_second_pivot": {
        "numerator": 102152,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 64225,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3490,
      "required_second_pivot": {
        "numerator": 99902,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 64045,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3491,
      "required_second_pivot": {
        "numerator": 99877,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 64043,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3492,
      "required_second_pivot": {
        "numerator": 99852,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 64041,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3493,
      "required_second_pivot": {
        "numerator": 99827,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 64039,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3494,
      "required_second_pivot": {
        "numerator": 99802,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 64037,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3495,
      "required_second_pivot": {
        "numerator": 99777,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 64035,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3550,
      "required_second_pivot": {
        "numerator": 98402,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63925,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3551,
      "required_second_pivot": {
        "numerator": 98377,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63923,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3552,
      "required_second_pivot": {
        "numerator": 98352,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63921,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3553,
      "required_second_pivot": {
        "numerator": 98327,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63919,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3554,
      "required_second_pivot": {
        "numerator": 4274,
        "denominator": 1
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 2779,
        "denominator": 1
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3555,
      "required_second_pivot": {
        "numerator": 98277,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63915,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3620,
      "required_second_pivot": {
        "numerator": 96652,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63785,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3621,
      "required_second_pivot": {
        "numerator": 96627,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63783,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3622,
      "required_second_pivot": {
        "numerator": 96602,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63781,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3623,
      "required_second_pivot": {
        "numerator": 4199,
        "denominator": 1
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 2773,
        "denominator": 1
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3624,
      "required_second_pivot": {
        "numerator": 96552,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63777,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3625,
      "required_second_pivot": {
        "numerator": 96527,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63775,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3680,
      "required_second_pivot": {
        "numerator": 95152,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63665,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3681,
      "required_second_pivot": {
        "numerator": 95127,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63663,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3682,
      "required_second_pivot": {
        "numerator": 95102,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63661,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3683,
      "required_second_pivot": {
        "numerator": 95077,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63659,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3684,
      "required_second_pivot": {
        "numerator": 95052,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63657,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3685,
      "required_second_pivot": {
        "numerator": 95027,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63655,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3705,
      "required_second_pivot": {
        "numerator": 94527,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63615,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3706,
      "required_second_pivot": {
        "numerator": 94502,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63613,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3707,
      "required_second_pivot": {
        "numerator": 94477,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63611,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3708,
      "required_second_pivot": {
        "numerator": 94452,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63609,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3709,
      "required_second_pivot": {
        "numerator": 94427,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63607,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3710,
      "required_second_pivot": {
        "numerator": 94402,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63605,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3765,
      "required_second_pivot": {
        "numerator": 93027,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63495,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3766,
      "required_second_pivot": {
        "numerator": 93002,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63493,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3767,
      "required_second_pivot": {
        "numerator": 92977,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63491,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3768,
      "required_second_pivot": {
        "numerator": 92952,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63489,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3769,
      "required_second_pivot": {
        "numerator": 92927,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63487,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3770,
      "required_second_pivot": {
        "numerator": 92902,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63485,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3835,
      "required_second_pivot": {
        "numerator": 91277,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63355,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3836,
      "required_second_pivot": {
        "numerator": 91252,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63353,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3837,
      "required_second_pivot": {
        "numerator": 91227,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63351,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3838,
      "required_second_pivot": {
        "numerator": 91202,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63349,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3839,
      "required_second_pivot": {
        "numerator": 91177,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63347,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3840,
      "required_second_pivot": {
        "numerator": 91152,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63345,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3895,
      "required_second_pivot": {
        "numerator": 89777,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63235,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3896,
      "required_second_pivot": {
        "numerator": 89752,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63233,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3897,
      "required_second_pivot": {
        "numerator": 89727,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63231,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3898,
      "required_second_pivot": {
        "numerator": 89702,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63229,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3899,
      "required_second_pivot": {
        "numerator": 3899,
        "denominator": 1
      },
      "second_pivot_sourced": true,
      "intermediate": {
        "numerator": 2749,
        "denominator": 1
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    },
    {
      "first_pivot": 3900,
      "required_second_pivot": {
        "numerator": 89652,
        "denominator": 23
      },
      "second_pivot_sourced": false,
      "intermediate": {
        "numerator": 63225,
        "denominator": 23
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    }
  ],
  "sourced_pivot_solutions": [
    {
      "first_pivot": 3899,
      "required_second_pivot": {
        "numerator": 3899,
        "denominator": 1
      },
      "second_pivot_sourced": true,
      "intermediate": {
        "numerator": 2749,
        "denominator": 1
      },
      "intermediate_sourced": false,
      "same_map_source_path": false
    }
  ],
  "prime_valuation_certificate": {
    "7": {
      "E": 0,
      "P": 1,
      "J": 0,
      "target": 0
    },
    "13": {
      "E": 0,
      "P": 0,
      "J": -1,
      "target": 0
    },
    "5": {
      "E": 2,
      "P": 1,
      "J": 2,
      "target": 4
    }
  },
  "integer_solution_form": "p=3899+23t, q=3899−25t, t integer. Both pivots≤3900 force t=0.",
  "positive_word_count_solution": {
    "E": 2,
    "P": 0,
    "J": 0
  },
  "counts": {
    "first_pivot_coordinates": 373,
    "sourced_two_pivot_solutions": 1,
    "complete_source_paths": 0
  },
  "interpretation": "Every finite positive E/P/J word equal as a whole affine map has exactly two E steps, by slope valuations. The only possible pair of pivots from even the entire regular coordinate menu is3899,3899, requiring absent2749. This excludes same-map source paths in that grammar. It does not exclude different slopes coinciding at one endpoint, inverse operators, or a future expanded source menu."
}