{
  "python_version": "3.14.7",
  "method": "Exact equivalence-relation enumeration; no external dependencies",
  "two_view_relations": {
    "cases": 327,
    "universal_exists": 9,
    "finest_quotient_count_histogram": {
      "1": 9,
      "2": 66,
      "3": 66,
      "4": 99,
      "5": 36,
      "6": 45,
      "8": 6
    },
    "axis_sizes": "1..3 each",
    "individually_surjective": true
  },
  "three_view_relations": {
    "cases": 193,
    "universal_exists": 1,
    "finest_quotient_count_histogram": {
      "1": 1,
      "2": 90,
      "3": 102
    },
    "axis_sizes": [
      2,
      2,
      2
    ],
    "individually_surjective": true
  },
  "repeated_fiber_full_search": {
    "cases": 194,
    "universal_exists": 25,
    "finest_quotient_count_histogram": {
      "1": 25,
      "2": 133,
      "3": 36
    },
    "source_sizes": "1..4",
    "axis_sizes": "1..3 each",
    "congruence_tuples_examined": 36042,
    "coverage": "Every joint-map multiset, up to permutation of source states, with surjective endpoint maps"
  },
  "arrow_full_search": {
    "cases": 121,
    "universal_exists": 51,
    "finest_quotient_count_histogram": {
      "1": 51,
      "2": 18,
      "3": 16,
      "4": 36
    },
    "source_sizes": "1..4",
    "target_sizes": "1..4",
    "congruence_pairs_examined": 11414,
    "coverage": "Every map up to permutation of source states; identity transitions"
  },
  "protected_overlap_full_search": {
    "cases": 49,
    "universal_exists": 1,
    "finest_quotient_count_histogram": {
      "1": 1,
      "2": 12,
      "4": 36
    },
    "axis_sizes": [
      4,
      4
    ],
    "base_size": 2,
    "base_maps": [
      [
        0,
        0,
        1,
        1
      ],
      [
        0,
        0,
        1,
        1
      ]
    ],
    "congruence_tuples_examined": 257728,
    "coverage": "Every full-projection subset of the eight-state fiber product; every endpoint partition refining the protected base"
  },
  "protected_overlap_dynamic_examples": [
    {
      "name": "diagonal_fibers_base_swap",
      "rows": [
        [
          0,
          0
        ],
        [
          1,
          1
        ],
        [
          2,
          2
        ],
        [
          3,
          3
        ]
      ],
      "base_maps": [
        [
          0,
          0,
          1,
          1
        ],
        [
          0,
          0,
          1,
          1
        ]
      ],
      "base_transition": [
        1,
        0
      ],
      "transitions": [
        [
          2,
          3,
          0,
          1
        ],
        [
          2,
          3,
          0,
          1
        ],
        [
          2,
          3,
          0,
          1
        ]
      ],
      "feasible_quotients": 3,
      "finest_quotients": 2,
      "universal_quotient_exists": false,
      "minimal_kernels": [
        [
          [
            0,
            1,
            2,
            3
          ],
          [
            0,
            0,
            1,
            1
          ],
          [
            0,
            1,
            2,
            3
          ]
        ],
        [
          [
            0,
            1,
            2,
            3
          ],
          [
            0,
            1,
            2,
            3
          ],
          [
            0,
            0,
            1,
            1
          ]
        ]
      ]
    },
    {
      "name": "full_fiber_product_base_swap",
      "rows": [
        [
          0,
          0
        ],
        [
          0,
          1
        ],
        [
          1,
          0
        ],
        [
          1,
          1
        ],
        [
          2,
          2
        ],
        [
          2,
          3
        ],
        [
          3,
          2
        ],
        [
          3,
          3
        ]
      ],
      "base_maps": [
        [
          0,
          0,
          1,
          1
        ],
        [
          0,
          0,
          1,
          1
        ]
      ],
      "base_transition": [
        1,
        0
      ],
      "transitions": [
        [
          4,
          5,
          6,
          7,
          0,
          1,
          2,
          3
        ],
        [
          2,
          3,
          0,
          1
        ],
        [
          2,
          3,
          0,
          1
        ]
      ],
      "feasible_quotients": 4,
      "finest_quotients": 1,
      "universal_quotient_exists": true,
      "minimal_kernels": [
        [
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            0,
            1,
            2,
            3
          ],
          [
            0,
            1,
            2,
            3
          ]
        ]
      ]
    }
  ],
  "nontrivial_process_examples": [
    {
      "axis_sizes": [
        2,
        2
      ],
      "source_states": 2,
      "distinct_joint_images": 2,
      "joint_surjective": false,
      "endpoint_surjective": true,
      "feasible_quotients": 3,
      "finest_quotients": 2,
      "universal_quotient_exists": false,
      "intersection_feasible": false,
      "minimal_kernels": [
        [
          [
            0,
            1
          ],
          [
            0,
            0
          ],
          [
            0,
            1
          ]
        ],
        [
          [
            0,
            1
          ],
          [
            0,
            1
          ],
          [
            0,
            0
          ]
        ]
      ],
      "intersection_kernel": [
        [
          0,
          1
        ],
        [
          0,
          1
        ],
        [
          0,
          1
        ]
      ],
      "name": "diagonal_span_bit_flip",
      "rows": [
        [
          0,
          0
        ],
        [
          1,
          1
        ]
      ],
      "transitions": [
        [
          1,
          0
        ],
        [
          1,
          0
        ],
        [
          1,
          0
        ]
      ],
      "all_equivalence_tuples_examined": 8,
      "process_quotient_kernels": [
        [
          [
            0,
            0
          ],
          [
            0,
            0
          ],
          [
            0,
            0
          ]
        ],
        [
          [
            0,
            1
          ],
          [
            0,
            0
          ],
          [
            0,
            1
          ]
        ],
        [
          [
            0,
            1
          ],
          [
            0,
            1
          ],
          [
            0,
            0
          ]
        ]
      ],
      "observations": "None; all transition systems have one common input."
