[
  {
    "name": "complete2_mm_-1,-1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 30500,
      "exhaustive_oracle": 28500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1077584,
      "exhaustive_oracle": 23917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 273917,
      "exhaustive_oracle": 22084
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 4
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13292,
      "exhaustive_oracle": 20959
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1048708,
      "exhaustive_oracle": 21834
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 271542,
      "exhaustive_oracle": 22250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6140,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1023,
        "edge_scans": 4092,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1049625,
      "exhaustive_oracle": 22667
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6140,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1023,
        "edge_scans": 4092,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1046208,
      "exhaustive_oracle": 20750
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 269458,
      "exhaustive_oracle": 21250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 4
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12791,
      "exhaustive_oracle": 20625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1044834,
      "exhaustive_oracle": 21292
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 271334,
      "exhaustive_oracle": 20542
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1037791,
      "exhaustive_oracle": 21666
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1143916,
      "exhaustive_oracle": 45708
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 298833,
      "exhaustive_oracle": 24000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 427000,
      "exhaustive_oracle": 21875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 432333,
      "exhaustive_oracle": 22834
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 271500,
      "exhaustive_oracle": 20833
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6140,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1023,
        "edge_scans": 4092,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1051125,
      "exhaustive_oracle": 21000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1091708,
      "exhaustive_oracle": 20666
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 321791,
      "exhaustive_oracle": 79250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 512375,
      "exhaustive_oracle": 23500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 433125,
      "exhaustive_oracle": 21291
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 272292,
      "exhaustive_oracle": 21209
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 269458,
      "exhaustive_oracle": 20334
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 269208,
      "exhaustive_oracle": 20833
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 271833,
      "exhaustive_oracle": 21084
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1046209,
      "exhaustive_oracle": 21959
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1074625,
      "exhaustive_oracle": 23458
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1562,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 260,
        "edge_scans": 1040,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 274084,
      "exhaustive_oracle": 21042
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1036500,
      "exhaustive_oracle": 20666
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1032584,
      "exhaustive_oracle": 29167
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1562,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 260,
        "edge_scans": 1040,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 271250,
      "exhaustive_oracle": 24792
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1060958,
      "exhaustive_oracle": 19375
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1041542,
      "exhaustive_oracle": 19709
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1562,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 260,
        "edge_scans": 1040,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 270416,
      "exhaustive_oracle": 20875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1051958,
      "exhaustive_oracle": 19959
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1041583,
      "exhaustive_oracle": 20000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 292875,
      "exhaustive_oracle": 20417
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1093583,
      "exhaustive_oracle": 18959
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "512",
      "transformed_max_weight": "1",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 3080,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 513,
        "edge_scans": 2052,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 524583,
      "exhaustive_oracle": 18875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 274208,
      "exhaustive_oracle": 22458
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 441584,
      "exhaustive_oracle": 22875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 428584,
      "exhaustive_oracle": 21375
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 282291,
      "exhaustive_oracle": 23750
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6140,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1023,
        "edge_scans": 4092,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1047667,
      "exhaustive_oracle": 19292
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1061209,
      "exhaustive_oracle": 19000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 280166,
      "exhaustive_oracle": 20500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 438250,
      "exhaustive_oracle": 31333
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 444333,
      "exhaustive_oracle": 20792
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 278333,
      "exhaustive_oracle": 20750
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 277042,
      "exhaustive_oracle": 19917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 286916,
      "exhaustive_oracle": 21875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 282167,
      "exhaustive_oracle": 20000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 267584,
      "exhaustive_oracle": 20625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1562,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 260,
        "edge_scans": 1040,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 283750,
      "exhaustive_oracle": 19500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 265833,
      "exhaustive_oracle": 19666
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 278500,
      "exhaustive_oracle": 19709
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 276208,
      "exhaustive_oracle": 21625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 295583,
      "exhaustive_oracle": 19750
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 281875,
      "exhaustive_oracle": 18959
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 272500,
      "exhaustive_oracle": 21750
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 273541,
      "exhaustive_oracle": 19292
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 274083,
      "exhaustive_oracle": 26709
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1562,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 260,
