{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261026-r01-finite-abelian-group-0012",
  "pilot_number": "R0023-12012",
  "number": 12012,
  "family_id": "qf-mod26-abelian-commutativity-audit",
  "title": {
    "zh-Hant": "第 12012 題：有限阿貝爾群",
    "en": "Question 12012: Audit Every Pair of a Qualified Finite Group"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261026-r01-finite-abelian-group-0012",
    "batch_id": "batch-20261013-advance-20261026-r01-finite-abelian-group-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-abelian-group"
    ],
    "title_zh": "有限阿貝爾群",
    "stem_zh": "完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(6, 0, [], 是)",
      "explanation_zh": "結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "有限阿貝爾群"
      ]
    },
    "variant": {
      "family_id": "qf-mod26-abelian-commutativity-audit",
      "seed": 202610261012,
      "parameters": {
        "n": 6,
        "table": [
          [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          [
            1,
            3,
            0,
            5,
            2,
            4
          ],
          [
            2,
            0,
            4,
            1,
            5,
            3
          ],
          [
            3,
            5,
            1,
            4,
            0,
            2
          ],
          [
            4,
            2,
            5,
            0,
            3,
            1
          ],
          [
            5,
            4,
            3,
            2,
            1,
            0
          ]
        ],
        "domain": "Complete_finite_abelian_group_table",
        "operation": "abelian-commutativity-audit",
        "semantic_template": "qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261026-r01-finite-abelian-group-0012",
    "batch_id": "batch-20261013-advance-20261026-r01-finite-abelian-group-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "d9ca28a0db6f44abb7eecb0dcd0af108c1d03db85c80f24a2603b2c3d56fbb54",
      "family_id": "qf-mod26-abelian-commutativity-audit",
      "seed": 202610261012,
      "semantic_template": "qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]]}"
    },
    "fields": {
      "title": "Audit Every Pair of a Qualified Finite Group",
      "stem": "Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).",
      "answer": {
        "kind": "expression",
        "value": "(6, 0, [], Yes)",
        "explanation": "Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes)."
      },
      "skills": [
        "Audit Every Pair of a Qualified Finite Group"
      ],
      "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": "完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。",
        "en": "Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "carrier Fin(6);complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]]",
      "after": "all entries lie in carrier;do not assume identity0 or commutativity",
      "relation": "implies",
      "rule_id": "complete_finite_group_typed_table",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先固定完整運算表與標籤，合型封閉只是第一條群前提；名字含Abelian不替代交換律檢查。",
        "en": "Fix the whole table and labels.Typed closure is only the first group premise;an Abelian family name does not replace the commutativity audit."
      },
      "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": "a=0;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=0;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[1, 3, 0, 5, 2, 4];a*(b*c) values=[1, 3, 0, 5, 2, 4];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=0;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[2, 0, 4, 1, 5, 3];a*(b*c) values=[2, 0, 4, 1, 5, 3];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=0;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[3, 5, 1, 4, 0, 2];a*(b*c) values=[3, 5, 1, 4, 0, 2];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=0;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[4, 2, 5, 0, 3, 1];a*(b*c) values=[4, 2, 5, 0, 3, 1];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=0;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[5, 4, 3, 2, 1, 0];a*(b*c) values=[5, 4, 3, 2, 1, 0];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=1;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[1, 3, 0, 5, 2, 4];a*(b*c) values=[1, 3, 0, 5, 2, 4];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=1;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[3, 5, 1, 4, 0, 2];a*(b*c) values=[3, 5, 1, 4, 0, 2];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=1;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=1;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[5, 4, 3, 2, 1, 0];a*(b*c) values=[5, 4, 3, 2, 1, 0];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=1;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[2, 0, 4, 1, 5, 3];a*(b*c) values=[2, 0, 4, 1, 5, 3];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=1;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[4, 2, 5, 0, 3, 1];a*(b*c) values=[4, 2, 5, 0, 3, 1];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=2;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[2, 0, 4, 1, 5, 3];a*(b*c) values=[2, 0, 4, 1, 5, 3];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=2;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=2;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[4, 2, 5, 0, 3, 1];a*(b*c) values=[4, 2, 5, 0, 3, 1];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=2;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[1, 3, 0, 5, 2, 4];a*(b*c) values=[1, 3, 0, 5, 2, 4];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=2;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[5, 4, 3, 2, 1, 0];a*(b*c) values=[5, 4, 3, 2, 1, 0];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=2;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[3, 5, 1, 4, 0, 2];a*(b*c) values=[3, 5, 1, 4, 0, 2];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=3;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[3, 5, 1, 4, 0, 2];a*(b*c) values=[3, 5, 1, 4, 0, 2];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=3;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[5, 4, 3, 2, 1, 0];a*(b*c) values=[5, 4, 3, 2, 1, 0];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=3;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[1, 