{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0083",
  "pilot_number": "R0045-23408",
  "number": 23408,
  "family_id": "qf-mod43-cauchy-composite-divisor-boundary",
  "title": {
    "zh-Hant": "第 23408 題：有限群的Cauchy定理",
    "en": "Question 23408: A Composite Divisor Need Not Be an Element Order"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0083",
    "batch_id": "batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-cauchy-theorem-finite-groups"
    ],
    "title_zh": "有限群的Cauchy定理",
    "stem_zh": "完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5],[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7],[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"divisor\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(8, 8, [[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)",
      "explanation_zh": "Wave6第六層核對=(8, 8, [[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有限群的Cauchy定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod43-cauchy-composite-divisor-boundary",
      "seed": 202611121083,
      "parameters": {
        "group_table": [
          [
            3,
            2,
            1,
            0,
            7,
            6,
            5,
            4
          ],
          [
            2,
            3,
            0,
            1,
            6,
            7,
            4,
            5
          ],
          [
            1,
            0,
            3,
            2,
            5,
            4,
            7,
            6
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            7,
            6,
            5,
            4,
            3,
            2,
            1,
            0
          ],
          [
            6,
            7,
            4,
            5,
            2,
            3,
            0,
            1
          ],
          [
            5,
            4,
            7,
            6,
            1,
            0,
            3,
            2
          ],
          [
            4,
            5,
            6,
            7,
            0,
            1,
            2,
            3
          ]
        ],
        "identity": 3,
        "divisor": 8,
        "domain": "Complete_finite_group_Cauchy_witness",
        "operation": "cauchy-composite-divisor-boundary",
        "semantic_template": "qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_table\":[[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5],[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7],[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3]],\"identity\":3}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0083",
    "batch_id": "batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "b25b500019846d3ae3959a9d9f935c3b8cfcfb848ad1ff59427db2dca030fb46",
      "family_id": "qf-mod43-cauchy-composite-divisor-boundary",
      "seed": 202611121083,
      "semantic_template": "qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_table\":[[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5],[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7],[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3]],\"identity\":3}"
    },
    "fields": {
      "title": "A Composite Divisor Need Not Be an Element Order",
      "stem": "Use complete finite Cayley table=[[3, 2, 1, 0, 7, 6, 5, 4], [2, 3, 0, 1, 6, 7, 4, 5], [1, 0, 3, 2, 5, 4, 7, 6], [0, 1, 2, 3, 4, 5, 6, 7], [7, 6, 5, 4, 3, 2, 1, 0], [6, 7, 4, 5, 2, 3, 0, 1], [5, 4, 7, 6, 1, 0, 3, 2], [4, 5, 6, 7, 0, 1, 2, 3]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=8. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(8, 8, [[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)",
        "explanation": "Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (8, 8, [[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)."
      },
      "skills": [
        "A Composite Divisor Need Not Be an Element Order"
      ],
      "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": "完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5],[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7],[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"divisor\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete finite Cayley table=[[3, 2, 1, 0, 7, 6, 5, 4], [2, 3, 0, 1, 6, 7, 4, 5], [1, 0, 3, 2, 5, 4, 7, 6], [0, 1, 2, 3, 4, 5, 6, 7], [7, 6, 5, 4, 3, 2, 1, 0], [6, 7, 4, 5, 2, 3, 0, 1], [5, 4, 7, 6, 1, 0, 3, 2], [4, 5, 6, 7, 0, 1, 2, 3]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=8. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=cauchy-composite-divisor-boundary;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面；此處直接列來源欄序，不依賴JavaScript。tag與No保聲稱範圍，固定來源排序不冒充數學證明。",
