{
  "checker": "Independent direct-equation enumeration; no imports from the main checker.",
  "checker_sha256": "fa0fa04bece8012412faf39d65f3afb45059a1e7939ef75f9a60e8ad4da352fb",
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      "clamped_solution_sets": [
        {
          "clamp": [
            -1,
            -1,
            -1
          ],
          "solutions": [
            [
              0,
              0,
              0
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        },
        {
          "clamp": [
            -1,
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          "solutions": [
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              0,
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              0
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          ]
        },
        {
          "clamp": [
            -1,
            -1,
            1
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          "solutions": [
            [
              0,
              1,
              1
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          ]
        },
        {
          "clamp": [
            -1,
            0,
            -1
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          "solutions": [
            [
              0,
              0,
              0
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          ]
        },
        {
          "clamp": [
            -1,
            0,
            0
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          "solutions": [
            [
              0,
              0,
              0
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          ]
        },
        {
          "clamp": [
            -1,
            0,
            1
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          "solutions": [
            [
              0,
              0,
              1
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          ]
        },
        {
          "clamp": [
            -1,
            1,
            -1
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          "solutions": [
            [
              1,
              1,
              0
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          ]
        },
        {
          "clamp": [
            -1,
            1,
            0
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          "solutions": [
            [
              1,
              1,
              0
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          ]
        },
        {
          "clamp": [
            -1,
            1,
            1
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          "solutions": [
            [
              0,
              1,
              1
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          ]
        },
        {
          "clamp": [
            0,
            -1,
            -1
          ],
          "solutions": [
            [
              0,
              0,
              0
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          ]
        },
        {
          "clamp": [
            0,
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            0
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          "solutions": [
            [
              0,
              0,
              0
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        },
        {
          "clamp": [
            0,
            -1,
            1
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          "solutions": [
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              0,
              1,
              1
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          ]
        },
        {
          "clamp": [
            0,
            0,
            -1
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          "solutions": [
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              0,
              0,
              0
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          ]
        },
        {
          "clamp": [
            0,
            0,
            0
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          "solutions": [
            [
              0,
              0,
              0
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          ]
        },
        {
          "clamp": [
            0,
            0,
            1
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          "solutions": [
            [
              0,
              0,
              1
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          ]
        },
        {
          "clamp": [
            0,
            1,
            -1
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          "solutions": [
            [
              0,
              1,
              0
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          ]
        },
        {
          "clamp": [
            0,
            1,
            0
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          "solutions": [
            [
              0,
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              0
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          ]
        },
        {
          "clamp": [
            0,
            1,
            1
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          "solutions": [
            [
              0,
              1,
              1
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          ]
        },
        {
          "clamp": [
            1,
            -1,
            -1
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          "solutions": [
            [
              1,
              0,
              1
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          ]
        },
        {
          "clamp": [
            1,
            -1,
            0