3, 0, 5, 2, 4];a*(b*c) values=[1, 3, 0, 5, 2, 4];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=3;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[4, 2, 5, 0, 3, 1];a*(b*c) values=[4, 2, 5, 0, 3, 1];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=3;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=3;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[2, 0, 4, 1, 5, 3];a*(b*c) values=[2, 0, 4, 1, 5, 3];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=4;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[4, 2, 5, 0, 3, 1];a*(b*c) values=[4, 2, 5, 0, 3, 1];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=4;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[2, 0, 4, 1, 5, 3];a*(b*c) values=[2, 0, 4, 1, 5, 3];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=4;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[5, 4, 3, 2, 1, 0];a*(b*c) values=[5, 4, 3, 2, 1, 0];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=4;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=4;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[3, 5, 1, 4, 0, 2];a*(b*c) values=[3, 5, 1, 4, 0, 2];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=4;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[1, 3, 0, 5, 2, 4];a*(b*c) values=[1, 3, 0, 5, 2, 4];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=5;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[5, 4, 3, 2, 1, 0];a*(b*c) values=[5, 4, 3, 2, 1, 0];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=5;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[4, 2, 5, 0, 3, 1];a*(b*c) values=[4, 2, 5, 0, 3, 1];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=5;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[3, 5, 1, 4, 0, 2];a*(b*c) values=[3, 5, 1, 4, 0, 2];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=5;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[2, 0, 4, 1, 5, 3];a*(b*c) values=[2, 0, 4, 1, 5, 3];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=5;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[1, 3, 0, 5, 2, 4];a*(b*c) values=[1, 3, 0, 5, 2, 4];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "a=5;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=是",
      "relation": "implies",
      "rule_id": "associativity_every_third_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定每個有序(a,b)後，逐一列全部c的兩側，所有列相等才是完整有限結合律核查。",
        "en": "For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation."
      },
      "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": "candidate identity=0;left products=[0, 1, 2, 3, 4, 5];right products=[0, 1, 2, 3, 4, 5]",
      "after": "both equal original carrier labels=是",
      "relation": "implies",
      "rule_id": "two_sided_identity_derive_not_zero_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同時要求e*x=x與x*e=x，標籤0沒有預設特權；回讀兩邊原表。",
        "en": "Require both e*x=x and x*e=x.Label0 has no special status;read both table directions."
      },
      "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": "candidate identity=1;left products=[1, 3, 0, 5, 2, 4];right products=[1, 3, 0, 5, 2, 4]",
      "after": "both equal original carrier labels=否",
      "relation": "implies",
      "rule_id": "two_sided_identity_derive_not_zero_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同時要求e*x=x與x*e=x，標籤0沒有預設特權；回讀兩邊原表。",
        "en": "Require both e*x=x and x*e=x.Label0 has no special status;read both table directions."
      },
      "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": "candidate identity=2;left products=[2, 0, 4, 1, 5, 3];right products=[2, 0, 4, 1, 5, 3]",
      "after": "both equal original carrier labels=否",
      "relation": "implies",
      "rule_id": "two_sided_identity_derive_not_zero_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同時要求e*x=x與x*e=x，標籤0沒有預設特權；回讀兩邊原表。",
        "en": "Require both e*x=x and x*e=x.Label0 has no special status;read both table directions."
      },
      "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": "candidate identity=3;left products=[3, 5, 1, 4, 0, 2];right products=[3, 5, 1, 4, 0, 2]",
      "after": "both equal original carrier labels=否",
      "relation": "implies",
      "rule_id": "two_sided_identity_derive_not_zero_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同時要求e*x=x與x*e=x，標籤0沒有預設特權；回讀兩邊原表。",
        "en": "Require both e*x=x and x*e=x.Label0 has no special status;read both table directions."
      },
      "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": "candidate identity=4;left products=[4, 2, 5, 0, 3, 1];right products=[4, 2, 5, 0, 3, 1]",
      "after": "both equal original carrier labels=否",
      "relation": "implies",
      "rule_id": "two_sided_identity_derive_not_zero_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同時要求e*x=x與x*e=x，標籤0沒有預設特權；回讀兩邊原表。",
        "en": "Require both e*x=x and x*e=x.Label0 has no special status;read both table directions."
      },
      "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": "candidate identity=5;left products=[5, 4, 3, 2, 1, 0];right products=[5, 4, 3, 2, 1, 0]",
      "after": "both equal original carrier labels=否",
      "relation": "implies",
      "rule_id": "two_sided_identity_derive_not_zero_label",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同時要求e*x=x與x*e=x，標籤0沒有預設特權；回讀兩邊原表。",
        "en": "Require both e*x=x and x*e=x.Label0 has no special status;read both table directions."
      },
      "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": "element=0;actual identity=0;two-sided inverse candidates=[0]",
      "after": "inverse=0",
      "relation": "implies",
      "rule_id": "group_inverse_qualified_at_actual_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個元素找兩側都回到實際單位元的逆元；缺任一逆元就不能沿用後續群論證。",
        "en": "Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning."
      },
      "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": "element=1;actual identity=0;two-sided inverse candidates=[2]",
      "after": "inverse=2",
      "relation": "implies",
      "rule_id": "group_inverse_qualified_at_actual_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個元素找兩側都回到實際單位元的逆元；缺任一逆元就不能沿用後續群論證。",
        "en": "Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning."