        "edge_scans": 1040,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 302875,
      "exhaustive_oracle": 22875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 298917,
      "exhaustive_oracle": 21459
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 324833,
      "exhaustive_oracle": 26500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 305709,
      "exhaustive_oracle": 20792
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 304542,
      "exhaustive_oracle": 33042
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 377084,
      "exhaustive_oracle": 25917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 318709,
      "exhaustive_oracle": 26042
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 270167,
      "exhaustive_oracle": 19667
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 289541,
      "exhaustive_oracle": 27917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1562,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 260,
        "edge_scans": 1040,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 276417,
      "exhaustive_oracle": 19125
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 291542,
      "exhaustive_oracle": 27291
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 269667,
      "exhaustive_oracle": 19000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 270541,
      "exhaustive_oracle": 20375
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 270250,
      "exhaustive_oracle": 20792
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 272708,
      "exhaustive_oracle": 19958
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 267459,
      "exhaustive_oracle": 20000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0,
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 302166,
      "exhaustive_oracle": 35209
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 20333,
      "exhaustive_oracle": 24542
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 14083,
      "exhaustive_oracle": 20583
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12000,
      "exhaustive_oracle": 35708
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11750,
      "exhaustive_oracle": 23167
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12875,
      "exhaustive_oracle": 21292
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12875,
      "exhaustive_oracle": 19708
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11208,
      "exhaustive_oracle": 21959
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1075292,
      "exhaustive_oracle": 21167
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6140,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1023,
        "edge_scans": 4092,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1075792,
      "exhaustive_oracle": 22542
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 14458,
      "exhaustive_oracle": 22666
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12167,
      "exhaustive_oracle": 20042
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12625,
      "exhaustive_oracle": 19291
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13792,
      "exhaustive_oracle": 21583
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1146042,
      "exhaustive_oracle": 19834
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1074167,
      "exhaustive_oracle": 19625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12417,
      "exhaustive_oracle": 20292
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1101958,
      "exhaustive_oracle": 21000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 427084,
      "exhaustive_oracle": 19625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-4"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 15041,
      "exhaustive_oracle": 21458
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6146,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1024,
        "edge_scans": 4096,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1047834,
      "exhaustive_oracle": 31208
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6140,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1023,
        "edge_scans": 4092,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1044708,
      "exhaustive_oracle": 20000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-4"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12208,
      "exhaustive_oracle": 19583
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1074875,
      "exhaustive_oracle": 20333
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 428791,
      "exhaustive_oracle": 18708
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-4"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12625,
      "exhaustive_oracle": 21209
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1037708,
      "exhaustive_oracle": 19500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 269083,
      "exhaustive_oracle": 18500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1036875,
      "exhaustive_oracle": 19375
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1079125,
      "exhaustive_oracle": 18333
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1073875,
      "exhaustive_oracle": 18083
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1095875,
      "exhaustive_oracle": 19917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1152125,
      "exhaustive_oracle": 19458
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1081917,
      "exhaustive_oracle": 18542
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1080833,
      "exhaustive_oracle": 20917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1083375,
      "exhaustive_oracle": 20250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1099750,
      "exhaustive_oracle": 20459
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1083125,
      "exhaustive_oracle": 30917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1096208,
      "exhaustive_oracle": 20125
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1098875,
      "exhaustive_oracle": 20791
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1112042,
      "exhaustive_oracle": 19500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "512",
      "transformed_max_weight": "1",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 3080,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 513,
        "edge_scans": 2052,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 541125,
      "exhaustive_oracle": 19667
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1104875,
      "exhaustive_oracle": 20375
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1100250,
      "exhaustive_oracle": 21875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1097416,
      "exhaustive_oracle": 20083
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 466167,
      "exhaustive_oracle": 20666
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 6134,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1022,
        "edge_scans": 4088,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1062292,
      "exhaustive_oracle": 20500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6146,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1024,
        "edge_scans": 4096,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1083667,
      "exhaustive_oracle": 18500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6140,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1023,
        "edge_scans": 4092,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1088291,
      "exhaustive_oracle": 26125
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 6134,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1022,