    },
    {
      "axis_sizes": [
        2,
        2
      ],
      "source_states": 3,
      "distinct_joint_images": 3,
      "joint_surjective": false,
      "endpoint_surjective": true,
      "feasible_quotients": 3,
      "finest_quotients": 2,
      "universal_quotient_exists": false,
      "intersection_feasible": false,
      "minimal_kernels": [
        [
          [
            0,
            0,
            1
          ],
          [
            0,
            1
          ],
          [
            0,
            0
          ]
        ],
        [
          [
            0,
            1,
            0
          ],
          [
            0,
            0
          ],
          [
            0,
            1
          ]
        ]
      ],
      "intersection_kernel": [
        [
          0,
          1,
          2
        ],
        [
          0,
          1
        ],
        [
          0,
          1
        ]
      ],
      "name": "connected_three_corner_reset",
      "rows": [
        [
          0,
          0
        ],
        [
          0,
          1
        ],
        [
          1,
          0
        ]
      ],
      "transitions": [
        [
          0,
          0,
          0
        ],
        [
          0,
          0
        ],
        [
          0,
          0
        ]
      ],
      "all_equivalence_tuples_examined": 20,
      "process_quotient_kernels": [
        [
          [
            0,
            0,
            0
          ],
          [
            0,
            0
          ],
          [
            0,
            0
          ]
        ],
        [
          [
            0,
            0,
            1
          ],
          [
            0,
            1
          ],
          [
            0,
            0
          ]
        ],
        [
          [
            0,
            1,
            0
          ],
          [
            0,
            0
          ],
          [
            0,
            1
          ]
        ]
      ],
      "observations": "None; all transition systems have one common input."
    },
    {
      "axis_sizes": [
        3
      ],
      "source_states": 2,
      "distinct_joint_images": 2,
      "joint_surjective": false,
      "endpoint_surjective": false,
      "feasible_quotients": 3,
      "finest_quotients": 2,
      "universal_quotient_exists": false,
      "intersection_feasible": false,
      "minimal_kernels": [
        [
          [
            0,
            1
          ],
          [
            0,
            1,
            0
          ]
        ],
        [
          [
            0,
            1
          ],
          [
            0,
            1,
            1
          ]
        ]
      ],
      "intersection_kernel": [
        [
          0,
          1
        ],
        [
          0,
          1,
          2
        ]
      ],
      "name": "arrow_identity_transition_counterexample",
      "rows": [
        [
          0
        ],
        [
          1
        ]
      ],
      "transitions": [
        [
          0,
          1
        ],
        [
          0,
          1,
          2
        ]
      ],
      "all_equivalence_tuples_examined": 10,
      "process_quotient_kernels": [
        [
          [
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ],
        [
          [
            0,
            1
          ],
          [
            0,
            1,
            0
          ]
        ],
        [
          [
            0,
            1
          ],
          [
            0,
            1,
            1
          ]
        ]
      ],
      "observations": "None; all transition systems have one common input."
    },
    {
      "axis_sizes": [
        3
      ],
      "source_states": 2,
      "distinct_joint_images": 2,
      "joint_surjective": false,
      "endpoint_surjective": false,
      "feasible_quotients": 2,
      "finest_quotients": 1,
      "universal_quotient_exists": true,
      "intersection_feasible": true,
      "minimal_kernels": [
        [
          [
            0,
            1
          ],
          [
            0,
            1,
            0
          ]
        ]
      ],
      "intersection_kernel": [
        [
          0,
          1
        ],
        [
          0,
          1,
          0
        ]
      ],
      "name": "arrow_dynamics_force_unique_attachment",
      "rows": [
        [
          0
        ],
        [
          1
        ]
      ],
      "transitions": [
        [
          0,
          1
        ],
        [
          0,
          1,
          0
        ]
      ],
      "all_equivalence_tuples_examined": 10,
      "process_quotient_kernels": [
        [
          [
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ],
        [
          [
            0,
            1
          ],
          [
            0,
            1,
            0
          ]
        ]
      ],
      "observations": "None; all transition systems have one common input."
    }
  ],
  "minimal_counterexamples_in_enumerated_families": {
    "arrow_total_states": 5,
    "surjective_span_total_states": 6,
    "connected_surjective_span_total_states": 7
  },
  "all_assertions_pass": true,
  "limits": [
    "Finite exhaustive evidence supports but does not replace the general proofs.",
    "Only constant observations or those factoring through the specified protected overlap are covered; extra restrictions are not covered.",
    "The walking-arrow criterion is asserted only with identity transitions."
  ]
}