{
  "quadratic_nodes": [
    "0",
    "1",
    "2"
  ],
  "quadratic_values": [
    "1",
    "2",
    "5"
  ],
  "barycentric_weights": [
    "1/2",
    "-1",
    "1/2"
  ],
  "seventeen_quarter_grid_queries_with_Lagrange_oracle": [
    {
      "query": "-1",
      "value": "2",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "-1/2",
          "numerator_contribution": "-1/2"
        },
        {
          "node": "1",
          "weight_over_distance": "1/2",
          "numerator_contribution": "1"
        },
        {
          "node": "2",
          "weight_over_distance": "-1/6",
          "numerator_contribution": "-5/6"
        }
      ]
    },
    {
      "query": "-3/4",
      "value": "25/16",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "-2/3",
          "numerator_contribution": "-2/3"
        },
        {
          "node": "1",
          "weight_over_distance": "4/7",
          "numerator_contribution": "8/7"
        },
        {
          "node": "2",
          "weight_over_distance": "-2/11",
          "numerator_contribution": "-10/11"
        }
      ]
    },
    {
      "query": "-1/2",
      "value": "5/4",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "-1",
          "numerator_contribution": "-1"
        },
        {
          "node": "1",
          "weight_over_distance": "2/3",
          "numerator_contribution": "4/3"
        },
        {
          "node": "2",
          "weight_over_distance": "-1/5",
          "numerator_contribution": "-1"
        }
      ]
    },
    {
      "query": "-1/4",
      "value": "17/16",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "-2",
          "numerator_contribution": "-2"
        },
        {
          "node": "1",
          "weight_over_distance": "4/5",
          "numerator_contribution": "8/5"
        },
        {
          "node": "2",
          "weight_over_distance": "-2/9",
          "numerator_contribution": "-10/9"
        }
      ]
    },
    {
      "query": "0",
      "value": "1",
      "status": "exact_node_return",
      "terms": []
    },
    {
      "query": "1/4",
      "value": "17/16",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "2",
          "numerator_contribution": "2"
        },
        {
          "node": "1",
          "weight_over_distance": "4/3",
          "numerator_contribution": "8/3"
        },
        {
          "node": "2",
          "weight_over_distance": "-2/7",
          "numerator_contribution": "-10/7"
        }
      ]
    },
    {
      "query": "1/2",
      "value": "5/4",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "1",
          "numerator_contribution": "1"
        },
        {
          "node": "1",
          "weight_over_distance": "2",
          "numerator_contribution": "4"
        },
        {
          "node": "2",
          "weight_over_distance": "-1/3",
          "numerator_contribution": "-5/3"
        }
      ]
    },
    {
      "query": "3/4",
      "value": "25/16",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "2/3",
          "numerator_contribution": "2/3"
        },
        {
          "node": "1",
          "weight_over_distance": "4",
          "numerator_contribution": "8"
        },
        {
          "node": "2",
          "weight_over_distance": "-2/5",
          "numerator_contribution": "-2"
        }
      ]
    },
    {
      "query": "1",
      "value": "2",
      "status": "exact_node_return",
      "terms": []
    },
    {
      "query": "5/4",
      "value": "41/16",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "2/5",
          "numerator_contribution": "2/5"
        },
        {
          "node": "1",
          "weight_over_distance": "-4",
          "numerator_contribution": "-8"
        },
        {
          "node": "2",
          "weight_over_distance": "-2/3",
          "numerator_contribution": "-10/3"
        }
      ]
    },
    {
      "query": "3/2",
      "value": "13/4",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "1/3",
          "numerator_contribution": "1/3"
        },
        {
          "node": "1",
          "weight_over_distance": "-2",
          "numerator_contribution": "-4"
        },
        {
          "node": "2",
          "weight_over_distance": "-1",