        "en": "Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "ORIGINAL closed Cayley table=[[3, 2, 1, 0, 7, 6, 5, 4], [2, 3, 0, 1, 6, 7, 4, 5], [1, 0, 3, 2, 5, 4, 7, 6], [0, 1, 2, 3, 4, 5, 6, 7], [7, 6, 5, 4, 3, 2, 1, 0], [6, 7, 4, 5, 2, 3, 0, 1], [5, 4, 7, 6, 1, 0, 3, 2], [4, 5, 6, 7, 0, 1, 2, 3]];ACTUAL declared group identity=3",
      "after": "ALL [g,e*g,g*e,both_equal_g] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是]];two-sided group identity=是",
      "relation": "implies",
      "rule_id": "groupaction_group_identity_both_sides_not_action_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "群表的e*g與g*e兩側都須等g；這裡不把作用點的e*x=x當群identity證書。",
        "en": "Both e*g and g*e in the group table must equal g.This is not certified by an identity-action row on points."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 0];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 1];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 0, 0, 是], [2, 3, 3, 是], [3, 2, 2, 是], [4, 5, 5, 是], [5, 4, 4, 是], [6, 7, 7, 是], [7, 6, 6, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 2];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 6, 6, 是], [5, 7, 7, 是], [6, 4, 4, 是], [7, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 3];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 2, 2, 是], [2, 1, 1, 是], [3, 0, 0, 是], [4, 7, 7, 是], [5, 6, 6, 是], [6, 5, 5, 是], [7, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 4];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 6, 6, 是], [3, 7, 7, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 2, 2, 是], [7, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 5];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 4, 4, 是], [2, 7, 7, 是], [3, 6, 6, 是], [4, 1, 1, 是], [5, 0, 0, 是], [6, 3, 3, 是], [7, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 6];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, 是], [1, 7, 7, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 0, 0, 是], [7, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 7];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, 是], [1, 6, 6, 是], [2, 5, 5, 是], [3, 4, 4, 是], [4, 3, 3, 是], [5, 2, 2, 是], [6, 1, 1, 是], [7, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 0, 0, 是], [2, 3, 3, 是], [3, 2, 2, 是], [4, 5, 5, 是], [5, 4, 4, 是], [6, 7, 7, 是], [7, 6, 6, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 2];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 2, 2, 是], [2, 1, 1, 是], [3, 0, 0, 是], [4, 7, 7, 是], [5, 6, 6, 是], [6, 5, 5, 是], [7, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 3];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 6, 6, 是], [5, 7, 7, 是], [6, 4, 4, 是], [7, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 4];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 4, 4, 是], [2, 7, 7, 是], [3, 6, 6, 是], [4, 1, 1, 是], [5, 0, 0, 是], [6, 3, 3, 是], [7, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 5];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 6, 6, 是], [3, 7, 7, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 2, 2, 是], [7, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 6];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, 是], [1, 6, 6, 是], [2, 5, 5, 是], [3, 4, 4, 是], [4, 3, 3, 是], [5, 2, 2, 是], [6, 1, 1, 是], [7, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 7];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, 是], [1, 7, 7, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 0, 0, 是], [7, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 0];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 6, 6, 是], [5, 7, 7, 是], [6, 4, 4, 是], [7, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 1];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 2, 2, 是], [2, 1, 1, 是], [3, 0, 0, 是], [4, 7, 7, 是], [5, 6, 6, 是], [6, 5, 5, 是], [7, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 2];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 3];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 0, 0, 是], [2, 3, 3, 是], [3, 2, 2, 是], [4, 5, 5, 是], [5, 4, 4, 是], [6, 7, 7, 是], [7, 6, 6, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 4];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, 是], [1, 7, 7, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 0, 0, 是], [7, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 5];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, 是], [1, 6, 6, 是], [2, 5, 5, 是], [3, 4, 4, 是], [4, 3, 3, 是], [5, 2, 2, 是], [6, 1, 1, 是], [7, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 