        "edge_scans": 4088,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1046125,
      "exhaustive_oracle": 20916
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1062417,
      "exhaustive_oracle": 20667
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 425000,
      "exhaustive_oracle": 19458
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 6134,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1022,
        "edge_scans": 4088,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1027500,
      "exhaustive_oracle": 19625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1068542,
      "exhaustive_oracle": 20167
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 1
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 278917,
      "exhaustive_oracle": 29541
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 289666,
      "exhaustive_oracle": 20167
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6146,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1024,
        "edge_scans": 4096,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1047083,
      "exhaustive_oracle": 19041
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 283500,
      "exhaustive_oracle": 17916
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 271625,
      "exhaustive_oracle": 19250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1045959,
      "exhaustive_oracle": 20834
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 268625,
      "exhaustive_oracle": 18083
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 267333,
      "exhaustive_oracle": 18917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1086459,
      "exhaustive_oracle": 20792
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 278875,
      "exhaustive_oracle": 18583
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 267500,
      "exhaustive_oracle": 18750
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6146,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1024,
        "edge_scans": 4096,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1088834,
      "exhaustive_oracle": 22125
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 279375,
      "exhaustive_oracle": 20583
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 279791,
      "exhaustive_oracle": 25666
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1071541,
      "exhaustive_oracle": 27250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 293167,
      "exhaustive_oracle": 18584
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 272625,
      "exhaustive_oracle": 20708
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1076667,
      "exhaustive_oracle": 32500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 286459,
      "exhaustive_oracle": 21042
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 319916,
      "exhaustive_oracle": 26375
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6146,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1024,
        "edge_scans": 4096,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1130292,
      "exhaustive_oracle": 28583
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 275750,
      "exhaustive_oracle": 20875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 270334,
      "exhaustive_oracle": 20208
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1049125,
      "exhaustive_oracle": 20250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 267583,
      "exhaustive_oracle": 18417
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "-1"
      ],
      "winning_vertices": [
        0
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 263375,
      "exhaustive_oracle": 19750
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1039250,
      "exhaustive_oracle": 19667
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "max",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "0": 0
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 272125,
      "exhaustive_oracle": 18250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11250,
      "exhaustive_oracle": 18000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "0"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1035166,
      "exhaustive_oracle": 26291
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "1"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 273083,
      "exhaustive_oracle": 19583
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12459,
      "exhaustive_oracle": 18334
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "0"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1018416,
      "exhaustive_oracle": 18791
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "1"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 263916,
      "exhaustive_oracle": 18625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-4",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12292,
      "exhaustive_oracle": 249417
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 6134,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1022,
        "edge_scans": 4088,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "0"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1028208,
      "exhaustive_oracle": 20625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "1"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 273500,
      "exhaustive_oracle": 19083
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11542,
      "exhaustive_oracle": 18250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "0"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1060459,
      "exhaustive_oracle": 19917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "1"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 264541,
      "exhaustive_oracle": 18875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 15750,
      "exhaustive_oracle": 17833
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "0"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1076000,
      "exhaustive_oracle": 19500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "1"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 268250,
      "exhaustive_oracle": 20125
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-4",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13417,
      "exhaustive_oracle": 19375
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 6134,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1022,
        "edge_scans": 4088,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "0"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1080959,
      "exhaustive_oracle": 21667
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "1"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 271291,
      "exhaustive_oracle": 18583
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11083,
      "exhaustive_oracle": 18416
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "0"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1073750,
      "exhaustive_oracle": 18958
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "1"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 270500,
      "exhaustive_oracle": 21500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13375,
      "exhaustive_oracle": 20708
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "0"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1093334,
      "exhaustive_oracle": 22000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "1"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 306833,
      "exhaustive_oracle": 31625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-4",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12667,
      "exhaustive_oracle": 18625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 6134,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1022,
