{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261026-r01-finite-abelian-group-0059",
  "pilot_number": "R0023-12084",
  "number": 12084,
  "family_id": "qf-mod26-abelian-integer-multiple",
  "title": {
    "zh-Hant": "第 12084 題：有限阿貝爾群",
    "en": "Question 12084: Repeated Group Operation with a Zero-Multiple Boundary"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261026-r01-finite-abelian-group-0059",
    "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-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"element\":5,\"multiplier\":11}。請依 [element,multiplier,identity,result] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(5, 11, 0, 1)",
      "explanation_zh": "結構深度核對=(5, 11, 0, 1)；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-integer-multiple",
      "seed": 202610261059,
      "parameters": {
        "n": 6,
        "table": [
          [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          [
            1,
            3,
            5,
            4,
            2,
            0
          ],
          [
            2,
            5,
            3,
            0,
            1,
            4
          ],
          [
            3,
            4,
            0,
            2,
            5,
            1
          ],
          [
            4,
            2,
            1,
            5,
            0,
            3
          ],
          [
            5,
            0,
            4,
            1,
            3,
            2
          ]
        ],
        "element": 5,
        "multiplier": 11,
        "domain": "Complete_finite_abelian_group_table",
        "operation": "abelian-integer-multiple",
        "semantic_template": "qf-mod26-abelian-integer-multiple/{\"element\":5,\"multiplier\":11,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]]}"
      }
    },
    "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-0059",
    "batch_id": "batch-20261013-advance-20261026-r01-finite-abelian-group-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "98d432bef943131273d4db5dc1dc9b8a6678fc44b0c633fec221506f182c44c6",
      "family_id": "qf-mod26-abelian-integer-multiple",
      "seed": 202610261059,
      "semantic_template": "qf-mod26-abelian-integer-multiple/{\"element\":5,\"multiplier\":11,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]]}"
    },
    "fields": {
      "title": "Repeated Group Operation with a Zero-Multiple Boundary",
      "stem": "Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] 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. element=5, nonnegative multiplier=11. Give (element,multiplier,identity,result).",
      "answer": {
        "kind": "expression",
        "value": "(5, 11, 0, 1)",
        "explanation": "Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 11, 0, 1)."
      },
      "skills": [
        "Repeated Group Operation with a Zero-Multiple Boundary"
      ],
      "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-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"element\":5,\"multiplier\":11}。請依 [element,multiplier,identity,result] 欄序回答。",
        "en": "Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] 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. element=5, nonnegative multiplier=11. Give (element,multiplier,identity,result)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "carrier Fin(6);complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]]",
      "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, 5, 4, 2, 0];a*(b*c) values=[1, 3, 5, 4, 2, 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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "a=0;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[2, 5, 3, 0, 1, 4];a*(b*c) values=[2, 5, 3, 0, 1, 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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "a=0;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[3, 4, 0, 2, 5, 1];a*(b*c) values=[3, 4, 0, 2, 5, 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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "a=0;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[4, 2, 1, 5, 0, 3];a*(b*c) values=[4, 2, 1, 5, 0, 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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "a=0;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[5, 0, 4, 1, 3, 2];a*(b*c) values=[5, 0, 4, 1, 3, 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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "a=1;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[1, 3, 5, 4, 2, 0];a*(b*c) values=[1, 3, 5, 4, 2, 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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "a=1;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[3, 4, 0, 2, 5, 1];a*(b*c) values=[3, 4, 0, 2, 5, 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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "a=1;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[5, 0, 4, 1, 3, 2];a*(b*c) values=[5, 0, 4, 1, 3, 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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "a=1;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[4, 2, 1, 5, 0, 3];a*(b*c) values=[4, 2, 1, 5, 0, 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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "a=1;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[2, 5, 3, 0, 1, 4];a*(b*c) values=[2, 5, 3, 0, 1, 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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "a=1;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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "a=2;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[2, 5, 3, 0, 1, 4];a*(b*c) values=[2, 5, 3, 0, 1, 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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "a=2;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[5, 0, 4, 1, 3, 2];a*(b*c) values=[5, 0, 4, 1, 3, 