6];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 6, 6, 是], [3, 7, 7, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 2, 2, 是], [7, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 7];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 4, 4, 是], [2, 7, 7, 是], [3, 6, 6, 是], [4, 1, 1, 是], [5, 0, 0, 是], [6, 3, 3, 是], [7, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 0];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 2, 2, 是], [2, 1, 1, 是], [3, 0, 0, 是], [4, 7, 7, 是], [5, 6, 6, 是], [6, 5, 5, 是], [7, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 1];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 6, 6, 是], [5, 7, 7, 是], [6, 4, 4, 是], [7, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 2];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 0, 0, 是], [2, 3, 3, 是], [3, 2, 2, 是], [4, 5, 5, 是], [5, 4, 4, 是], [6, 7, 7, 是], [7, 6, 6, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 3];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 4];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, 是], [1, 6, 6, 是], [2, 5, 5, 是], [3, 4, 4, 是], [4, 3, 3, 是], [5, 2, 2, 是], [6, 1, 1, 是], [7, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 5];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, 是], [1, 7, 7, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 0, 0, 是], [7, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 6];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 4, 4, 是], [2, 7, 7, 是], [3, 6, 6, 是], [4, 1, 1, 是], [5, 0, 0, 是], [6, 3, 3, 是], [7, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 7];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 6, 6, 是], [3, 7, 7, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 2, 2, 是], [7, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 0];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 6, 6, 是], [3, 7, 7, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 2, 2, 是], [7, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 1];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 4, 4, 是], [2, 7, 7, 是], [3, 6, 6, 是], [4, 1, 1, 是], [5, 0, 0, 是], [6, 3, 3, 是], [7, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 2];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, 是], [1, 7, 7, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 0, 0, 是], [7, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 3];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, 是], [1, 6, 6, 是], [2, 5, 5, 是], [3, 4, 4, 是], [4, 3, 3, 是], [5, 2, 2, 是], [6, 1, 1, 是], [7, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 4];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 5];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 0, 0, 是], [2, 3, 3, 是], [3, 2, 2, 是], [4, 5, 5, 是], [5, 4, 4, 是], [6, 7, 7, 是], [7, 6, 6, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 6];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 6, 6, 是], [5, 7, 7, 是], [6, 4, 4, 是], [7, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 7];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 2, 2, 是], [2, 1, 1, 是], [3, 0, 0, 是], [4, 7, 7, 是], [5, 6, 6, 是], [6, 5, 5, 是], [7, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 0];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 4, 4, 是], [2, 7, 7, 是], [3, 6, 6, 是], [4, 1, 1, 是], [5, 0, 0, 是], [6, 3, 3, 是], [7, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 1];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 6, 6, 是], [3, 7, 7, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 2, 2, 是], [7, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 2];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, 是], [1, 6, 6, 是], [2, 5, 5, 是], [3, 4, 4, 是], [4, 3, 3, 是], [5, 2, 2, 是], [6, 1, 1, 是], [7, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 3];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, 是], [1, 7, 7, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 0, 0, 是], [7, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 4];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 0, 0, 是], [2, 3, 3, 是], [3, 2, 2, 是], [4, 5, 5, 是], [5, 4, 4, 是], [6, 7, 7, 是], [7, 6, 6, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 5];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 6];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 2, 2, 是], [2, 1, 1, 是], [3, 0, 0, 是], [4, 7, 7, 是], [5, 6, 6, 是], [6, 5, 5, 是], [7, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 7];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 6, 6, 是], [5, 7, 7, 是], [6, 4, 4, 是], [7, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "ORIGINAL group