{
  "grid": [
    [
      1,
      2,
      3,
      4
    ],
    [
      5,
      6,
      7,
      8
    ],
    [
      9,
      10,
      11,
      12
    ]
  ],
  "zero_padded_prefix": [
    [
      0,
      0,
      0,
      0,
      0
    ],
    [
      0,
      1,
      3,
      6,
      10
    ],
    [
      0,
      6,
      14,
      24,
      36
    ],
    [
      0,
      15,
      33,
      54,
      78
    ]
  ],
  "main_rectangle": {
    "rows": [
      1,
      3
    ],
    "cols": [
      1,
      4
    ],
    "sum": 54,
    "wrong_only_prefix_bottom_right": 78
  },
  "all150_rectangles": [
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        0,
        0
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        0,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        0,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        0,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        0,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        1,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        1,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        1,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        1,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        2,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        2,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        2,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        3,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        3,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        0
      ],
      "cols": [
        4,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        0,
        0
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        0,
        1
      ],
      "sum": 1
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        0,
        2
      ],
      "sum": 3
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        0,
        3
      ],
      "sum": 6
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        0,
        4
      ],
      "sum": 10
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        1,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        1,
        2
      ],
      "sum": 2
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        1,
        3
      ],
      "sum": 5
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        1,
        4
      ],
      "sum": 9
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        2,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        2,
        3
      ],
      "sum": 3
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        2,
        4
      ],
      "sum": 7
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        3,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        3,
        4
      ],
      "sum": 4
    },
    {
      "rows": [
        0,
        1
      ],
      "cols": [
        4,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        0,
        0
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        0,
        1
      ],
      "sum": 6
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        0,
        2
      ],
      "sum": 14
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        0,
        3
      ],
      "sum": 24
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        0,
        4
      ],
      "sum": 36
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        1,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        1,
        2
      ],
      "sum": 8
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        1,
        3
      ],
      "sum": 18
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        1,
        4
      ],
      "sum": 30
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        2,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        2,
        3
      ],
      "sum": 10
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        2,
        4
      ],
      "sum": 22
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        3,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        3,
        4
      ],
      "sum": 12
    },
    {
      "rows": [
        0,
        2
      ],
      "cols": [
        4,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        0,
        0
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        0,
        1
      ],
      "sum": 15
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        0,
        2
      ],
      "sum": 33
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        0,
        3
      ],
      "sum": 54
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        0,
        4
      ],
      "sum": 78
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        1,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        1,
        2
      ],
      "sum": 18
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        1,
        3
      ],
      "sum": 39
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        1,
        4
      ],
      "sum": 63
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        2,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        2,
        3
      ],
      "sum": 21
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        2,
        4
      ],
      "sum": 45
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        3,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        3,
        4
      ],
      "sum": 24
    },
    {
      "rows": [
        0,
        3
      ],
      "cols": [
        4,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        0,
        0
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        0,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        0,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        0,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        0,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        1,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        1,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        1,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        1,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        2,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        2,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        2,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        3,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        3,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        1
      ],
      "cols": [
        4,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        0,
        0
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        0,
        1
      ],
      "sum": 5
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        0,
        2
      ],
      "sum": 11
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        0,
        3
      ],
      "sum": 18
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        0,
        4
      ],
      "sum": 26
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        1,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        1,
        2
      ],
      "sum": 6
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        1,
        3
      ],
      "sum": 13
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        1,
        4
      ],
      "sum": 21
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        2,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        2,
        3
      ],
      "sum": 7
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        2,
        4
      ],
      "sum": 15
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        3,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        3,
        4
      ],
      "sum": 8
    },
    {
      "rows": [
        1,
        2
      ],
      "cols": [
        4,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        0,
        0
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        0,
        1
      ],
      "sum": 14
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        0,
        2
      ],
      "sum": 30
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        0,
        3
      ],
      "sum": 48
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        0,
        4
      ],
      "sum": 68
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        1,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        1,
        2
      ],
      "sum": 16
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        1,
        3
      ],
      "sum": 34
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        1,
        4
      ],
      "sum": 54
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        2,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        2,
        3
      ],
      "sum": 18
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        2,
        4
      ],
      "sum": 38
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        3,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        3,
        4
      ],
      "sum": 20
    },
    {
      "rows": [
        1,
        3
      ],
      "cols": [
        4,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        0,
        0
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        0,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        0,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        0,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        0,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        1,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        1,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        1,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        1,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        2,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        2,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        2,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        3,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        3,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        2
      ],
      "cols": [
        4,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        0,
        0
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        0,
        1
      ],
      "sum": 9
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        0,
        2
      ],
      "sum": 19
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        0,
