{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0050",
  "pilot_number": "R0043-22325",
  "number": 22325,
  "family_id": "qf-mod41-orbitstab-action-fibers",
  "title": {
    "zh-Hant": "第 22325 題：軌道－穩定子定理",
    "en": "Question 22325: All Action-Map Fibers Have the Same Stabilizer Cardinality"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0050",
    "batch_id": "batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-orbit-stabilizer-theorem"
    ],
    "title_zh": "軌道－穩定子定理",
    "stem_zh": "同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":10,\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"point\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[0, 4], [1, 5], [2, 0], [3, 1], [4, 2], [5, 3]], [[0, [2]], [1, [3]], [2, [4]], [3, [5]], [4, [0]], [5, [1]]], 是, [0, 1, 2, 3, 4, 5], action_map_surjective_onto_orbit)",
      "explanation_zh": "Wave6第四層核對=([[0, 4], [1, 5], [2, 0], [3, 1], [4, 2], [5, 3]], [[0, [2]], [1, [3]], [2, [4]], [3, [5]], [4, [0]], [5, [1]]], 是, [0, 1, 2, 3, 4, 5], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "軌道－穩定子定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod41-orbitstab-action-fibers",
      "seed": 202611101050,
      "parameters": {
        "group_table": [
          [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          [
            1,
            2,
            3,
            4,
            5,
            0
          ],
          [
            2,
            3,
            4,
            5,
            0,
            1
          ],
          [
            3,
            4,
            5,
            0,
            1,
            2
          ],
          [
            4,
            5,
            0,
            1,
            2,
            3
          ],
          [
            5,
            0,
            1,
            2,
            3,
            4
          ]
        ],
        "identity": 0,
        "point_count": 10,
        "action_table": [
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7,
            8,
            9
          ],
          [
            1,
            2,
            3,
            4,
            5,
            0,
            6,
            7,
            8,
            9
          ],
          [
            2,
            3,
            4,
            5,
            0,
            1,
            6,
            7,
            8,
            9
          ],
          [
            3,
            4,
            5,
            0,
            1,
            2,
            6,
            7,
            8,
            9
          ],
          [
            4,
            5,
            0,
            1,
            2,
            3,
            6,
            7,
            8,
            9
          ],
          [
            5,
            0,
            1,
            2,
            3,
            4,
            6,
            7,
            8,
            9
          ]
        ],
        "point": 4,
        "domain": "Complete_finite_action_orbit_stabilizer_bijection",
        "operation": "orbitstab-action-fibers",
        "semantic_template": "qf-mod41-orbitstab-action-fibers/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":4,\"point_count\":10}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0050",
    "batch_id": "batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f3f7c9df4b0f0b95343f863736f5a90462860a4ef3d05fb15751752fb8d8145c",
      "family_id": "qf-mod41-orbitstab-action-fibers",
      "seed": 202611101050,
      "semantic_template": "qf-mod41-orbitstab-action-fibers/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":4,\"point_count\":10}"
    },
    "fields": {
      "title": "All Action-Map Fibers Have the Same Stabilizer Cardinality",
      "stem": "Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, point carrier Fin(10)={0,…,9}, total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]] and specified point=4. Row g/column h gives g*h, and action row g/column x gives g*x. Certify every group/action axiom, then enumerate all group elements with the same action and point binding. Free, faithful or transitive action is not required. Give (action_map, fibers, equal_fiber_sizes, orbit, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([[0, 4], [1, 5], [2, 0], [3, 1], [4, 2], [5, 3]], [[0, [2]], [1, [3]], [2, [4]], [3, [5]], [4, [0]], [5, [1]]], Yes, [0, 1, 2, 3, 4, 5], action_map_surjective_onto_orbit)",
        "explanation": "List every [g,g*x] in group-index order and every [orbit point,fiber] in point order. The action map is surjective onto its orbit, not necessarily onto the whole point carrier. Each fiber has the same size as this point stabilizer. The exact result, retaining the requested order, is ([[0, 4], [1, 5], [2, 0], [3, 1], [4, 2], [5, 3]], [[0, [2]], [1, [3]], [2, [4]], [3, [5]], [4, [0]], [5, [1]]], Yes, [0, 1, 2, 3, 4, 5], action_map_surjective_onto_orbit)."
