{
  "A": [
    [
      "1",
      "1"
    ],
    [
      "2",
      "2"
    ]
  ],
  "b": [
    "1",
    "3"
  ],
  "rank_one_formula_denominator": "10",
  "exact_pseudoinverse": [
    [
      "1/10",
      "1/5"
    ],
    [
      "1/10",
      "1/5"
    ]
  ],
  "four_exact_MP_identities": {
    "A_P_A_equals_A": true,
    "P_A_P_equals_P": true,
    "A_P_symmetric": true,
    "P_A_symmetric": true
  },
  "minimum_norm_x": [
    "7/10",
    "7/10"
  ],
  "all17_family_checks": [
    {
      "t": "-2",
      "x": [
        "-13/10",
        "27/10"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "449/50"
    },
    {
      "t": "-7/4",
      "x": [
        "-21/20",
        "49/20"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "1421/200"
    },
    {
      "t": "-3/2",
      "x": [
        "-4/5",
        "11/5"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "137/25"
    },
    {
      "t": "-5/4",
      "x": [
        "-11/20",
        "39/20"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "821/200"
    },
    {
      "t": "-1",
      "x": [
        "-3/10",
        "17/10"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "149/50"
    },
    {
      "t": "-3/4",
      "x": [
        "-1/20",
        "29/20"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "421/200"
    },
    {
      "t": "-1/2",
      "x": [
        "1/5",
        "6/5"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "37/25"
    },
    {
      "t": "-1/4",
      "x": [
        "9/20",
        "19/20"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "221/200"
    },
    {
      "t": "0",
      "x": [
        "7/10",
        "7/10"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "49/50"
    },
    {
      "t": "1/4",
      "x": [
        "19/20",
        "9/20"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "221/200"
    },
    {
      "t": "1/2",
      "x": [
        "6/5",
        "1/5"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "37/25"
    },
    {
      "t": "3/4",
      "x": [
        "29/20",
        "-1/20"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "421/200"
    },
    {
      "t": "1",
      "x": [
        "17/10",
        "-3/10"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "149/50"
    },
    {
      "t": "5/4",
      "x": [
        "39/20",
        "-11/20"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "821/200"
    },
    {
      "t": "3/2",
      "x": [
        "11/5",
        "-4/5"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "137/25"
    },
    {
      "t": "7/4",
      "x": [
        "49/20",
        "-21/20"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "1421/200"
    },
    {
      "t": "2",
      "x": [
        "27/10",
        "-13/10"
      ],
      "residual_squared": "1/5",
      "solution_norm_squared": "449/50"
    }
  ],
  "actual_NumPy_comparison": {
    "rcond": 1e-12,
    "pinv": [
      [
        0.09999999999999995,
        0.19999999999999993
      ],
      [
        0.09999999999999995,
        0.19999999999999993
      ]
    ],
    "pinv_b": [
      0.6999999999999997,
      0.6999999999999997
    ],
    "lstsq_x": [
      0.6999999999999995,
      0.6999999999999995
    ],
    "reported_rank": 1,
    "singular_values": [
      3.1622776601683795,
      8.226773080236186e-17
    ],
    "lstsq_residual_array": [],
    "residual_array_empty_is_not_zero_residual_claim": true
  },
  "underdetermined_example": {
    "A": [
      [
        "1",
        "1",
        "0"
      ],
      [
        "0",
        "0",
        "1"
      ]
    ],
    "exact_pinv": [
      [
        "1/2",
        "0"
      ],
      [
        "1/2",
        "0"
      ],
      [
        "0",
        "1"
      ]
    ],
    "actual": {
      "rcond": 1e-12,
      "pinv": [
        [
          0.4999999999999998,
          0.0
        ],
        [
          0.4999999999999999,
          0.0
        ],
        [
          0.0,
          1.0
        ]
      ],
      "pinv_b": [
        0.9999999999999996,
        0.9999999999999998,
        3.0
      ],
      "lstsq_x": [
        0.9999999999999996,
        0.9999999999999999,
        3.0
      ],
      "reported_rank": 2,
      "singular_values": [
        1.4142135623730951,
        1.0
      ],
      "lstsq_residual_array": [],
      "residual_array_empty_is_not_zero_residual_claim": true
    }
  },
  "cutoff_diagonal_1_and_1e_minus8": [
    {
      "rcond": 1e-06,
      "pinv": [
        [
          1.0,
          0.0
        ],
        [
          0.0,
          0.0
        ]
      ]
    },
    {
      "rcond": 1e-10,
      "pinv": [
        [
          1.0,
          0.0
        ],
        [
          0.0,
          100000000.0
        ]
      ]
    }
  ],
  "zero_matrix_actual": {
    "rcond": 1e-12,
    "pinv": [
      [
        0.0,
        0.0
      ],
      [
        0.0,
        0.0
      ]
    ],
    "pinv_b": [
      0.0,
      0.0
    ],
    "lstsq_x": [
      0.0,
      0.0
    ],
    "reported_rank": 0,
    "singular_values": [
      0.0,
      0.0
    ],
    "lstsq_residual_array": [],
    "residual_array_empty_is_not_zero_residual_claim": true
  },
  "invalid_cases_rejected": [
    "shape_b",
    "ragged",
    "NaN",
    "bool",
    "empty",
    "negative_cutoff",
    "infinite_cutoff",
    "overbudget"
  ],
  "numpy_version": "2.3.5",
  "scope": "own exact rank-one/underdetermined finite examples versus actual existing NumPy; numeric cutoff is not true-rank discovery, residual-array empty is not zero SSE, no normal-equation inverse/general numerical stability/model accuracy/Opera claim"
}