        "edge_scans": 4088,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "0"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1070166,
      "exhaustive_oracle": 21250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "1"
      ],
      "winning_vertices": [
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {
        "0": 0
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            1
          ],
          "distance_potential": {
            "1": "0"
          },
          "retained_edges": 1
        },
        {
          "player": "min",
          "vertices": [
            0
          ],
          "distance_potential": {
            "0": "0"
          },
          "retained_edges": 1
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 280834,
      "exhaustive_oracle": 23208
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11208,
      "exhaustive_oracle": 18166
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1094708,
      "exhaustive_oracle": 20709
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6146,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1024,
        "edge_scans": 4096,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1105917,
      "exhaustive_oracle": 21917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12542,
      "exhaustive_oracle": 19250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1096000,
      "exhaustive_oracle": 22000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6146,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1024,
        "edge_scans": 4096,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1096458,
      "exhaustive_oracle": 22541
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6146,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1024,
        "edge_scans": 4096,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1113250,
      "exhaustive_oracle": 21042
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6146,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1024,
        "edge_scans": 4096,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1091875,
      "exhaustive_oracle": 18083
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6146,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1024,
        "edge_scans": 4096,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1092167,
      "exhaustive_oracle": 19166
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12083,
      "exhaustive_oracle": 18833
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1113417,
      "exhaustive_oracle": 20209
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1110084,
      "exhaustive_oracle": 21375
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1096625,
      "exhaustive_oracle": 20583
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "512",
      "transformed_max_weight": "1",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 3080,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 513,
        "edge_scans": 2052,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 573834,
      "exhaustive_oracle": 29417
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1102833,
      "exhaustive_oracle": 22500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1099250,
      "exhaustive_oracle": 19875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1157000,
      "exhaustive_oracle": 28083
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1058500,
      "exhaustive_oracle": 25167
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1042542,
      "exhaustive_oracle": 19250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1058958,
      "exhaustive_oracle": 19083
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1052083,
      "exhaustive_oracle": 19458
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1031250,
      "exhaustive_oracle": 18042
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1032083,
      "exhaustive_oracle": 18833
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1032750,
      "exhaustive_oracle": 18667
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1070583,
      "exhaustive_oracle": 19667
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1059208,
      "exhaustive_oracle": 20500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1064416,
      "exhaustive_oracle": 19541
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11334,
      "exhaustive_oracle": 22250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1067916,
      "exhaustive_oracle": 19458
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 273667,
      "exhaustive_oracle": 21500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13917,
      "exhaustive_oracle": 29208
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1065542,
      "exhaustive_oracle": 23250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 287958,
      "exhaustive_oracle": 19625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6140,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1023,
        "edge_scans": 4092,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1074959,
      "exhaustive_oracle": 18417
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6140,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1023,
        "edge_scans": 4092,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1071083,
      "exhaustive_oracle": 18417
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 283708,
      "exhaustive_oracle": 24791
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 3
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 16334,
      "exhaustive_oracle": 20250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1070333,
      "exhaustive_oracle": 19875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 289834,
      "exhaustive_oracle": 20458
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1081583,
      "exhaustive_oracle": 18666
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1085584,
      "exhaustive_oracle": 18209
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 273625,
      "exhaustive_oracle": 21584
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 458458,
      "exhaustive_oracle": 18666
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 425291,
      "exhaustive_oracle": 20208
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1550,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 258,
        "edge_scans": 1032,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 287791,
      "exhaustive_oracle": 21416
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6140,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1023,
        "edge_scans": 4092,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1102667,
      "exhaustive_oracle": 20167
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1086375,
      "exhaustive_oracle": 18792
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 264791,
      "exhaustive_oracle": 27291
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 449250,
      "exhaustive_oracle": 20667
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 437125,
      "exhaustive_oracle": 18208
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 3
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 277917,
      "exhaustive_oracle": 18917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 286333,
      "exhaustive_oracle": 20750
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 270042,
      "exhaustive_oracle": 20167
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "max"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {
        "1": 2
      },
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 3