pair(g,h)=[6, 0];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, 是], [1, 7, 7, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 0, 0, 是], [7, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "ORIGINAL group pair(g,h)=[6, 1];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, 是], [1, 6, 6, 是], [2, 5, 5, 是], [3, 4, 4, 是], [4, 3, 3, 是], [5, 2, 2, 是], [6, 1, 1, 是], [7, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "ORIGINAL group pair(g,h)=[6, 2];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 6, 6, 是], [3, 7, 7, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 2, 2, 是], [7, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "ORIGINAL group pair(g,h)=[6, 3];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 4, 4, 是], [2, 7, 7, 是], [3, 6, 6, 是], [4, 1, 1, 是], [5, 0, 0, 是], [6, 3, 3, 是], [7, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "ORIGINAL group pair(g,h)=[6, 4];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 6, 6, 是], [5, 7, 7, 是], [6, 4, 4, 是], [7, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "ORIGINAL group pair(g,h)=[6, 5];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 2, 2, 是], [2, 1, 1, 是], [3, 0, 0, 是], [4, 7, 7, 是], [5, 6, 6, 是], [6, 5, 5, 是], [7, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "ORIGINAL group pair(g,h)=[6, 6];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "ORIGINAL group pair(g,h)=[6, 7];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 0, 0, 是], [2, 3, 3, 是], [3, 2, 2, 是], [4, 5, 5, 是], [5, 4, 4, 是], [6, 7, 7, 是], [7, 6, 6, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "ORIGINAL group pair(g,h)=[7, 0];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, 是], [1, 6, 6, 是], [2, 5, 5, 是], [3, 4, 4, 是], [4, 3, 3, 是], [5, 2, 2, 是], [6, 1, 1, 是], [7, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "ORIGINAL group pair(g,h)=[7, 1];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, 是], [1, 7, 7, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 2, 2, 是], [5, 3, 3, 是], [6, 0, 0, 是], [7, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "ORIGINAL group pair(g,h)=[7, 2];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 4, 4, 是], [2, 7, 7, 是], [3, 6, 6, 是], [4, 1, 1, 是], [5, 0, 0, 是], [6, 3, 3, 是], [7, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "ORIGINAL group pair(g,h)=[7, 3];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 6, 6, 是], [3, 7, 7, 是], [4, 0, 0, 是], [5, 1, 1, 是], [6, 2, 2, 是], [7, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "ORIGINAL group pair(g,h)=[7, 4];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 2, 2, 是], [2, 1, 1, 是], [3, 0, 0, 是], [4, 7, 7, 是], [5, 6, 6, 是], [6, 5, 5, 是], [7, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "ORIGINAL group pair(g,h)=[7, 5];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 6, 6, 是], [5, 7, 7, 是], [6, 4, 4, 是], [7, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "ORIGINAL group pair(g,h)=[7, 6];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 0, 0, 是], [2, 3, 3, 是], [3, 2, 2, 是], [4, 5, 5, 是], [5, 4, 4, 是], [6, 7, 7, 是], [7, 6, 6, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "ORIGINAL group pair(g,h)=[7, 7];check ALL k in Fin(8)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "ORIGINAL group element g=0;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 3, 3, 是], [1, 2, 2, 否], [2, 1, 1, 否], [3, 0, 0, 否], [4, 7, 7, 否], [5, 6, 6, 否], [6, 5, 5, 否], [7, 4, 4, 否]]",
      "after": "first actual two-sided inverse witness=0;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "ORIGINAL group element g=1;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 2, 2, 否], [1, 3, 3, 是], [2, 0, 0, 否], [3, 1, 1, 否], [4, 6, 6, 否], [5, 7, 7, 否], [6, 4, 4, 否], [7, 5, 5, 否]]",
      "after": "first actual two-sided inverse witness=1;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "ORIGINAL group element g=2;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 1, 1, 否], [1, 0, 0, 否], [2, 3, 3, 是], [3, 2, 2, 否], [4, 5, 5, 否], [5, 4, 4, 否], [6, 7, 7, 否], [7, 6, 6, 否]]",
      "after": "first actual two-sided inverse witness=2;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "ORIGINAL group element g=3;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 0, 0, 否], [1, 1, 1, 否], [2, 2, 2, 否], [3, 3, 3, 是], [4, 4, 4, 否], [5, 5, 5, 否], [6, 6, 6, 否], [7, 7, 7, 否]]",
      "after": "first actual two-sided inverse witness=3;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "ORIGINAL group element g=4;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 7, 7, 否], [1, 6, 6, 否], [2, 5, 5, 否], [3, 4, 4, 否], [4, 3, 3, 是], [5, 2, 2, 否], [6, 1, 1, 否], [7, 0, 0, 否]]",