      },
      "skills": [
        "All Action-Map Fibers Have the Same Stabilizer Cardinality"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":10,\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"point\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, point carrier Fin(10)={0,…,9}, total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]] and specified point=4. Row g/column h gives g*h, and action row g/column x gives g*x. Certify every group/action axiom, then enumerate all group elements with the same action and point binding. Free, faithful or transitive action is not required. Give (action_map, fibers, equal_fiber_sizes, orbit, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=orbitstab-action-fibers;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[action_map, fibers, equal_fiber_sizes, orbit, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記保留，直接列來源有序欄位，不需JavaScript；MVT由原mode選欄序，不補空欄或看答案形狀猜。tag與固定No保原聲稱範圍。",
        "en": "Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "ORIGINAL complete group-label carrier Fin(6);declared identity label=0;Cayley table row g,column h gives g*h=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]",
      "after": "ORIGINAL complete point carrier Fin(10);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]];all entries in range,so structural closure/totality checked;GROUP AXIOMS not assumed from shape",
      "relation": "implies",
      "rule_id": "groupaction_two_table_roles_complete_candidate_not_group_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩張表的column角色不同，一張h是群標籤，一張x是作用點。in-range完整表只保closed／total，群候選可不滿足公理，不能因叫group_table就先填True。",
        "en": "The two tables have different column roles:group label h versus action point x.Complete in-range tables establish closure/totality only.A candidate group may fail its axioms;its table name does not certify them."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "ORIGINAL closed Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]];ACTUAL declared group identity=0",
      "after": "ALL [g,e*g,g*e,both_equal_g] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]];two-sided group identity=是",
      "relation": "implies",
      "rule_id": "groupaction_group_identity_both_sides_not_action_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "群表的e*g與g*e兩側都須等g；這裡不把作用點的e*x=x當群identity證書。",
        "en": "Both e*g and g*e in the group table must equal g.This is not certified by an identity-action row on points."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 0];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 1];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 2];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 3];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 4];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 5];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 2];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 3];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 4];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 5];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 0];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 1];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 2];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 3];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 4];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 5];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 0];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 1];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 2];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 3];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 4];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 5];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 0];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 1];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 2];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 3];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 4];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 5];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 0];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 1];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 2];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 3];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 4];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 5];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "ORIGINAL group element g=0;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 0, 0, 是], [1, 1, 1, 否], [2, 2, 2, 否], [3, 3, 3, 否], [4, 4, 4, 否], [5, 5, 5, 否]]",
      "after": "first actual two-sided inverse witness=0;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "ORIGINAL group element g=1;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 1, 1, 否], [1, 2, 2, 否], [2, 3, 3, 否], [3, 4, 4, 否], [4, 5, 5, 否], [5, 0, 0, 是]]",
      "after": "first actual two-sided inverse witness=5;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "ORIGINAL group element g=2;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 2, 2, 否], [1, 3, 3, 否], [2, 4, 4, 否], [3, 5, 5, 否], [4, 0, 0, 是], [5, 1, 1, 否]]",