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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "a=2;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[3, 4, 0, 2, 5, 1];a*(b*c) values=[3, 4, 0, 2, 5, 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=[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": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "a=2;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[1, 3, 5, 4, 2, 0];a*(b*c) values=[1, 3, 5, 4, 2, 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=[4, 2, 1, 5, 0, 3];a*(b*c) values=[4, 2, 1, 5, 0, 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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "a=3;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[3, 4, 0, 2, 5, 1];a*(b*c) values=[3, 4, 0, 2, 5, 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": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "a=3;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[4, 2, 1, 5, 0, 3];a*(b*c) values=[4, 2, 1, 5, 0, 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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "a=3;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": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "a=3;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[2, 5, 3, 0, 1, 4];a*(b*c) values=[2, 5, 3, 0, 1, 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": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "a=3;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[5, 0, 4, 1, 3, 2];a*(b*c) values=[5, 0, 4, 1, 3, 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": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "a=3;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[1, 3, 5, 4, 2, 0];a*(b*c) values=[1, 3, 5, 4, 2, 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": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "a=4;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[4, 2, 1, 5, 0, 3];a*(b*c) values=[4, 2, 1, 5, 0, 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": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "a=4;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[2, 5, 3, 0, 1, 4];a*(b*c) values=[2, 5, 3, 0, 1, 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": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "a=4;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[1, 3, 5, 4, 2, 0];a*(b*c) values=[1, 3, 5, 4, 2, 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=[5, 0, 4, 1, 3, 2];a*(b*c) values=[5, 0, 4, 1, 3, 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": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "a=4;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": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "a=4;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[3, 4, 0, 2, 5, 1];a*(b*c) values=[3, 4, 0, 2, 5, 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": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "a=5;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[5, 0, 4, 1, 3, 2];a*(b*c) values=[5, 0, 4, 1, 3, 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": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "a=5;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": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "a=5;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[4, 2, 1, 5, 0, 3];a*(b*c) values=[4, 2, 1, 5, 0, 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": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "a=5;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[1, 3, 5, 4, 2, 0];a*(b*c) values=[1, 3, 5, 4, 2, 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": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "a=5;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[3, 4, 0, 2, 5, 1];a*(b*c) values=[3, 4, 0, 2, 5, 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": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "a=5;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "(a*b)*c values=[2, 5, 3, 0, 1, 4];a*(b*c) values=[2, 5, 3, 0, 1, 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": "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, 5, 4, 2, 0];right products=[1, 3, 5, 4, 2, 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": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "candidate identity=2;left products=[2, 5, 3, 0, 1, 4];right products=[2, 5, 3, 0, 1, 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": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "candidate identity=3;left products=[3, 4, 0, 2, 5, 1];right products=[3, 4, 0, 2, 5, 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": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "candidate identity=4;left products=[4, 2, 1, 5, 0, 3];right products=[4, 2, 1, 5, 0, 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": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "candidate identity=5;left products=[5, 0, 4, 1, 3, 2];right products=[5, 0, 4, 1, 3, 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": "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=[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": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "element=2;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": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "element=3;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": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "element=4;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": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "element=5;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": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "element=5;actual identity=0",
      "after": "zero repetitions give=0",
      "relation": "implies",
      "rule_id": "power_starts_at_actual_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "冪從單位元起算，0次就是此群的單位元；目前沒有額外假設交換律。",
        "en": "Start powers at the actual identity.Zero repetitions return it;no extra commutativity premise has been assumed."