        3
      ],
      "sum": 30
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        0,
        4
      ],
      "sum": 42
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        1,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        1,
        2
      ],
      "sum": 10
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        1,
        3
      ],
      "sum": 21
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        1,
        4
      ],
      "sum": 33
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        2,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        2,
        3
      ],
      "sum": 11
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        2,
        4
      ],
      "sum": 23
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        3,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        3,
        4
      ],
      "sum": 12
    },
    {
      "rows": [
        2,
        3
      ],
      "cols": [
        4,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        0,
        0
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        0,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        0,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        0,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        0,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        1,
        1
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        1,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        1,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        1,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        2,
        2
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        2,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        2,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        3,
        3
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        3,
        4
      ],
      "sum": 0
    },
    {
      "rows": [
        3,
        3
      ],
      "cols": [
        4,
        4
      ],
      "sum": 0
    }
  ],
  "negative_grid": {
    "grid": [
      [
        -2,
        0,
        3
      ],
      [
        4,
        -5,
        1
      ]
    ],
    "prefix": [
      [
        0,
        0,
        0,
        0
      ],
      [
        0,
        -2,
        -2,
        1
      ],
      [
        0,
        2,
        -3,
        1
      ]
    ],
    "all_rectangles": [
      {
        "rows": [
          0,
          0
        ],
        "cols": [
          0,
          0
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          0
        ],
        "cols": [
          0,
          1
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          0
        ],
        "cols": [
          0,
          2
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          0
        ],
        "cols": [
          0,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          0
        ],
        "cols": [
          1,
          1
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          0
        ],
        "cols": [
          1,
          2
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          0
        ],
        "cols": [
          1,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          0
        ],
        "cols": [
          2,
          2
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          0
        ],
        "cols": [
          2,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          0
        ],
        "cols": [
          3,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          1
        ],
        "cols": [
          0,
          0
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          1
        ],
        "cols": [
          0,
          1
        ],
        "sum": -2
      },
      {
        "rows": [
          0,
          1
        ],
        "cols": [
          0,
          2
        ],
        "sum": -2
      },
      {
        "rows": [
          0,
          1
        ],
        "cols": [
          0,
          3
        ],
        "sum": 1
      },
      {
        "rows": [
          0,
          1
        ],
        "cols": [
          1,
          1
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          1
        ],
        "cols": [
          1,
          2
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          1
        ],
        "cols": [
          1,
          3
        ],
        "sum": 3
      },
      {
        "rows": [
          0,
          1
        ],
        "cols": [
          2,
          2
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          1
        ],
        "cols": [
          2,
          3
        ],
        "sum": 3
      },
      {
        "rows": [
          0,
          1
        ],
        "cols": [
          3,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          2
        ],
        "cols": [
          0,
          0
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          2
        ],
        "cols": [
          0,
          1
        ],
        "sum": 2
      },
      {
        "rows": [
          0,
          2
        ],
        "cols": [
          0,
          2
        ],
        "sum": -3
      },
      {
        "rows": [
          0,
          2
        ],
        "cols": [
          0,
          3
        ],
        "sum": 1
      },
      {
        "rows": [
          0,
          2
        ],
        "cols": [
          1,
          1
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          2
        ],
        "cols": [
          1,
          2
        ],
        "sum": -5
      },
      {
        "rows": [
          0,
          2
        ],
        "cols": [
          1,
          3
        ],
        "sum": -1
      },
      {
        "rows": [
          0,
          2
        ],
        "cols": [
          2,
          2
        ],
        "sum": 0
      },
      {
        "rows": [
          0,
          2
        ],
        "cols": [
          2,
          3
        ],
        "sum": 4
      },
      {
        "rows": [
          0,
          2
        ],
        "cols": [
          3,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          1
        ],
        "cols": [
          0,
          0
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          1
        ],
        "cols": [
          0,
          1
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          1
        ],
        "cols": [
          0,
          2
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          1
        ],
        "cols": [
          0,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          1
        ],
        "cols": [
          1,
          1
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          1
        ],
        "cols": [
          1,
          2
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          1
        ],
        "cols": [
          1,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          1
        ],
        "cols": [
          2,
          2
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          1
        ],
        "cols": [
          2,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          1
        ],
        "cols": [
          3,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          2
        ],
        "cols": [
          0,
          0
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          2
        ],
        "cols": [
          0,
          1
        ],
        "sum": 4
      },
      {
        "rows": [
          1,
          2
        ],
        "cols": [
          0,
          2
        ],
        "sum": -1
      },
      {
        "rows": [
          1,
          2
        ],
        "cols": [
          0,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          2
        ],
        "cols": [
          1,
          1
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          2
        ],
        "cols": [
          1,
          2
        ],
        "sum": -5
      },
      {
        "rows": [
          1,
          2
        ],
        "cols": [
          1,
          3
        ],
        "sum": -4
      },
      {
        "rows": [
          1,
          2
        ],
        "cols": [
          2,
          2
        ],
        "sum": 0
      },
      {
        "rows": [
          1,
          2
        ],
        "cols": [
          2,
          3
        ],
        "sum": 1
      },
      {
        "rows": [
          1,
          2
        ],
        "cols": [
          3,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          2,
          2
        ],
        "cols": [
          0,
          0
        ],
        "sum": 0
      },
      {
        "rows": [
          2,
          2
        ],
        "cols": [
          0,
          1
        ],
        "sum": 0
      },
      {
        "rows": [
          2,
          2
        ],
        "cols": [
          0,
          2
        ],
        "sum": 0
      },
      {
        "rows": [
          2,
          2
        ],
        "cols": [
          0,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          2,
          2
        ],
        "cols": [
          1,
          1
        ],
        "sum": 0
      },
      {
        "rows": [
          2,
          2
        ],
        "cols": [
          1,
          2
        ],
        "sum": 0
      },
      {
        "rows": [
          2,
          2
        ],
        "cols": [
          1,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          2,
          2
        ],
        "cols": [
          2,
          2
        ],
        "sum": 0
      },
      {
        "rows": [
          2,
          2
        ],
        "cols": [
          2,
          3
        ],
        "sum": 0
      },
      {
        "rows": [
          2,
          2
        ],
        "cols": [
          3,
          3
        ],
        "sum": 0
      }
    ]
  },
  "changed_pixel_stale_counterexample": {
    "row": 1,
    "col": 1,
    "increment": 10,
    "stale": 54,
    "rebuild_fresh": 64
  },
  "invalid_cases_rejected": [
    "empty",
    "ragged",
    "bool_pixel",
    "float_pixel",
    "over_dimension",
    "reversed_rectangle",
    "out_of_bounds",
    "bool_boundary"
  ],
  "scope": "static Python-int grid with no fixed-width overflow; preprocessing O(HW) plus padded memory, four-lookup query distinct; no OpenCV/image/OCR/rotated integral/live update or benchmark claim"
}