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 286125,
      "exhaustive_oracle": 20708
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12375,
      "exhaustive_oracle": 26666
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11625,
      "exhaustive_oracle": 18792
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11083,
      "exhaustive_oracle": 19375
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 10750,
      "exhaustive_oracle": 18417
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12625,
      "exhaustive_oracle": 20125
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13833,
      "exhaustive_oracle": 20750
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-4",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13250,
      "exhaustive_oracle": 20584
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 32,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 5,
        "edge_scans": 20,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-4",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 16208,
      "exhaustive_oracle": 20416
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,-1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-4",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13792,
      "exhaustive_oracle": 20125
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11458,
      "exhaustive_oracle": 19917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 10250,
      "exhaustive_oracle": 17583
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 19834,
      "exhaustive_oracle": 20250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 10583,
      "exhaustive_oracle": 17833
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11958,
      "exhaustive_oracle": 17875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11709,
      "exhaustive_oracle": 18709
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-4",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 10209,
      "exhaustive_oracle": 18375
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 32,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 5,
        "edge_scans": 20,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-4",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 14250,
      "exhaustive_oracle": 18208
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,0,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-4",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11709,
      "exhaustive_oracle": 18959
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 10750,
      "exhaustive_oracle": 19167
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 10458,
      "exhaustive_oracle": 19041
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 10417,
      "exhaustive_oracle": 18833
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 10333,
      "exhaustive_oracle": 18292
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11334,
      "exhaustive_oracle": 19458
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11584,
      "exhaustive_oracle": 18000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-4",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11500,
      "exhaustive_oracle": 17916
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 32,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 5,
        "edge_scans": 20,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-4",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 14667,
      "exhaustive_oracle": 18042
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_-1,1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          -1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 0,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-4",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11459,
      "exhaustive_oracle": 17542
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 10584,
      "exhaustive_oracle": 18458
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12416,
      "exhaustive_oracle": 26750
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11208,
      "exhaustive_oracle": 23625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 10833,
      "exhaustive_oracle": 19500
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13042,
      "exhaustive_oracle": 19625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11709,
      "exhaustive_oracle": 24542
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 10500,
      "exhaustive_oracle": 19542
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1087625,
      "exhaustive_oracle": 19166
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,-1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6146,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1024,
        "edge_scans": 4096,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1084709,
      "exhaustive_oracle": 18542
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12375,
      "exhaustive_oracle": 19917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11375,
      "exhaustive_oracle": 18542
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11875,
      "exhaustive_oracle": 18541
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "2",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11625,
      "exhaustive_oracle": 20875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "512",
      "transformed_max_weight": "1",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 3080,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 513,
        "edge_scans": 2052,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 554250,
      "exhaustive_oracle": 20333
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1082875,
      "exhaustive_oracle": 20042
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12250,
      "exhaustive_oracle": 19375
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1099209,
      "exhaustive_oracle": 20250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,0,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1125750,
      "exhaustive_oracle": 20416
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 32,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 5,
        "edge_scans": 20,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-4"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 17125,
      "exhaustive_oracle": 26875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1094958,
      "exhaustive_oracle": 19000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1078667,
      "exhaustive_oracle": 19292
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 32,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 5,
        "edge_scans": 20,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-4"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 15083,
      "exhaustive_oracle": 19334
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1081292,
      "exhaustive_oracle": 21667
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1124250,
      "exhaustive_oracle": 20250
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 32,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 5,
        "edge_scans": 20,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-4"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 15917,
      "exhaustive_oracle": 19917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1086291,
      "exhaustive_oracle": 19333
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_0,1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1088291,
      "exhaustive_oracle": 18208
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11084,
      "exhaustive_oracle": 20000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11500,
      "exhaustive_oracle": 19166