      "after": "first actual two-sided inverse witness=4;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "ORIGINAL group element g=3;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 3, 3, 否], [1, 4, 4, 否], [2, 5, 5, 否], [3, 0, 0, 是], [4, 1, 1, 否], [5, 2, 2, 否]]",
      "after": "first actual two-sided inverse witness=3;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "ORIGINAL group element g=4;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 4, 4, 否], [1, 5, 5, 否], [2, 0, 0, 是], [3, 1, 1, 否], [4, 2, 2, 否], [5, 3, 3, 否]]",
      "after": "first actual two-sided inverse witness=2;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "ORIGINAL group element g=5;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 5, 5, 否], [1, 0, 0, 是], [2, 1, 1, 否], [3, 2, 2, 否], [4, 3, 3, 否], [5, 4, 4, 否]]",
      "after": "first actual two-sided inverse witness=1;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "structurally closed COMPLETE group candidate;two-sided identity=是;ALL-triple associativity=是;ALL-element inverses=是",
      "after": "group_valid=是 is the CONJUNCTION;False remains a reported candidate result,not a silent repair",
      "relation": "implies",
      "rule_id": "groupaction_group_axiom_conjunction_candidate_may_fail",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整三項責任合取才是群資格。即使作用表看似排列，壞group table也不會因此變群；不自行換identity或修表。",
        "en": "Group qualification is the conjunction of these complete obligations.A permutation-looking action table cannot repair a bad group table;do not replace the identity or modify the table."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "ACTUAL declared identity label=0;ORIGINAL identity action row=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];complete point carrier Fin(10)",
      "after": "ALL ordered [point,identity_image,equality] rows=[[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是], [8, 8, 是], [9, 9, 是]];identity-action axiom=是",
      "relation": "implies",
      "rule_id": "groupaction_every_point_identity_action_not_group_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐作用點驗e*x=x，e是來源宣告的真標籤，不預設0；此axiom與群乘法表的雙邊identity責任分開。",
        "en": "Check e*x=x at every action point,using the actual declared identity label rather than assuming0.This action axiom is distinct from the two-sided identity law in the group table."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[0, 0];actual group product=0;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[0, 1];actual group product=1;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[0, 2];actual group product=2;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[0, 3];actual group product=3;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[0, 4];actual group product=4;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[0, 5];actual group product=5;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[1, 0];actual group product=1;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[1, 1];actual group product=2;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[1, 2];actual group product=3;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[1, 3];actual group product=4;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[1, 4];actual group product=5;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[1, 5];actual group product=0;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[2, 0];actual group product=2;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[2, 1];actual group product=3;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[2, 2];actual group product=4;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[2, 3];actual group product=5;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[2, 4];actual group product=0;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[2, 5];actual group product=1;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[3, 0];actual group product=3;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[3, 1];actual group product=4;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[3, 2];actual group product=5;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[3, 3];actual group product=0;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[3, 4];actual group product=1;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[3, 5];actual group product=2;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[4, 0];actual group product=4;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[4, 1];actual group product=5;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[4, 2];actual group product=0;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[4, 3];actual group product=1;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[4, 4];actual group product=2;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[4, 5];actual group product=3;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[5, 0];actual group product=5;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[5, 1];actual group product=0;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[5, 2];actual group product=1;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[5, 3];actual group product=2;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[5, 4];actual group product=3;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[5, 5];actual group product=4;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "group action ordered group-pair count := mul(6, 6)",