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          "solutions": [
            [
              1,
              0,
              0
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          ]
        },
        {
          "clamp": [
            1,
            -1,
            1
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          "solutions": [
            [
              1,
              0,
              1
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          ]
        },
        {
          "clamp": [
            1,
            0,
            -1
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          "solutions": [
            [
              1,
              0,
              1
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          ]
        },
        {
          "clamp": [
            1,
            0,
            0
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          "solutions": [
            [
              1,
              0,
              0
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          ]
        },
        {
          "clamp": [
            1,
            0,
            1
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          "solutions": [
            [
              1,
              0,
              1
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          ]
        },
        {
          "clamp": [
            1,
            1,
            -1
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          "solutions": [
            [
              1,
              1,
              0
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          ]
        },
        {
          "clamp": [
            1,
            1,
            0
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          "solutions": [
            [
              1,
              1,
              0
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          ]
        },
        {
          "clamp": [
            1,
            1,
            1
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          "solutions": [
            [
              1,
              1,
              1
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          ]
        }
      ],
      "cycle": [
        [
          0,
          0,
          1
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        [
          0,
          1,
          1
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        [
          0,
          1,
          0
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        [
          1,
          1,
          0
        ],
        [
          1,
          0,
          0
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        [
          1,
          0,
          1
        ]
      ],
      "mechanisms": [
        "y*(1-z)",
        "z*(1-x)",
        "x*(1-y)"
      ],
      "periodic_schedule_is_fair": true,
      "zero_based_update_coordinates": [
        1,
        2,
        0,
        1,
        2,
        0
      ]
    },
    "three_by_three_simple_counterexample": {
      "all_fine_interventions_uniquely_solvable": true,
      "f_B_to_A": [
        0,
        1,
        2
      ],
      "failures": [
        {
          "coarse_intervention": [
            -1,
            -1
          ],
          "coarse_solutions": [
            [
              0,
              0
            ],
            [
              1,
              1
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          ],
          "fine_intervention": [
            -1,
            -1
          ],
          "fine_solutions": [
            [
              0,
              0
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          ],
          "projected_solutions": [
            [
              0,
              0
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          ]
        }
      ],
      "fine_interventions": 16,
      "full_retention_audit": {
        "candidates": [
          {
            "actual_descent": true,
            "partial_graph_acyclic": false,
            "predecessor_closed": true,
            "retained": []
          },
          {
            "actual_descent": false,
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            "predecessor_closed": false,
            "retained": [
              0
            ]
          },
          {
            "actual_descent": false,
            "partial_graph_acyclic": true,
            "predecessor_closed": false,
            "retained": [
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          },
          {
            "actual_descent": true,
            "partial_graph_acyclic": true,
            "predecessor_closed": true,
            "retained": [
              0,
              1
            ]
          }
        ],
        "descending_certificate_minimum": 2,
        "graph_minimum": 1,
        "hidden_edges": [
          [
            0,
            1
          ],
          [
            1,
            0
          ]
        ]
      },
      "g_A_to_B": [
        0,
        2,
        1
      ],
      "quotients": [
        [
          0,
          1,
          1
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        [
          0,
          1,
          1
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      ]
    },
    "unique_unintervened_equilibrium_does_not_replace_simplicity": {
      "fine_intervention_solution_count_distribution": {
        "0": 1,
        "1": 33,
        "2": 2
      },
      "fine_solutions_after_do_i_1": [],
      "fine_unintervened_solutions": [
        [
          0,
          0,
          0
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      ],
      "macro_unintervened_solutions": [
        [
          0,
          0,
          0
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        [
          1,
          0,
          1
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      ],
      "mechanisms": [
