{
  "scope": "Six finite controller cases; binary-depth family n=1..8; all lift edges and reachability; bounded capacity enumeration as specified. Not an infinite-source proof.",
  "cases": [
    {
      "name": "fork",
      "forward_residual_states": 4,
      "residual_blocks": [
        [
          "p"
        ],
        [
          "q"
        ],
        [
          "r"
        ],
        [
          "s"
        ]
      ],
      "possible": true,
      "weights": {
        "3": 1,
        "0": 1,
        "1": 1,
        "2": 2
      },
      "states": 5,
      "arrows": 4,
      "bounded_capacity_search": {
        "coordinate_range": [
          1,
          3
        ],
        "vectors_checked": 81,
        "feasible_vectors": 4,
        "scope": "corroboration within this box; not a proof outside it"
      }
    },
    {
      "name": "merge_then_different_letter_loop",
      "forward_residual_states": 4,
      "residual_blocks": [
        [
          "p"
        ],
        [
          "q"
        ],
        [
          "r"
        ],
        [
          "s"
        ]
      ],
      "possible": true,
      "weights": {
        "3": 1,
        "0": 1,
        "1": 1,
        "2": 2
      },
      "states": 5,
      "arrows": 6,
      "bounded_capacity_search": {
        "coordinate_range": [
          1,
          3
        ],
        "vectors_checked": 81,
        "feasible_vectors": 4,
        "scope": "corroboration within this box; not a proof outside it"
      }
    },
    {
      "name": "merge_into_same_letter_loop",
      "forward_residual_states": 4,
      "residual_blocks": [
        [
          "p"
        ],
        [
          "q"
        ],
        [
          "r"
        ],
        [
          "s"
        ]
      ],
      "possible": false,
      "witness": {
        "other_source": "0",
        "cyclic_source": "2",
        "letter": "a",
        "target": "2"
      },
      "bounded_capacity_search": {
        "coordinate_range": [
          1,
          3
        ],
        "vectors_checked": 81,
        "feasible_vectors": 0,
        "scope": "corroboration within this box; not a proof outside it"
      }
    },
    {
      "name": "merge_into_two_state_component",
      "forward_residual_states": 5,
      "residual_blocks": [
        [
          "p"
        ],
        [
          "q"
        ],
        [
          "r"
        ],
        [
          "s"
        ],
        [
          "t"
        ]
      ],
      "possible": true,
      "weights": {
        "3": 1,
        "0": 1,
        "1": 1,
        "4": 2,
        "2": 2
      },
      "states": 7,
      "arrows": 8,
      "bounded_capacity_search": {
        "coordinate_range": [
          1,
          3
        ],
        "vectors_checked": 243,
        "feasible_vectors": 4,
        "scope": "corroboration within this box; not a proof outside it"
      }
    },
    {
      "name": "parity_with_legal_pop_promise",
      "forward_residual_states": 2,
      "residual_blocks": [
        [
          "s"
        ],
        [
          "t"
        ]
      ],
      "possible": true,
      "weights": {
        "1": 1,
        "0": 1
      },
      "states": 2,
      "arrows": 2,
      "bounded_capacity_search": {
        "coordinate_range": [
          1,
          3
        ],
        "vectors_checked": 9,
        "feasible_vectors": 3,
        "scope": "corroboration within this box; not a proof outside it"
      }
    },
    {
      "name": "a_star_b_star_constant_output",
      "forward_residual_states": 2,
      "residual_blocks": [
        [
          "s"
        ],
        [
          "t"
        ]
      ],
      "possible": false,
      "witness": {
        "other_source": "0",
        "cyclic_source": "1",
        "letter": "b",
        "target": "1"
      },
      "bounded_capacity_search": {
        "coordinate_range": [
          1,
          6
        ],
        "vectors_checked": 36,
        "feasible_vectors": 0,
        "scope": "corroboration within this box; not a proof outside it"
      }
    }
  ],
  "binary_depth_family": [
    {
      "depth": 1,
      "forward_states": 3,
      "restoration_states": 3,
      "all_legal_histories_checked": 3,
      "all_restoration_edges_checked": 2
    },
    {
      "depth": 2,
      "forward_states": 5,
      "restoration_states": 7,
      "all_legal_histories_checked": 7,
      "all_restoration_edges_checked": 6
    },
    {
      "depth": 3,
      "forward_states": 7,
      "restoration_states": 15,
      "all_legal_histories_checked": 15,
      "all_restoration_edges_checked": 14
    },
    {
      "depth": 4,
      "forward_states": 9,
      "restoration_states": 31,
      "all_legal_histories_checked": 31,
      "all_restoration_edges_checked": 30
    },
    {
      "depth": 5,
      "forward_states": 11,
      "restoration_states": 63,
      "all_legal_histories_checked": 63,
      "all_restoration_edges_checked": 62
    },
    {
      "depth": 6,
      "forward_states": 13,
      "restoration_states": 127,
      "all_legal_histories_checked": 127,
      "all_restoration_edges_checked": 126
    },
    {
      "depth": 7,
      "forward_states": 15,
      "restoration_states": 255,
      "all_legal_histories_checked": 255,
      "all_restoration_edges_checked": 254
    },
    {
      "depth": 8,
      "forward_states": 17,
      "restoration_states": 511,
      "all_legal_histories_checked": 511,
      "all_restoration_edges_checked": 510
    }
  ],
  "forced_merge_mutation_rejected": true
}