          "numerator_contribution": "-5"
        }
      ]
    },
    {
      "query": "7/4",
      "value": "65/16",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "2/7",
          "numerator_contribution": "2/7"
        },
        {
          "node": "1",
          "weight_over_distance": "-4/3",
          "numerator_contribution": "-8/3"
        },
        {
          "node": "2",
          "weight_over_distance": "-2",
          "numerator_contribution": "-10"
        }
      ]
    },
    {
      "query": "2",
      "value": "5",
      "status": "exact_node_return",
      "terms": []
    },
    {
      "query": "9/4",
      "value": "97/16",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "2/9",
          "numerator_contribution": "2/9"
        },
        {
          "node": "1",
          "weight_over_distance": "-4/5",
          "numerator_contribution": "-8/5"
        },
        {
          "node": "2",
          "weight_over_distance": "2",
          "numerator_contribution": "10"
        }
      ]
    },
    {
      "query": "5/2",
      "value": "29/4",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "1/5",
          "numerator_contribution": "1/5"
        },
        {
          "node": "1",
          "weight_over_distance": "-2/3",
          "numerator_contribution": "-4/3"
        },
        {
          "node": "2",
          "weight_over_distance": "1",
          "numerator_contribution": "5"
        }
      ]
    },
    {
      "query": "11/4",
      "value": "137/16",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "2/11",
          "numerator_contribution": "2/11"
        },
        {
          "node": "1",
          "weight_over_distance": "-4/7",
          "numerator_contribution": "-8/7"
        },
        {
          "node": "2",
          "weight_over_distance": "2/3",
          "numerator_contribution": "10/3"
        }
      ]
    },
    {
      "query": "3",
      "value": "10",
      "status": "barycentric_ratio",
      "terms": [
        {
          "node": "0",
          "weight_over_distance": "1/6",
          "numerator_contribution": "1/6"
        },
        {
          "node": "1",
          "weight_over_distance": "-1/2",
          "numerator_contribution": "-1"
        },
        {
          "node": "2",
          "weight_over_distance": "1/2",
          "numerator_contribution": "5/2"
        }
      ]
    }
  ],
  "all_six_node_permutations_checked": 6,
  "interior_Runge_counterexample": {
    "function": "1/(1+25*x*x)",
    "nodes": [
      "-1",
      "-1/2",
      "0",
      "1/2",
      "1"
    ],
    "values": [
      "1/26",
      "4/29",
      "1",
      "4/29",
      "1/26"
    ],
    "query": "3/4",
    "polynomial_interpolant": "-17219/48256",
    "true_function": "16/241",
    "all_five_nodes_match": [
      {
        "x": "-1",
        "exact_y_returned": "1/26"
      },
      {
        "x": "-1/2",
        "exact_y_returned": "4/29"
      },
      {
        "x": "0",
        "exact_y_returned": "1"
      },
      {
        "x": "1/2",
        "exact_y_returned": "4/29"
      },
      {
        "x": "1",
        "exact_y_returned": "1/26"
      }
    ],
    "terms": [
      {
        "node": "-1",
        "weight_over_distance": "8/21",
        "numerator_contribution": "4/273"
      },
      {
        "node": "-1/2",
        "weight_over_distance": "-32/15",
        "numerator_contribution": "-128/435"
      },
      {
        "node": "0",
        "weight_over_distance": "16/3",
        "numerator_contribution": "16/3"
      },
      {
        "node": "1/2",
        "weight_over_distance": "-32/3",
        "numerator_contribution": "-128/87"
      },
      {
        "node": "1",
        "weight_over_distance": "-8/3",
        "numerator_contribution": "-4/39"
      }
    ]
  },
  "invalid_inputs_rejected": [
    "duplicate_same_y",
    "duplicate_different_y",
    "mismatch",
    "empty",
    "nonfinite",
    "bool",
    "too_many"
  ],
  "scope": "exact finite interpolation, not regression/SMOTE/statistical confidence or SciPy execution; barycentric representation does not remove intrinsic polynomial interpolation error/Runge phenomenon; node equality branch avoids division by0"
}