      "after": "group action ordered group-pair count = 36",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 group action ordered group-pair count，不以小數近似替代。",
          "en": "Recompute group action ordered group-pair count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "6"
        ],
        "output": "36",
        "label": "group action ordered group-pair count"
      }
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "group action complete triple-domain count := mul(36, 10)",
      "after": "group action complete triple-domain count = 360",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 group action complete triple-domain count，不以小數近似替代。",
          "en": "Recompute group action complete triple-domain count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "36",
          "10"
        ],
        "output": "360",
        "label": "group action complete triple-domain count"
      }
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "COMPLETE triple domain order^2*point_count=360;this derivation ACTUALLY checked all360 triples",
      "after": "compatibility=是;FIRST source-order failure[g,h,x,left,right]=null;null means NO failure anywhere",
      "relation": "implies",
      "rule_id": "groupaction_full_domain_count_first_failure_not_early_stop_measurement",
      "conditions": [],
      "explanation": {
        "zh-Hant": "triple_count是完整量詞域大小，不是把原型在第一錯誤處停止的迭代數冒充測量。本推導確實列了每對g,h的全部點，所以null給全域通過，首失敗則保確切資料與順序。",
        "en": "The triple_count is the complete quantified domain size,not a measurement of an early-stopping prototype.This derivation actually checks every point for every g,h pair,so null certifies full compatibility and a first failure retains exact source data/order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "ORIGINAL group-row label=0;ORIGINAL point-indexed action row=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]",
      "after": "sorted image MULTISET=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];equals full point carrier=是;original row NOT reordered",
      "relation": "implies",
      "rule_id": "groupaction_every_row_permutation_actual_images_not_validity_converse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "排序只核image多重集是否完整無重複，原row按點索引的資料不改。單獨每列是permutation仍不足完整action，須原group及兩個action axiom。",
        "en": "Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "ORIGINAL group-row label=1;ORIGINAL point-indexed action row=[1, 2, 3, 4, 5, 0, 6, 7, 8, 9]",
      "after": "sorted image MULTISET=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];equals full point carrier=是;original row NOT reordered",
      "relation": "implies",
      "rule_id": "groupaction_every_row_permutation_actual_images_not_validity_converse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "排序只核image多重集是否完整無重複，原row按點索引的資料不改。單獨每列是permutation仍不足完整action，須原group及兩個action axiom。",
        "en": "Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "ORIGINAL group-row label=2;ORIGINAL point-indexed action row=[2, 3, 4, 5, 0, 1, 6, 7, 8, 9]",
      "after": "sorted image MULTISET=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];equals full point carrier=是;original row NOT reordered",
      "relation": "implies",
      "rule_id": "groupaction_every_row_permutation_actual_images_not_validity_converse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "排序只核image多重集是否完整無重複，原row按點索引的資料不改。單獨每列是permutation仍不足完整action，須原group及兩個action axiom。",
        "en": "Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "ORIGINAL group-row label=3;ORIGINAL point-indexed action row=[3, 4, 5, 0, 1, 2, 6, 7, 8, 9]",
      "after": "sorted image MULTISET=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];equals full point carrier=是;original row NOT reordered",
      "relation": "implies",
      "rule_id": "groupaction_every_row_permutation_actual_images_not_validity_converse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "排序只核image多重集是否完整無重複，原row按點索引的資料不改。單獨每列是permutation仍不足完整action，須原group及兩個action axiom。",
        "en": "Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "ORIGINAL group-row label=4;ORIGINAL point-indexed action row=[4, 5, 0, 1, 2, 3, 6, 7, 8, 9]",
      "after": "sorted image MULTISET=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];equals full point carrier=是;original row NOT reordered",
      "relation": "implies",
      "rule_id": "groupaction_every_row_permutation_actual_images_not_validity_converse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "排序只核image多重集是否完整無重複，原row按點索引的資料不改。單獨每列是permutation仍不足完整action，須原group及兩個action axiom。",
        "en": "Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "ORIGINAL group-row label=5;ORIGINAL point-indexed action row=[5, 0, 1, 2, 3, 4, 6, 7, 8, 9]",