      "after": "first actual two-sided inverse witness=4;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "ORIGINAL group element g=5;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 6, 6, 否], [1, 7, 7, 否], [2, 4, 4, 否], [3, 5, 5, 否], [4, 2, 2, 否], [5, 3, 3, 是], [6, 0, 0, 否], [7, 1, 1, 否]]",
      "after": "first actual two-sided inverse witness=5;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "ORIGINAL group element g=6;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 5, 5, 否], [1, 4, 4, 否], [2, 7, 7, 否], [3, 6, 6, 否], [4, 1, 1, 否], [5, 0, 0, 否], [6, 3, 3, 是], [7, 2, 2, 否]]",
      "after": "first actual two-sided inverse witness=6;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "ORIGINAL group element g=7;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 4, 4, 否], [1, 5, 5, 否], [2, 6, 6, 否], [3, 7, 7, 否], [4, 0, 0, 否], [5, 1, 1, 否], [6, 2, 2, 否], [7, 3, 3, 是]]",
      "after": "first actual two-sided inverse witness=7;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "structurally closed COMPLETE group candidate;two-sided identity=是;ALL-triple associativity=是;ALL-element inverses=是",
      "after": "group_valid=是 is the CONJUNCTION;False remains a reported candidate result,not a silent repair",
      "relation": "implies",
      "rule_id": "groupaction_group_axiom_conjunction_candidate_may_fail",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整三項責任合取才是群資格。即使作用表看似排列，壞group table也不會因此變群；不自行換identity或修表。",
        "en": "Group qualification is the conjunction of these complete obligations.A permutation-looking action table cannot repair a bad group table;do not replace the identity or modify the table."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "CURRENT ORIGINAL complete Cayley table=[[3, 2, 1, 0, 7, 6, 5, 4], [2, 3, 0, 1, 6, 7, 4, 5], [1, 0, 3, 2, 5, 4, 7, 6], [0, 1, 2, 3, 4, 5, 6, 7], [7, 6, 5, 4, 3, 2, 1, 0], [6, 7, 4, 5, 2, 3, 0, 1], [5, 4, 7, 6, 1, 0, 3, 2], [4, 5, 6, 7, 0, 1, 2, 3]];actual identity=3;order=8",
      "after": "ALL group axioms qualified;actual inverse list=[0, 1, 2, 3, 4, 5, 6, 7];no point carrier or group-action assumption is used",
      "relation": "implies",
      "rule_id": "group_CURRENT_complete_certified_Cayley_NOT_action_wrapper",
      "conditions": [],
      "explanation": {
        "zh-Hant": "群表本身先完整驗閉合、identity、associativity、雙邊inverse；不套舊作用表的8元素限制，也不假設cyclic或commutative。純舊函式只代入本題表與trace。",
        "en": "Certify closure,identity,associativity and two-sided inverses in the group table itself.Do not import an old8-element action-table limit or assume cyclicity/commutativity.The frozen pure helper receives only this question's table and trace."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "Cauchy ORIGINAL element0:ORIGINAL element=0;positive exponent=1;previous power=3;right multiply ORIGINAL g via table",
      "after": "g^1=0;identity=3;is_identity=否;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "Cauchy ORIGINAL element0:ORIGINAL element=0;positive exponent=2;previous power=0;right multiply ORIGINAL g via table",
      "after": "g^2=3;identity=3;is_identity=是;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "Cauchy ORIGINAL element0:CURRENT certified group order=8,element=0;ALL powers through first return=[[1, 0], [2, 3]]",
      "after": "exact positive order=2;cyclic powers before repetition form a subgroup of this size;its equal-size left cosets partition G,so this order divides group order",
      "relation": "implies",
      "rule_id": "cauchy_finite_order_exists_and_cyclic_subgroup_divisibility",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限群中e,g,…,g^|G|必重複，消去較小冪得正回返≤|G|。最小回返之前的冪互異，按exponent mod order乘法／inverse封閉；其left cosets給整除理由，不是從一個冪的等式猜order。",
        "en": "Among e,g,...,g^|G|,a repetition cancels to a positive return at most|G|.Powers before the minimal return are distinct and close under multiplication/inverse modulo that order.Their left cosets prove divisibility;one power equation alone does not."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "Cauchy ORIGINAL element1:ORIGINAL element=1;positive exponent=1;previous power=3;right multiply ORIGINAL g via table",
      "after": "g^1=1;identity=3;is_identity=否;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "Cauchy ORIGINAL element1:ORIGINAL element=1;positive exponent=2;previous power=1;right multiply ORIGINAL g via table",
      "after": "g^2=3;identity=3;is_identity=是;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "Cauchy ORIGINAL element1:CURRENT certified group order=8,element=1;ALL powers through first return=[[1, 1], [2, 3]]",
