{
  "cycle_adjacent": {
    "laplacian": [
      [
        "2",
        "-1",
        "0",
        "-1"
      ],
      [
        "-1",
        "2",
        "-1",
        "0"
      ],
      [
        "0",
        "-1",
        "2",
        "-1"
      ],
      [
        "-1",
        "0",
        "-1",
        "2"
      ]
    ],
    "ground": "B",
    "voltages": {
      "A": "3/4",
      "B": "0",
      "C": "1/4",
      "D": "1/2"
    },
    "flows": [
      {
        "edge": [
          "A",
          "B"
        ],
        "conductance": 1,
        "current_a_to_b": "3/4"
      },
      {
        "edge": [
          "B",
          "C"
        ],
        "conductance": 1,
        "current_a_to_b": "-1/4"
      },
      {
        "edge": [
          "C",
          "D"
        ],
        "conductance": 1,
        "current_a_to_b": "-1/4"
      },
      {
        "edge": [
          "D",
          "A"
        ],
        "conductance": 1,
        "current_a_to_b": "-1/4"
      }
    ],
    "kcl": [
      "1",
      "-1",
      "0",
      "0"
    ],
    "effective_resistance": "3/4",
    "energy": "3/4"
  },
  "cycle_opposite": {
    "laplacian": [
      [
        "2",
        "-1",
        "0",
        "-1"
      ],
      [
        "-1",
        "2",
        "-1",
        "0"
      ],
      [
        "0",
        "-1",
        "2",
        "-1"
      ],
      [
        "-1",
        "0",
        "-1",
        "2"
      ]
    ],
    "ground": "C",
    "voltages": {
      "A": "1",
      "B": "1/2",
      "C": "0",
      "D": "1/2"
    },
    "flows": [
      {
        "edge": [
          "A",
          "B"
        ],
        "conductance": 1,
        "current_a_to_b": "1/2"
      },
      {
        "edge": [
          "B",
          "C"
        ],
        "conductance": 1,
        "current_a_to_b": "1/2"
      },
      {
        "edge": [
          "C",
          "D"
        ],
        "conductance": 1,
        "current_a_to_b": "-1/2"
      },
      {
        "edge": [
          "D",
          "A"
        ],
        "conductance": 1,
        "current_a_to_b": "-1/2"
      }
    ],
    "kcl": [
      "1",
      "0",
      "-1",
      "0"
    ],
    "effective_resistance": "1",
    "energy": "1"
  },
  "diagonal_added_adjacent": {
    "laplacian": [
      [
        "3",
        "-1",
        "-1",
        "-1"
      ],
      [
        "-1",
        "2",
        "-1",
        "0"
      ],
      [
        "-1",
        "-1",
        "3",
        "-1"
      ],
      [
        "-1",
        "0",
        "-1",
        "2"
      ]
    ],
    "ground": "B",
    "voltages": {
      "A": "5/8",
      "B": "0",
      "C": "3/8",
      "D": "1/2"
    },
    "flows": [
      {
        "edge": [
          "A",
          "B"
        ],
        "conductance": 1,
        "current_a_to_b": "5/8"
      },
      {
        "edge": [
          "B",
          "C"
        ],
        "conductance": 1,
        "current_a_to_b": "-3/8"
      },
      {
        "edge": [
          "C",
          "D"
        ],
        "conductance": 1,
        "current_a_to_b": "-1/8"
      },
      {
        "edge": [
          "D",
          "A"
        ],
        "conductance": 1,
        "current_a_to_b": "-1/8"
      },
      {
        "edge": [
          "A",
          "C"
        ],
        "conductance": 1,
        "current_a_to_b": "1/4"
      }
    ],
    "kcl": [
      "1",
      "-1",
      "0",
      "0"
    ],
    "effective_resistance": "5/8",
    "energy": "5/8"
  },
  "diagonal_added_opposite": {
    "laplacian": [
      [
        "3",
        "-1",
        "-1",
        "-1"
      ],
      [
        "-1",
        "2",
        "-1",
        "0"
      ],
      [
        "-1",
        "-1",
        "3",
        "-1"
      ],
      [
        "-1",
        "0",
        "-1",
        "2"
      ]
    ],
    "ground": "C",
    "voltages": {
      "A": "1/2",
      "B": "1/4",
      "C": "0",
      "D": "1/4"
    },
    "flows": [
      {
        "edge": [
          "A",
          "B"
        ],
        "conductance": 1,
        "current_a_to_b": "1/4"
      },
      {
        "edge": [
          "B",
          "C"
        ],
        "conductance": 1,
        "current_a_to_b": "1/4"
      },
      {
        "edge": [
          "C",
          "D"
        ],
        "conductance": 1,
        "current_a_to_b": "-1/4"
      },
      {
        "edge": [
          "D",
          "A"
        ],
        "conductance": 1,
        "current_a_to_b": "-1/4"
      },
      {
        "edge": [
          "A",
          "C"
        ],
        "conductance": 1,
        "current_a_to_b": "1/2"
      }
    ],
    "kcl": [
      "1",
      "0",
      "-1",
      "0"
    ],
    "effective_resistance": "1/2",
    "energy": "1/2"
  },
  "sixteen_ground_gauge_checks": [
    {
      "destination": "B",
      "ground": "A",
      "resistance": "3/4"
    },
    {
      "destination": "B",
      "ground": "B",
      "resistance": "3/4"
    },
    {
      "destination": "B",
      "ground": "C",
      "resistance": "3/4"
    },
    {
      "destination": "B",
      "ground": "D",
      "resistance": "3/4"
    },
    {
      "destination": "C",
      "ground": "A",
      "resistance": "1"
    },
    {
      "destination": "C",
      "ground": "B",
      "resistance": "1"
    },
    {
      "destination": "C",
      "ground": "C",
      "resistance": "1"
    },
    {
      "destination": "C",
      "ground": "D",
      "resistance": "1"
    },
    {
      "destination": "B",
      "ground": "A",
      "resistance": "5/8"
    },
    {
      "destination": "B",
      "ground": "B",
      "resistance": "5/8"
    },
    {
      "destination": "B",
      "ground": "C",
      "resistance": "5/8"
    },
    {
      "destination": "B",
      "ground": "D",
      "resistance": "5/8"
    },
    {
      "destination": "C",
      "ground": "A",
      "resistance": "1/2"
    },
    {
      "destination": "C",
      "ground": "B",
      "resistance": "1/2"
    },
    {
      "destination": "C",
      "ground": "C",
      "resistance": "1/2"
    },
    {
      "destination": "C",
      "ground": "D",
      "resistance": "1/2"
    }
  ],
  "shortest_unit_edge_path_lengths": {
    "AB": 1,
    "AC_cycle": 2,
    "AC_diagonal": 1
  },
  "invalid_cases_rejected": [
    "disconnected",
    "nonpositive",
    "unknown",
    "selfloop",
    "parallel",
    "bad_edge_tuple",
    "budget"
  ],
  "scope": "Exact Fraction Gaussian grounded solve plus separate series/parallel references, KCL and energy. Whole local graph connected positive simple undirected. Full L singular; no ordinary inverse, no real circuit/safety/Opera routing claim."
}