      "after": "sorted image MULTISET=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];equals full point carrier=是;original row NOT reordered",
      "relation": "implies",
      "rule_id": "groupaction_every_row_permutation_actual_images_not_validity_converse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "排序只核image多重集是否完整無重複，原row按點索引的資料不改。單獨每列是permutation仍不足完整action，須原group及兩個action axiom。",
        "en": "Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "CURRENT qualified group/action T=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]],identity=0,inverse list=[0, 5, 4, 3, 2, 1];ORIGINAL SAME point=4",
      "after": "ALL [g,g*point]=[[0, 4], [1, 5], [2, 0], [3, 1], [4, 2], [5, 3]];full orbit=[0, 1, 2, 3, 4, 5];SAME-point stabilizer=[0];ALL [target,fiber] rows=[[0, [2]], [1, [3]], [2, [4]], [3, [5]], [4, [0]], [5, [1]]];action map onto orbit,NOT necessarily whole carrier",
      "relation": "implies",
      "rule_id": "orbitstab_qualified_SAME_action_point_full_fibers",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗真群與兩個作用公理，再由同一point枚舉全部g。不同point的stabilizer或proper sample不能替換；不需作用free、faithful或transitive。",
        "en": "Qualify the actual group and both action axioms,then enumerate all g at the same point.Another point's stabilizer or a proper sample cannot replace these bindings.No free,faithful or transitive premise is needed."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "ACTUAL SAME-point stabilizer=[0];identity=0",
      "after": "ALL [g,h,product]=[[0, 0, 0]];ALL [g,inverse]=[[0, 0]];identity/product/inverse stay in SAME stabilizer",
      "relation": "implies",
      "rule_id": "orbitstab_SAME_point_stabilizer_actual_subgroup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "e固定point，兩個fixer的乘積仍固定，逆元作用回去亦固定；完整表逐member核，所以左陪集是這個真subgroup的陪集，不是任意相同大小列表。",
        "en": "The identity fixes the point;products of fixers fix it,and acting back by an inverse also fixes it.Check every member in the actual table,so the left cosets use this genuine subgroup,not any equally sized list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "Orbit-stabilizer fiber target0 complete size := sum(1)",
      "after": "Orbit-stabilizer fiber target0 complete size = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Orbit-stabilizer fiber target0 complete size，不以小數近似替代。",
          "en": "Recompute Orbit-stabilizer fiber target0 complete size with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "Orbit-stabilizer fiber target0 complete size"
      }
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "ORIGINAL orbit target=0;LEAST group-index representative g0=2;SAME stabilizer=[0];full fiber=[2]",
      "after": "ALL [h,g0*h,(g0*h)*point]=[[0, 2, 0]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[2, 0, 2]];LEFT coset sorted=[2] equals fiber;fiber size=1",
      "relation": "implies",
      "rule_id": "orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps",
      "conditions": [],
      "explanation": {
        "zh-Hant": "h↦g0*h把同point的Stab送進fiber，反向g↦g0逆*g由作用相容落回Stab；兩個合成皆identity，故雙射。不可改右陪集，非交換群也用相同乘法方向。",
        "en": "The map h->g0*h sends this point's stabilizer into the fibre;g->g0_inverse*g lands back in it by action compatibility.The compositions are identities,so the maps are inverse bijections.Do not substitute a right coset;the multiplication order works for nonabelian groups too."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "Orbit-stabilizer fiber target1 complete size := sum(1)",
      "after": "Orbit-stabilizer fiber target1 complete size = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Orbit-stabilizer fiber target1 complete size，不以小數近似替代。",
          "en": "Recompute Orbit-stabilizer fiber target1 complete size with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "Orbit-stabilizer fiber target1 complete size"
      }
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "ORIGINAL orbit target=1;LEAST group-index representative g0=3;SAME stabilizer=[0];full fiber=[3]",
      "after": "ALL [h,g0*h,(g0*h)*point]=[[0, 3, 1]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[3, 0, 3]];LEFT coset sorted=[3] equals fiber;fiber size=1",
      "relation": "implies",
      "rule_id": "orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps",
      "conditions": [],
      "explanation": {
        "zh-Hant": "h↦g0*h把同point的Stab送進fiber，反向g↦g0逆*g由作用相容落回Stab；兩個合成皆identity，故雙射。不可改右陪集，非交換群也用相同乘法方向。",
        "en": "The map h->g0*h sends this point's stabilizer into the fibre;g->g0_inverse*g lands back in it by action compatibility.The compositions are identities,so the maps are inverse bijections.Do not substitute a right coset;the multiplication order works for nonabelian groups too."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "Orbit-stabilizer fiber target2 complete size := sum(1)",
      "after": "Orbit-stabilizer fiber target2 complete size = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Orbit-stabilizer fiber target2 complete size，不以小數近似替代。",
          "en": "Recompute Orbit-stabilizer fiber target2 complete size with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "Orbit-stabilizer fiber target2 complete size"
