{
  "function": "x^3-2x+2",
  "bisection_eight_steps": [
    {
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      "before": [
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      ],
      "mid": "-3/2",
      "f_mid": "13/8",
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      "width": "1/2",
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    {
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    },
    {
      "step": 3,
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      ],
      "mid": "-15/8",
      "f_mid": "-431/512",
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        "-15/8",
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      "width": "1/8",
      "endpoint_sign_product": "-3879/32768"
    },
    {
      "step": 4,
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      "mid": "-29/16",
      "f_mid": "-1349/4096",
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      "width": "1/16",
      "endpoint_sign_product": "-12141/262144"
    },
    {
      "step": 5,
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      "mid": "-57/32",
      "f_mid": "-2921/32768",
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        "-57/32",
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      "width": "1/32",
      "endpoint_sign_product": "-26289/2097152"
    },
    {
      "step": 6,
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        "-57/32",
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      ],
      "mid": "-113/64",
      "f_mid": "7087/262144",
      "after": [
        "-57/32",
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      "width": "1/64",
      "endpoint_sign_product": "-20701127/8589934592"
    },
    {
      "step": 7,
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      "mid": "-227/128",
      "f_mid": "-64443/2097152",
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        "-227/128",
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      "width": "1/128",
      "endpoint_sign_product": "-456707541/549755813888"
    },
    {
      "step": 8,
      "before": [
        "-227/128",
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      "mid": "-453/256",
      "f_mid": "-29629/16777216",
      "after": [
        "-453/256",
        "-113/64"
      ],
      "width": "1/256",
      "endpoint_sign_product": "-209980723/4398046511104"
    }
  ],
  "final_bracket": [
    "-453/256",
    "-113/64"
  ],
  "width": "1/256",
  "status": "budget_exhausted",
  "Newton_from_zero_exact_cycle": [
    "0",
    "1",
    "0",
    "1",
    "0",
    "1",
    "0"
  ],
  "Newton_status": "budget_exhausted",
  "separate_Newton_zero_derivative": {
    "points": [
      "0"
    ],
    "status": "zero_derivative"
  },
  "endpoint_root": {
    "value": "1",
    "status": "endpoint_root"
  },
  "separate_width_stop": {
    "bracket": [
      "181/128",
      "1449/1024"
    ],
    "status": "width_tolerance"
  },
  "discontinuous_one_over_x_counterexample": {
    "bracket": [
      "-1/256",
      "1/128"
    ],
    "last_midpoint_residual": "-256",
    "status": "budget_exhausted",
    "continuous_precondition_false": true,
    "root_claimed": false
  },
  "rejected": [
    "no_bracket",
    "reversed",
    "zero_tol",
    "invalid_cap",
    "nonfinite_function"
  ],
  "scope": "no SciPy run, no universal Newton convergence, bracket sign alone requires continuity for root existence"
}