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 12667,
      "exhaustive_oracle": 20791
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 10000,
      "exhaustive_oracle": 18875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11916,
      "exhaustive_oracle": 22583
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13000,
      "exhaustive_oracle": 21166
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 8,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1,
        "edge_scans": 4,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11708,
      "exhaustive_oracle": 21667
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6158,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1026,
        "edge_scans": 4104,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1086041,
      "exhaustive_oracle": 20041
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,-1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          -1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6140,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1023,
        "edge_scans": 4092,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1084333,
      "exhaustive_oracle": 21167
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13417,
      "exhaustive_oracle": 32166
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13250,
      "exhaustive_oracle": 20625
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1/2",
        "-1/2"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 2
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13166,
      "exhaustive_oracle": 20417
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11834,
      "exhaustive_oracle": 19667
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1084041,
      "exhaustive_oracle": 21416
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1086416,
      "exhaustive_oracle": 18875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-1"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 11875,
      "exhaustive_oracle": 18917
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1085875,
      "exhaustive_oracle": 21916
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,0,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 451542,
      "exhaustive_oracle": 21583
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,-1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-4"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13041,
      "exhaustive_oracle": 21417
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,-1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6146,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1024,
        "edge_scans": 4096,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1100333,
      "exhaustive_oracle": 20875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,-1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          -1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6140,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1023,
        "edge_scans": 4092,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "-1",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1096542,
      "exhaustive_oracle": 22875
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,0,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-4"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13250,
      "exhaustive_oracle": 20208
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,0,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1045083,
      "exhaustive_oracle": 20667
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,0,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 2468,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 411,
        "edge_scans": 1644,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1/2",
        "1/2"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 430125,
      "exhaustive_oracle": 20000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,1,-1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          -1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [],
      "counts": {
        "work": 14,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 2,
        "edge_scans": 8,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "-1",
        "-1"
      ],
      "winning_vertices": [],
      "max_edges": {},
      "min_edges": {
        "0": 1,
        "1": 3
      },
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [],
      "checks": [
        {
          "player": "max",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        },
        {
          "player": "min",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "-4"
          },
          "retained_edges": 2
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 13459,
      "exhaustive_oracle": 20042
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,1,0",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          0
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 6152,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 1025,
        "edge_scans": 4100,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "0",
        "0"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 1031416,
      "exhaustive_oracle": 20000
    },
    "work_limit": 200000
  },
  {
    "name": "complete2_mm_1,1,1,1",
    "input": {
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "owners": [
        "min",
        "min"
      ],
      "edges": [
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          1
        ],
        [
          1,
          1,
          1
        ]
      ]
    },
    "reference": {
      "algorithm": "deterministic_paper_procedure_4_1_bounded_reference",
      "model": {
        "arena": "turn_based_deterministic",
        "reward": "signed_integer_edge",
        "objective": "liminf_mean_at_least_zero",
        "opponents": "full_history"
      },
      "vertices": 2,
      "initial_width": "1024",
      "transformed_max_weight": "4",
      "limits": {
        "work": 200000,
        "depth": 128
      },
      "source_proof_accepted": false,
      "outputs_values_or_strategies": false,
      "status": "complete_labels_source_conditional",
      "winning_vertices": [
        0,
        1
      ],
      "counts": {
        "work": 1544,
        "calls": 1,
        "nonbase_calls": 0,
        "mass_base_calls": 0,
        "width_base_calls": 1,
        "operator_evaluations": 257,
        "edge_scans": 1028,
        "pivot_updates": 0,
        "max_depth": 0
      },
      "counter_scope": "vertex visits at call/pivot and edge+vertex visits at operator evaluations; excludes bit cost, preprocessing, allocation, parsing and wall time"
    },
    "oracle": {
      "values": [
        "1",
        "1"
      ],
      "winning_vertices": [
        0,
        1
      ],
      "max_edges": {},
      "min_edges": {},
      "positional_pairs": 4
    },
    "certificate": {
      "accepted": true,
      "status": "threshold_regions_finitely_certified",
      "winning_vertices": [
        0,
        1
      ],
      "checks": [
        {
          "player": "max",
          "vertices": [
            0,
            1
          ],
          "distance_potential": {
            "0": "0",
            "1": "0"
          },
          "retained_edges": 4
        },
        {
          "player": "min",
          "vertices": [],
          "distance_potential": {},
          "retained_edges": 0
        }
      ],
      "scope": "zero-threshold region strategies only; no exact value-optimality, stochastic, parity, or initial-credit claim"
    },
    "timings_ns": {
      "source_reference": 265459,
      "exhaustive_oracle": 18875
    },
    "work_limit": 200000
  }
]