      }
    },
    {
      "id": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "ORIGINAL orbit target=2;LEAST group-index representative g0=4;SAME stabilizer=[0];full fiber=[4]",
      "after": "ALL [h,g0*h,(g0*h)*point]=[[0, 4, 2]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[4, 0, 4]];LEFT coset sorted=[4] equals fiber;fiber size=1",
      "relation": "implies",
      "rule_id": "orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps",
      "conditions": [],
      "explanation": {
        "zh-Hant": "h↦g0*h把同point的Stab送進fiber，反向g↦g0逆*g由作用相容落回Stab；兩個合成皆identity，故雙射。不可改右陪集，非交換群也用相同乘法方向。",
        "en": "The map h->g0*h sends this point's stabilizer into the fibre;g->g0_inverse*g lands back in it by action compatibility.The compositions are identities,so the maps are inverse bijections.Do not substitute a right coset;the multiplication order works for nonabelian groups too."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "Orbit-stabilizer fiber target3 complete size := sum(1)",
      "after": "Orbit-stabilizer fiber target3 complete size = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Orbit-stabilizer fiber target3 complete size，不以小數近似替代。",
          "en": "Recompute Orbit-stabilizer fiber target3 complete size with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "Orbit-stabilizer fiber target3 complete size"
      }
    },
    {
      "id": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "ORIGINAL orbit target=3;LEAST group-index representative g0=5;SAME stabilizer=[0];full fiber=[5]",
      "after": "ALL [h,g0*h,(g0*h)*point]=[[0, 5, 3]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[5, 0, 5]];LEFT coset sorted=[5] equals fiber;fiber size=1",
      "relation": "implies",
      "rule_id": "orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps",
      "conditions": [],
      "explanation": {
        "zh-Hant": "h↦g0*h把同point的Stab送進fiber，反向g↦g0逆*g由作用相容落回Stab；兩個合成皆identity，故雙射。不可改右陪集，非交換群也用相同乘法方向。",
        "en": "The map h->g0*h sends this point's stabilizer into the fibre;g->g0_inverse*g lands back in it by action compatibility.The compositions are identities,so the maps are inverse bijections.Do not substitute a right coset;the multiplication order works for nonabelian groups too."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "Orbit-stabilizer fiber target4 complete size := sum(1)",
      "after": "Orbit-stabilizer fiber target4 complete size = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Orbit-stabilizer fiber target4 complete size，不以小數近似替代。",
          "en": "Recompute Orbit-stabilizer fiber target4 complete size with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "Orbit-stabilizer fiber target4 complete size"
      }
    },
    {
      "id": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "ORIGINAL orbit target=4;LEAST group-index representative g0=0;SAME stabilizer=[0];full fiber=[0]",
      "after": "ALL [h,g0*h,(g0*h)*point]=[[0, 0, 4]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[0, 0, 0]];LEFT coset sorted=[0] equals fiber;fiber size=1",
      "relation": "implies",
      "rule_id": "orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps",
      "conditions": [],
      "explanation": {
        "zh-Hant": "h↦g0*h把同point的Stab送進fiber，反向g↦g0逆*g由作用相容落回Stab；兩個合成皆identity，故雙射。不可改右陪集，非交換群也用相同乘法方向。",
        "en": "The map h->g0*h sends this point's stabilizer into the fibre;g->g0_inverse*g lands back in it by action compatibility.The compositions are identities,so the maps are inverse bijections.Do not substitute a right coset;the multiplication order works for nonabelian groups too."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "Orbit-stabilizer fiber target5 complete size := sum(1)",
      "after": "Orbit-stabilizer fiber target5 complete size = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Orbit-stabilizer fiber target5 complete size，不以小數近似替代。",
          "en": "Recompute Orbit-stabilizer fiber target5 complete size with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "Orbit-stabilizer fiber target5 complete size"
      }
    },
    {
      "id": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "ORIGINAL orbit target=5;LEAST group-index representative g0=1;SAME stabilizer=[0];full fiber=[1]",
      "after": "ALL [h,g0*h,(g0*h)*point]=[[0, 1, 5]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[1, 0, 1]];LEFT coset sorted=[1] equals fiber;fiber size=1",
      "relation": "implies",
      "rule_id": "orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps",
      "conditions": [],
      "explanation": {
        "zh-Hant": "h↦g0*h把同point的Stab送進fiber，反向g↦g0逆*g由作用相容落回Stab；兩個合成皆identity，故雙射。不可改右陪集，非交換群也用相同乘法方向。",
        "en": "The map h->g0*h sends this point's stabilizer into the fibre;g->g0_inverse*g lands back in it by action compatibility.The compositions are identities,so the maps are inverse bijections.Do not substitute a right coset;the multiplication order works for nonabelian groups too."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "Orbit-stabilizer complete orbit count := sum(1, 1, 1, 1, 1, 1)",