        "0 if i=0 else 1+c",
        "indicator(p=1)",
        "indicator(p!=0)"
      ],
      "ordinary_edges": [
        [
          0,
          1
        ],
        [
          0,
          2
        ],
        [
          1,
          0
        ],
        [
          2,
          0
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      ],
      "partial_coordinates": [
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      "quotients": [
        [
          0,
          1,
          1
        ],
        [
          0,
          0
        ],
        [
          0,
          1
        ]
      ],
      "sizes": [
        3,
        2,
        2
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    }
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  "python_version": "3.14.7",
  "reporting_convention": "A map on a singleton is classified as injective; it is also constant. Partial means neither.",
  "three_bit": {
    "bounds": "All 4096 no-self-dependence three-bit networks and all 8 identity/erasure quotient products.",
    "by_quotient_mask_I_identity_E_erasure": {
      "EEE": {
        "compatible_pairs": 4096,
        "matched_micro_interventions_checked": 20088,
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        "simple_compatible_pairs": 744,
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "EEI": {
        "compatible_pairs": 512,
        "matched_micro_interventions_checked": 7776,
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        "simple_compatible_pairs": 288,
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "EIE": {
        "compatible_pairs": 512,
        "matched_micro_interventions_checked": 7776,
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        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "EII": {
        "compatible_pairs": 256,
        "matched_micro_interventions_checked": 5184,
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        "simple_compatible_pairs": 192,
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "IEE": {
        "compatible_pairs": 512,
        "matched_micro_interventions_checked": 7776,
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        "simple_compatible_pairs": 288,
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "IEI": {
        "compatible_pairs": 256,
        "matched_micro_interventions_checked": 5184,
        "safe_simple_pairs": 192,
        "simple_compatible_pairs": 192,
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "IIE": {
        "compatible_pairs": 256,
        "matched_micro_interventions_checked": 5184,
        "safe_simple_pairs": 192,
        "simple_compatible_pairs": 192,
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "III": {
        "compatible_pairs": 4096,
        "matched_micro_interventions_checked": 20088,
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        "simple_compatible_pairs": 744,
        "spurious_macro_solutions_across_matched_interventions": 0,
        "unsafe_simple_pairs": 0
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    },
    "compatible_pairs": 10496,
    "matched_micro_interventions_checked": 79056,
    "raw_systems": 4096,
    "safe_simple_pairs": 2928,
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    "simple_raw_systems": 744,
    "spurious_macro_solutions_across_matched_interventions": 0,
    "unsafe_simple_pairs": 0
  },
  "two_component": {
    "bounds": "All 1 <= |A|, |B| <= 3, all f:B->A and g:A->B, all canonical set partitions.",
    "by_domain": [
      {
        "canonical_partition_pairs": 1,
        "compatible_pairs": 1,
        "matched_micro_interventions_checked": 4,
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        "sizes": [
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          1
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        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      {
        "canonical_partition_pairs": 2,
        "compatible_pairs": 4,
        "matched_micro_interventions_checked": 24,
        "raw_systems": 2,
        "safe_simple_pairs": 4,
        "simple_compatible_pairs": 4,
        "simple_raw_systems": 2,
        "sizes": [
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          2
        ],
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      {
        "canonical_partition_pairs": 2,
        "compatible_pairs": 4,
        "matched_micro_interventions_checked": 24,
        "raw_systems": 2,
        "safe_simple_pairs": 4,
        "simple_compatible_pairs": 4,
        "simple_raw_systems": 2,
        "sizes": [
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          1
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        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      {
        "canonical_partition_pairs": 5,
        "compatible_pairs": 15,
        "matched_micro_interventions_checked": 120,
        "raw_systems": 3,
        "safe_simple_pairs": 15,
        "simple_compatible_pairs": 15,
        "simple_raw_systems": 3,
        "sizes": [
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          3
        ],
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      {
        "canonical_partition_pairs": 5,
        "compatible_pairs": 15,
        "matched_micro_interventions_checked": 120,
        "raw_systems": 3,
        "safe_simple_pairs": 15,
        "simple_compatible_pairs": 15,
        "simple_raw_systems": 3,
        "sizes": [
          3,
          1
        ],
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      {
        "canonical_partition_pairs": 4,
        "compatible_pairs": 48,
        "matched_micro_interventions_checked": 360,
        "raw_systems": 16,
        "safe_simple_pairs": 40,
        "simple_compatible_pairs": 40,
        "simple_raw_systems": 12,
        "sizes": [
          2,
          2
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        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      {
        "canonical_partition_pairs": 10,