      },
      "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": "repetition=1;previous=0;element=5;table row=[0, 1, 2, 3, 4, 5]",
      "after": "new value=table[0][5]=5",
      "relation": "implies",
      "rule_id": "ordered_group_power_table_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按當前累計值與指定元素查原表，保留迭代次序；不能把標籤當整數加法。",
        "en": "Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not 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": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "repetition=2;previous=5;element=5;table row=[5, 0, 4, 1, 3, 2]",
      "after": "new value=table[5][5]=2",
      "relation": "implies",
      "rule_id": "ordered_group_power_table_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按當前累計值與指定元素查原表，保留迭代次序；不能把標籤當整數加法。",
        "en": "Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not 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": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "repetition=3;previous=2;element=5;table row=[2, 5, 3, 0, 1, 4]",
      "after": "new value=table[2][5]=4",
      "relation": "implies",
      "rule_id": "ordered_group_power_table_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按當前累計值與指定元素查原表，保留迭代次序；不能把標籤當整數加法。",
        "en": "Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not 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": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "repetition=4;previous=4;element=5;table row=[4, 2, 1, 5, 0, 3]",
      "after": "new value=table[4][5]=3",
      "relation": "implies",
      "rule_id": "ordered_group_power_table_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按當前累計值與指定元素查原表，保留迭代次序；不能把標籤當整數加法。",
        "en": "Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not 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": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "repetition=5;previous=3;element=5;table row=[3, 4, 0, 2, 5, 1]",
      "after": "new value=table[3][5]=1",
      "relation": "implies",
      "rule_id": "ordered_group_power_table_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按當前累計值與指定元素查原表，保留迭代次序；不能把標籤當整數加法。",
        "en": "Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not 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": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "repetition=6;previous=1;element=5;table row=[1, 3, 5, 4, 2, 0]",
      "after": "new value=table[1][5]=0",
      "relation": "implies",
      "rule_id": "ordered_group_power_table_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按當前累計值與指定元素查原表，保留迭代次序；不能把標籤當整數加法。",
        "en": "Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not 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": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "repetition=7;previous=0;element=5;table row=[0, 1, 2, 3, 4, 5]",
      "after": "new value=table[0][5]=5",
      "relation": "implies",
      "rule_id": "ordered_group_power_table_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按當前累計值與指定元素查原表，保留迭代次序；不能把標籤當整數加法。",
        "en": "Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not 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": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "repetition=8;previous=5;element=5;table row=[5, 0, 4, 1, 3, 2]",
      "after": "new value=table[5][5]=2",
      "relation": "implies",
      "rule_id": "ordered_group_power_table_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按當前累計值與指定元素查原表，保留迭代次序；不能把標籤當整數加法。",
        "en": "Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not 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": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "repetition=9;previous=2;element=5;table row=[2, 5, 3, 0, 1, 4]",
      "after": "new value=table[2][5]=4",
      "relation": "implies",
      "rule_id": "ordered_group_power_table_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按當前累計值與指定元素查原表，保留迭代次序；不能把標籤當整數加法。",
        "en": "Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not 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": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "repetition=10;previous=4;element=5;table row=[4, 2, 1, 5, 0, 3]",
      "after": "new value=table[4][5]=3",
      "relation": "implies",
      "rule_id": "ordered_group_power_table_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按當前累計值與指定元素查原表，保留迭代次序；不能把標籤當整數加法。",
        "en": "Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not 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": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "repetition=11;previous=3;element=5;table row=[3, 4, 0, 2, 5, 1]",
      "after": "new value=table[3][5]=1",
      "relation": "implies",
      "rule_id": "ordered_group_power_table_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按當前累計值與指定元素查原表，保留迭代次序；不能把標籤當整數加法。",
        "en": "Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not 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
    }
  ],
  "math_model": {
    "result": [
      5,
      11,
      0,
      1
    ]
  },
  "conclusion": {
    "canonical_value": "(5, 11, 0, 1)",
    "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": 59,
      "original_line_sha256": "b9055fa10b5e8c2adec73109540f266d437706fcd529a9ce82f86a0aadf3c938",
      "question_compact_sha256": "98d432bef943131273d4db5dc1dc9b8a6678fc44b0c633fec221506f182c44c6",
      "english_row_compact_sha256": "f8cc7880676528e043c3782f58785d0fc292a31b44e3162b30832cbcfb2f7223",
      "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": "480a1ccd29d9db7830b5de9e0a84599a816c0284fa8c3c82095d42d57d1160e8",
    "original_candidate_line_sha256": "480a1ccd29d9db7830b5de9e0a84599a816c0284fa8c3c82095d42d57d1160e8"
  },
  "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": 61,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "611d91e5edc7db8610f12152afd57fd2ecaacae7852ed39c7b344f3417c2d297",
      "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."
  }
}