      "after": "exact positive order=2;cyclic powers before repetition form a subgroup of this size;its equal-size left cosets partition G,so this order divides group order",
      "relation": "implies",
      "rule_id": "cauchy_finite_order_exists_and_cyclic_subgroup_divisibility",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限群中e,g,…,g^|G|必重複，消去較小冪得正回返≤|G|。最小回返之前的冪互異，按exponent mod order乘法／inverse封閉；其left cosets給整除理由，不是從一個冪的等式猜order。",
        "en": "Among e,g,...,g^|G|,a repetition cancels to a positive return at most|G|.Powers before the minimal return are distinct and close under multiplication/inverse modulo that order.Their left cosets prove divisibility;one power equation alone does not."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "Cauchy ORIGINAL element2:ORIGINAL element=2;positive exponent=1;previous power=3;right multiply ORIGINAL g via table",
      "after": "g^1=2;identity=3;is_identity=否;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "Cauchy ORIGINAL element2:ORIGINAL element=2;positive exponent=2;previous power=2;right multiply ORIGINAL g via table",
      "after": "g^2=3;identity=3;is_identity=是;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "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": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "Cauchy ORIGINAL element2:CURRENT certified group order=8,element=2;ALL powers through first return=[[1, 2], [2, 3]]",
      "after": "exact positive order=2;cyclic powers before repetition form a subgroup of this size;its equal-size left cosets partition G,so this order divides group order",
      "relation": "implies",
      "rule_id": "cauchy_finite_order_exists_and_cyclic_subgroup_divisibility",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限群中e,g,…,g^|G|必重複，消去較小冪得正回返≤|G|。最小回返之前的冪互異，按exponent mod order乘法／inverse封閉；其left cosets給整除理由，不是從一個冪的等式猜order。",
        "en": "Among e,g,...,g^|G|,a repetition cancels to a positive return at most|G|.Powers before the minimal return are distinct and close under multiplication/inverse modulo that order.Their left cosets prove divisibility;one power equation alone does not."
      },
      "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": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "Cauchy ORIGINAL element3:ORIGINAL element=3;positive exponent=1;previous power=3;right multiply ORIGINAL g via table",
      "after": "g^1=3;identity=3;is_identity=是;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "Cauchy ORIGINAL element3:CURRENT certified group order=8,element=3;ALL powers through first return=[[1, 3]]",
      "after": "exact positive order=1;cyclic powers before repetition form a subgroup of this size;its equal-size left cosets partition G,so this order divides group order",
      "relation": "implies",
      "rule_id": "cauchy_finite_order_exists_and_cyclic_subgroup_divisibility",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限群中e,g,…,g^|G|必重複，消去較小冪得正回返≤|G|。最小回返之前的冪互異，按exponent mod order乘法／inverse封閉；其left cosets給整除理由，不是從一個冪的等式猜order。",
        "en": "Among e,g,...,g^|G|,a repetition cancels to a positive return at most|G|.Powers before the minimal return are distinct and close under multiplication/inverse modulo that order.Their left cosets prove divisibility;one power equation alone does not."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "Cauchy ORIGINAL element4:ORIGINAL element=4;positive exponent=1;previous power=3;right multiply ORIGINAL g via table",
      "after": "g^1=4;identity=3;is_identity=否;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "Cauchy ORIGINAL element4:ORIGINAL element=4;positive exponent=2;previous power=4;right multiply ORIGINAL g via table",
      "after": "g^2=3;identity=3;is_identity=是;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "Cauchy ORIGINAL element4:CURRENT certified group order=8,element=4;ALL powers through first return=[[1, 4], [2, 3]]",
      "after": "exact positive order=2;cyclic powers before repetition form a subgroup of this size;its equal-size left cosets partition G,so this order divides group order",
      "relation": "implies",
      "rule_id": "cauchy_finite_order_exists_and_cyclic_subgroup_divisibility",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限群中e,g,…,g^|G|必重複，消去較小冪得正回返≤|G|。最小回返之前的冪互異，按exponent mod order乘法／inverse封閉；其left cosets給整除理由，不是從一個冪的等式猜order。",