      },
      "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": "element=2;actual identity=0;two-sided inverse candidates=[1]",
      "after": "inverse=1",
      "relation": "implies",
      "rule_id": "group_inverse_qualified_at_actual_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個元素找兩側都回到實際單位元的逆元；缺任一逆元就不能沿用後續群論證。",
        "en": "Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning."
      },
      "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": "element=3;actual identity=0;two-sided inverse candidates=[4]",
      "after": "inverse=4",
      "relation": "implies",
      "rule_id": "group_inverse_qualified_at_actual_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個元素找兩側都回到實際單位元的逆元；缺任一逆元就不能沿用後續群論證。",
        "en": "Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning."
      },
      "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": "element=4;actual identity=0;two-sided inverse candidates=[3]",
      "after": "inverse=3",
      "relation": "implies",
      "rule_id": "group_inverse_qualified_at_actual_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個元素找兩側都回到實際單位元的逆元；缺任一逆元就不能沿用後續群論證。",
        "en": "Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning."
      },
      "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": "element=5;actual identity=0;two-sided inverse candidates=[5]",
      "after": "inverse=5",
      "relation": "implies",
      "rule_id": "group_inverse_qualified_at_actual_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個元素找兩側都回到實際單位元的逆元；缺任一逆元就不能沿用後續群論證。",
        "en": "Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning."
      },
      "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": "unordered distinct labels=[0, 1];a*b=1;b*a=1",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "unordered distinct labels=[0, 2];a*b=2;b*a=2",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "unordered distinct labels=[0, 3];a*b=3;b*a=3",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "unordered distinct labels=[0, 4];a*b=4;b*a=4",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "unordered distinct labels=[0, 5];a*b=5;b*a=5",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "unordered distinct labels=[1, 2];a*b=0;b*a=0",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "unordered distinct labels=[1, 3];a*b=5;b*a=5",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "unordered distinct labels=[1, 4];a*b=2;b*a=2",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "unordered distinct labels=[1, 5];a*b=4;b*a=4",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "unordered distinct labels=[2, 3];a*b=1;b*a=1",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "unordered distinct labels=[2, 4];a*b=5;b*a=5",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "unordered distinct labels=[2, 5];a*b=3;b*a=3",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "unordered distinct labels=[3, 4];a*b=0;b*a=0",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "unordered distinct labels=[3, 5];a*b=2;b*a=2",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "unordered distinct labels=[4, 5];a*b=1;b*a=1",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "every_off_diagonal_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對角同項自動相等；每個不同元素對核兩種次序，失敗記錄保留a<b原標籤順序。",
        "en": "Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label 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": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "all noncommuting records=[]",
      "after": "specified finite group abelian=是",
      "relation": "implies",
      "rule_id": "finite_abelian_all_pairs_not_general_classification",
      "conditions": [],
      "explanation": {
        "zh-Hant": "所有元素對相同才是此群交換；只分類這份完整有限表，不推廣到任意群。",
        "en": "All pairs must agree to make this group abelian.This classifies only the supplied finite table,not arbitrary groups."
      },
      "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": [
      6,
      0,
      [],
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(6, 0, [], 是)",
    "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-20261013-advance-20261026-r01-finite-abelian-group-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261026-r01-finite-abelian-group-foundations/questions.jsonl",
      "batch_sha256": "8932481b89961bcb52ef93bc0819294fcdc2300c8e5e421a8e75c167285cbe42",
      "original_line": 12,
      "original_line_sha256": "8e2790dcf68d36220651d95aa40c81e7f5f5c9f3e10953827d698d8f1da8c1c8",
      "question_compact_sha256": "d9ca28a0db6f44abb7eecb0dcd0af108c1d03db85c80f24a2603b2c3d56fbb54",
      "english_row_compact_sha256": "388ff9df8d1cf2423351f2bc2ce59a552a30584c00add4622b3e43448dc3eaa3",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261026-r01-finite-abelian-group-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261026-r01-finite-abelian-group-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-abelian-group": {
        "sha256": "1486476569952aee082941d6f91d2921526de39dd732a11bacce98c252b95415",
        "english_sha256": "7b784df7deb082eb20d9dc02a3cc71c900becdfafb7f8a2828225c8c53f99407",
        "original_path": "public/data/mko/finite-abelian-group.json",
        "english_path": "public/data/i18n/editions/mko-finite-abelian-group.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c6ae588be9bc8b3dd8461be32c073abfa25a0f827b4602cf28bf291ce532d507",
    "original_candidate_sha256": "a5c69f34a383ddafebbba3ca25c022e7df3e21e5762c60ae491c956345584e61",
    "original_candidate_line_sha256": "a5c69f34a383ddafebbba3ca25c022e7df3e21e5762c60ae491c956345584e61"
  },
  "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-20261026-006-p24",
    "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": "f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff",
      "question_first_witness_sha256": "49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 65,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "962853bb0a20eefeee1b6b6d74e839fdeb6b0a959e4c31dacac85c170ce2be70",
      "arithmetic_steps": 0
    },
    "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."
  }
}