      "after": "Orbit-stabilizer complete orbit count = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Orbit-stabilizer complete orbit count，不以小數近似替代。",
          "en": "Recompute Orbit-stabilizer complete orbit count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1",
          "1",
          "1",
          "1",
          "1"
        ],
        "output": "6",
        "label": "Orbit-stabilizer complete orbit count"
      }
    },
    {
      "id": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "Orbit-stabilizer SAME-point stabilizer count := sum(1)",
      "after": "Orbit-stabilizer SAME-point stabilizer count = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Orbit-stabilizer SAME-point stabilizer count，不以小數近似替代。",
          "en": "Recompute Orbit-stabilizer SAME-point stabilizer count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "Orbit-stabilizer SAME-point stabilizer count"
      }
    },
    {
      "id": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "Orbit-stabilizer SAME-bound orbit times stabilizer sizes := mul(6, 1)",
      "after": "Orbit-stabilizer SAME-bound orbit times stabilizer sizes = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Orbit-stabilizer SAME-bound orbit times stabilizer sizes，不以小數近似替代。",
          "en": "Recompute Orbit-stabilizer SAME-bound orbit times stabilizer sizes with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "1"
        ],
        "output": "6",
        "label": "Orbit-stabilizer SAME-bound orbit times stabilizer sizes"
      }
    },
    {
      "id": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "ALL original group elements partitioned by actual action-map image;fibers=[[0, [2]], [1, [3]], [2, [4]], [3, [5]], [4, [0]], [5, [1]]]",
      "after": "each fiber bijective with SAME stabilizer of size1;orbit size=6;group order=6=6*1=6",
      "relation": "implies",
      "rule_id": "orbitstab_partition_group_into_equal_size_fibers_count",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g有唯一像，因此fibers互不交且覆蓋全G；每個fiber的雙射才給同大小。這是完整計數理由，不是偶然乘積相等；不從orbit滿射推到整個point carrier滿射。",
        "en": "Each g has exactly one image,so these fibres are disjoint and cover G.The explicit bijections establish equal sizes;this is not merely a coincidental numerical product.Surjectivity onto the orbit does not imply surjectivity onto the entire point carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        [
          0,
          4
        ],
        [
          1,
          5
        ],
        [
          2,
          0
        ],
        [
          3,
          1
        ],
        [
          4,
          2
        ],
        [
          5,
          3
        ]
      ],
      [
        [
          0,
          [
            2
          ]
        ],
        [
          1,
          [
            3
          ]
        ],
        [
          2,
          [
            4
          ]
        ],
        [
          3,
          [
            5
          ]
        ],
        [
          4,
          [
            0
          ]
        ],
        [
          5,
          [
            1
          ]
        ]
      ],
      true,
      [
        0,
        1,
        2,
        3,
        4,
        5
      ],
      "action_map_surjective_onto_orbit"
    ]
  },
  "conclusion": {
    "canonical_value": "([[0, 4], [1, 5], [2, 0], [3, 1], [4, 2], [5, 3]], [[0, [2]], [1, [3]], [2, [4]], [3, [5]], [4, [0]], [5, [1]]], 是, [0, 1, 2, 3, 4, 5], action_map_surjective_onto_orbit)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations/questions.jsonl",
      "batch_sha256": "fc144d65674e79368c3afa23bb92f9851b0e454f837d26281983fc2e7ac6d723",
      "original_line": 50,
      "original_line_sha256": "fdae455782fa3cf81ec3af1aba735cfb5275a8f86b376aadeab9149e21d43196",
      "question_compact_sha256": "f3f7c9df4b0f0b95343f863736f5a90462860a4ef3d05fb15751752fb8d8145c",
      "english_row_compact_sha256": "ec750cb78740f9c13378d4cf736ae6f5ed36c8c67c92c8a735496ccd12c423dd",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-orbit-stabilizer-theorem": {
        "sha256": "0ae3563c72f6bb6e1702aa5a64f490a1f2b0ca0741f19d17ac3790775e66c38f",
        "english_sha256": "d40f8220b027fb6f66d44947a84f3e7b76317d3a768f913574f6cb6651f5bae7",
        "original_path": "public/data/mko/orbit-stabilizer-theorem.json",
        "english_path": "public/data/i18n/editions/mko-orbit-stabilizer-theorem.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "747b98e75085a6b55251b2d7e04bbf7c60cb5500e5047439b2798fb5f45538fa",
    "original_candidate_sha256": "a58f028c9e42004f13796c9d035c8b4c5cddbdc88000e2fae27e3e7cfd0716dd",
    "original_candidate_line_sha256": "a58f028c9e42004f13796c9d035c8b4c5cddbdc88000e2fae27e3e7cfd0716dd"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-009-p44",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5",
      "question_first_witness_sha256": "10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 110,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "98429187c158b861e084759e534712336a681786cbdbf4e5e7043ed7e06ba0e9",
      "arithmetic_steps": 11
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}