        "en": "Among e,g,...,g^|G|,a repetition cancels to a positive return at most|G|.Powers before the minimal return are distinct and close under multiplication/inverse modulo that order.Their left cosets prove divisibility;one power equation alone does not."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "Cauchy ORIGINAL element5:ORIGINAL element=5;positive exponent=1;previous power=3;right multiply ORIGINAL g via table",
      "after": "g^1=5;identity=3;is_identity=否;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "Cauchy ORIGINAL element5:ORIGINAL element=5;positive exponent=2;previous power=5;right multiply ORIGINAL g via table",
      "after": "g^2=3;identity=3;is_identity=是;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "Cauchy ORIGINAL element5:CURRENT certified group order=8,element=5;ALL powers through first return=[[1, 5], [2, 3]]",
      "after": "exact positive order=2;cyclic powers before repetition form a subgroup of this size;its equal-size left cosets partition G,so this order divides group order",
      "relation": "implies",
      "rule_id": "cauchy_finite_order_exists_and_cyclic_subgroup_divisibility",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限群中e,g,…,g^|G|必重複，消去較小冪得正回返≤|G|。最小回返之前的冪互異，按exponent mod order乘法／inverse封閉；其left cosets給整除理由，不是從一個冪的等式猜order。",
        "en": "Among e,g,...,g^|G|,a repetition cancels to a positive return at most|G|.Powers before the minimal return are distinct and close under multiplication/inverse modulo that order.Their left cosets prove divisibility;one power equation alone does not."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "Cauchy ORIGINAL element6:ORIGINAL element=6;positive exponent=1;previous power=3;right multiply ORIGINAL g via table",
      "after": "g^1=6;identity=3;is_identity=否;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "Cauchy ORIGINAL element6:ORIGINAL element=6;positive exponent=2;previous power=6;right multiply ORIGINAL g via table",
      "after": "g^2=3;identity=3;is_identity=是;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "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": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "Cauchy ORIGINAL element6:CURRENT certified group order=8,element=6;ALL powers through first return=[[1, 6], [2, 3]]",
      "after": "exact positive order=2;cyclic powers before repetition form a subgroup of this size;its equal-size left cosets partition G,so this order divides group order",
      "relation": "implies",
      "rule_id": "cauchy_finite_order_exists_and_cyclic_subgroup_divisibility",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限群中e,g,…,g^|G|必重複，消去較小冪得正回返≤|G|。最小回返之前的冪互異，按exponent mod order乘法／inverse封閉；其left cosets給整除理由，不是從一個冪的等式猜order。",
        "en": "Among e,g,...,g^|G|,a repetition cancels to a positive return at most|G|.Powers before the minimal return are distinct and close under multiplication/inverse modulo that order.Their left cosets prove divisibility;one power equation alone does not."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "Cauchy ORIGINAL element7:ORIGINAL element=7;positive exponent=1;previous power=3;right multiply ORIGINAL g via table",
      "after": "g^1=7;identity=3;is_identity=否;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "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": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "Cauchy ORIGINAL element7:ORIGINAL element=7;positive exponent=2;previous power=7;right multiply ORIGINAL g via table",
      "after": "g^2=3;identity=3;is_identity=是;all earlier POSITIVE powers are nonidentity",
      "relation": "implies",
      "rule_id": "cauchy_EVERY_positive_power_until_FIRST_identity_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從e開始只把正指數1、2、…列入階的候選；第一次回e才是精確order。exponent0不列入，不能只驗g^p=e便宣稱階為p。",
        "en": "Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "Cauchy ORIGINAL element7:CURRENT certified group order=8,element=7;ALL powers through first return=[[1, 7], [2, 3]]",
      "after": "exact positive order=2;cyclic powers before repetition form a subgroup of this size;its equal-size left cosets partition G,so this order divides group order",
      "relation": "implies",
      "rule_id": "cauchy_finite_order_exists_and_cyclic_subgroup_divisibility",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限群中e,g,…,g^|G|必重複，消去較小冪得正回返≤|G|。最小回返之前的冪互異，按exponent mod order乘法／inverse封閉；其left cosets給整除理由，不是從一個冪的等式猜order。",
        "en": "Among e,g,...,g^|G|,a repetition cancels to a positive return at most|G|.Powers before the minimal return are distinct and close under multiplication/inverse modulo that order.Their left cosets prove divisibility;one power equation alone does not."