        "compatible_pairs": 414,
        "matched_micro_interventions_checked": 3816,
        "raw_systems": 72,
        "safe_simple_pairs": 318,
        "simple_compatible_pairs": 318,
        "simple_raw_systems": 48,
        "sizes": [
          2,
          3
        ],
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      {
        "canonical_partition_pairs": 10,
        "compatible_pairs": 414,
        "matched_micro_interventions_checked": 3816,
        "raw_systems": 72,
        "safe_simple_pairs": 318,
        "simple_compatible_pairs": 318,
        "simple_raw_systems": 48,
        "sizes": [
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          2
        ],
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      {
        "canonical_partition_pairs": 25,
        "compatible_pairs": 6561,
        "matched_micro_interventions_checked": 71856,
        "raw_systems": 729,
        "safe_simple_pairs": 4473,
        "simple_compatible_pairs": 4491,
        "simple_raw_systems": 423,
        "sizes": [
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          3
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        "spurious_macro_solutions_across_matched_interventions": 18,
        "unsafe_simple_pairs": 18
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    ],
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      "constant/constant": {
        "compatible_pairs": 889,
        "matched_micro_interventions_checked": 8028,
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        "simple_compatible_pairs": 531,
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "constant/injective": {
        "compatible_pairs": 136,
        "matched_micro_interventions_checked": 1908,
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        "simple_compatible_pairs": 136,
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "constant/partial": {
        "compatible_pairs": 849,
        "matched_micro_interventions_checked": 9360,
        "safe_simple_pairs": 609,
        "simple_compatible_pairs": 609,
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "injective/constant": {
        "compatible_pairs": 136,
        "matched_micro_interventions_checked": 1908,
        "safe_simple_pairs": 136,
        "simple_compatible_pairs": 136,
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "injective/injective": {
        "compatible_pairs": 900,
        "matched_micro_interventions_checked": 8104,
        "safe_simple_pairs": 542,
        "simple_compatible_pairs": 542,
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "injective/partial": {
        "compatible_pairs": 846,
        "matched_micro_interventions_checked": 9288,
        "safe_simple_pairs": 606,
        "simple_compatible_pairs": 606,
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "partial/constant": {
        "compatible_pairs": 849,
        "matched_micro_interventions_checked": 9360,
        "safe_simple_pairs": 609,
        "simple_compatible_pairs": 609,
        "spurious_macro_solutions_across_matched_interventions": 0,
        "unsafe_simple_pairs": 0
      },
      "partial/injective": {
        "compatible_pairs": 846,
        "matched_micro_interventions_checked": 9288,
        "safe_simple_pairs": 606,
        "simple_compatible_pairs": 606,
        "spurious_macro_solutions_across_matched_interventions": 0,
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      },
      "partial/partial": {
        "compatible_pairs": 2025,
        "matched_micro_interventions_checked": 22896,
        "safe_simple_pairs": 1413,
        "simple_compatible_pairs": 1431,
        "spurious_macro_solutions_across_matched_interventions": 18,
        "unsafe_simple_pairs": 18
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    },
    "compatible_pairs": 7476,
    "full_retention_checks": {
      "base_pairs_with_graph_vs_descent_cost_gap": 324,
      "compatible_base_pairs": 7476,
      "descending_candidates": 17980,
      "refinement_candidates_checked": 23276,
      "simple_base_pairs_with_cost_gap": 18
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    "matched_micro_interventions_checked": 80140,
    "minimal_failure_classification": {
      "all_unsafe_pairs_checked_against_identity_swap_shape": 18,
      "canonical_quotient_pairs_with_failure": 9,
      "isomorphism_classes_at_size_3_by_3": 1,
      "meaning_of_isomorphism": "Independent relabellings of each fine coordinate and the induced coarse labels, preserving both mechanisms and quotient maps.",
      "unsafe_systems_per_such_quotient_pair": 2
    },
    "raw_systems": 900,
    "safe_simple_pairs": 5188,
    "simple_compatible_pairs": 5206,
    "simple_raw_systems": 542,
    "smallest_failure_in_search": {
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        1,
        2
      ],
      "failures": [
        {
          "coarse_intervention": [
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            -1
          ],
          "coarse_solutions": [
            [
              0,
              0
            ],
            [
              1,
              1
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          ],
          "fine_intervention": [
            -1,
            -1
          ],
          "fine_solutions": [
            [
              0,
              0
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          ],
          "projected_solutions": [
            [
              0,
              0
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          ]
        }
      ],
      "g_A_to_B": [
        0,
        2,
        1
      ],
      "quotients": [
        [
          0,
          1,
          1
        ],
        [
          0,
          1,
          1
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      ],
      "sizes": [
        3,
        3
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    },
    "spurious_macro_solutions_across_matched_interventions": 18,
    "unsafe_simple_pairs": 18
  }
}