      },
      "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": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "Cauchy ORIGINAL composite divisor:ORIGINAL integer=8;complete possible trial divisors=[2]",
      "after": "ALL [divisor,remainder] rows=[[2, 0]];prime=否;a composite>1 has a factor at most its square root",
      "relation": "implies",
      "rule_id": "finite_prime_ALL_trial_divisors_through_square_root",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整平方根範圍核整除；prime2的範圍為空而通過，0/1不通過。不能由少數取樣、平方表或Euler判準代替此資格。",
        "en": "Check the entire square-root divisor range.The empty range qualifies prime2,but not0 or1.Samples,square tables or an Euler criterion do not replace primality qualification."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "CURRENT group order=8;ORIGINAL composite divisor=8;COMPLETE element-order rows=[[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2]]",
      "after": "divisor divides order=是;exists element of requested order=否 in THIS complete group;not a claim composite element orders never occur",
      "relation": "implies",
      "rule_id": "cauchy_composite_counterexample_ACTUAL_absence_not_general_negation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "必須真整除且完整表真找不到該階，才是反例；其他群可以有合數階元素，不由此模型否定它們。這也不在討論同階subgroup是否存在。",
        "en": "Require actual divisibility and exhaustive absence of that order in this complete group.Other groups may have composite-order elements;this model does not deny them or settle existence of subgroups of that size."
      },
      "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": [
      8,
      8,
      [
        [
          0,
          2
        ],
        [
          1,
          2
        ],
        [
          2,
          2
        ],
        [
          3,
          1
        ],
        [
          4,
          2
        ],
        [
          5,
          2
        ],
        [
          6,
          2
        ],
        [
          7,
          2
        ]
      ],
      true,
      false,
      "composite_divisor_need_not_be_an_element_order"
    ]
  },
  "conclusion": {
    "canonical_value": "(8, 8, [[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations/questions.jsonl",
      "batch_sha256": "fab0187f15949aba06bf6ee6e60f81983e81907a1fb62931b50bc84c8521e4c6",
      "original_line": 83,
      "original_line_sha256": "516b1dff20e84becfa253ff4483f6e9073e6667d158db6dfcf2f40dec666a44d",
      "question_compact_sha256": "b25b500019846d3ae3959a9d9f935c3b8cfcfb848ad1ff59427db2dca030fb46",
      "english_row_compact_sha256": "9345e7cd0b2a462f1c35c0c9e90c2515fb8726ce80c09bcacc8b9979f4b61d0a",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-cauchy-theorem-finite-groups": {
        "sha256": "779e8ef831907c123fc1e7039941b328b38b54af646d6a171eab7e28b476927d",
        "english_sha256": "557db8e818ba9af78d5bbcd15fb786238f5f2099b296f105fa26a68e62fd821f",
        "original_path": "public/data/mko/cauchy-theorem-finite-groups.json",
        "english_path": "public/data/i18n/editions/mko-cauchy-theorem-finite-groups.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "9732ef02cb296d2e89f9b6cade5a76a83e9d37bdcf6a3de7f8613ebce02311e8",
    "original_candidate_sha256": "4c7347446811389737992642aa9a24f99343822c3b61d50bb8373e27bdc79a7f",
    "original_candidate_line_sha256": "4c7347446811389737992642aa9a24f99343822c3b61d50bb8373e27bdc79a7f"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-009-p46",
    "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": "357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438",
      "question_first_witness_sha256": "99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 101,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "5c569dea47f831298b178682f36092e7fe8ad8ab94be7848ef948516f91540dd",
      "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."
  }
}