{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261107-r01-finite-group-action-0068",
  "pilot_number": "R0039-20118",
  "number": 20118,
  "family_id": "qf-mod38-groupaction-compatibility-audit",
  "title": {
    "zh-Hant": "第 20118 題：有限群作用",
    "en": "Question 20118: All Group-Group-Point Triples Must Satisfy the Same Action Law"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261107-r01-finite-group-action-0068",
    "batch_id": "batch-20261014-advance-20261107-r01-finite-group-action-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-group-action"
    ],
    "title_zh": "有限群作用",
    "stem_zh": "完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(112, null, 是, all_g_h_x_checked)",
      "explanation_zh": "Wave6根層核對=(112, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限群作用"
      ]
    },
    "variant": {
      "family_id": "qf-mod38-groupaction-compatibility-audit",
      "seed": 202611071068,
      "parameters": {
        "group_table": [
          [
            0,
            1,
            2,
            3
          ],
          [
            1,
            2,
            3,
            0
          ],
          [
            2,
            3,
            0,
            1
          ],
          [
            3,
            0,
            1,
            2
          ]
        ],
        "identity": 0,
        "point_count": 7,
        "action_table": [
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6
          ],
          [
            1,
            2,
            3,
            0,
            4,
            5,
            6
          ],
          [
            2,
            3,
            0,
            1,
            4,
            5,
            6
          ],
          [
            3,
            0,
            1,
            2,
            4,
            5,
            6
          ]
        ],
        "domain": "Complete_finite_group_action_tables",
        "operation": "groupaction-compatibility-audit",
        "semantic_template": "qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7}"
      }
    },
    "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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0068",
    "batch_id": "batch-20261014-advance-20261107-r01-finite-group-action-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "64d262a49a2609930ad12daefc6fd1e7c6080bacbb59943ae1b65661af9249c5",
      "family_id": "qf-mod38-groupaction-compatibility-audit",
      "seed": 202611071068,
      "semantic_template": "qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7}"
    },
    "fields": {
      "title": "All Group-Group-Point Triples Must Satisfy the Same Action Law",
      "stem": "Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(112, null, Yes, all_g_h_x_checked)",
        "explanation": "Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (112, null, Yes, all_g_h_x_checked)."
      },
      "skills": [
        "All Group-Group-Point Triples Must Satisfy the Same Action Law"
      ],
      "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Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=groupaction-compatibility-audit;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[triple_count, first_failure, compatible, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。欄位標籤不是額外數學證據，No保持原題的聲稱範圍。",
        "en": "Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "ORIGINAL complete group-label carrier Fin(4);declared identity label=0;Cayley table row g,column h gives g*h=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]]",
      "after": "ORIGINAL complete point carrier Fin(7);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]];all entries in range,so structural closure/totality checked;GROUP AXIOMS not assumed from shape",
      "relation": "implies",
      "rule_id": "groupaction_two_table_roles_complete_candidate_not_group_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩張表的column角色不同，一張h是群標籤，一張x是作用點。in-range完整表只保closed／total，群候選可不滿足公理，不能因叫group_table就先填True。",
        "en": "The two tables have different column roles:group label h versus action point x.Complete in-range tables establish closure/totality only.A candidate group may fail its axioms;its table name does not certify them."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[0, 0];actual group product=0;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[0, 1];actual group product=1;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 0, 0, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[0, 2];actual group product=2;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[0, 3];actual group product=3;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[1, 0];actual group product=1;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 0, 0, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[1, 1];actual group product=2;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[1, 2];actual group product=3;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[1, 3];actual group product=0;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[2, 0];actual group product=2;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[2, 1];actual group product=3;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[2, 2];actual group product=0;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[2, 3];actual group product=1;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 0, 0, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[3, 0];actual group product=3;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[3, 1];actual group product=0;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[3, 2];actual group product=1;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 0, 0, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "ORIGINAL ordered group pair(g,h)=[3, 3];actual group product=2;action table roles unchanged",
      "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是]];earliest failure retained,not overwritten",
      "relation": "implies",
      "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。",
        "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "group action ordered group-pair count := mul(4, 4)",
      "after": "group action ordered group-pair count = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 group action ordered group-pair count，不以小數近似替代。",
          "en": "Recompute group action ordered group-pair count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "4"
        ],
        "output": "16",
        "label": "group action ordered group-pair count"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "group action complete triple-domain count := mul(16, 7)",
      "after": "group action complete triple-domain count = 112",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 group action complete triple-domain count，不以小數近似替代。",
          "en": "Recompute group action complete triple-domain count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "7"
        ],
        "output": "112",
        "label": "group action complete triple-domain count"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "COMPLETE triple domain order^2*point_count=112;this derivation ACTUALLY checked all112 triples",
      "after": "compatibility=是;FIRST source-order failure[g,h,x,left,right]=null;null means NO failure anywhere",
      "relation": "implies",
      "rule_id": "groupaction_full_domain_count_first_failure_not_early_stop_measurement",
      "conditions": [],
      "explanation": {
        "zh-Hant": "triple_count是完整量詞域大小，不是把原型在第一錯誤處停止的迭代數冒充測量。本推導確實列了每對g,h的全部點，所以null給全域通過，首失敗則保確切資料與順序。",
        "en": "The triple_count is the complete quantified domain size,not a measurement of an early-stopping prototype.This derivation actually checks every point for every g,h pair,so null certifies full compatibility and a first failure retains exact source data/order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "this component audited ALL group-group-point compatibility triples",
      "after": "compatibility flag alone does NOT establish identity-action or group qualification",
      "relation": "implies",
      "rule_id": "groupaction_compatibility_component_not_full_action_claim",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整相容性是其中一項責任，不能代替e*x=x或群表資格。即使COMPAT通過，這題仍不額外聲稱整體action有效。",
        "en": "Complete compatibility is one obligation,not a replacement for e*x=x or group qualification.Even passing this component does not additionally claim a valid action."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      112,
      null,
      true,
      "all_g_h_x_checked"
    ]
  },
  "conclusion": {
    "canonical_value": "(112, null, 是, all_g_h_x_checked)",
    "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-20261107-r01-finite-group-action-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261107-r01-finite-group-action-foundations/questions.jsonl",
      "batch_sha256": "fa5acca0cbac7a9c6e87334ae4261fa3e0ac4c1cb2ba0e64150ffc4f96129b7a",
      "original_line": 68,
      "original_line_sha256": "e6231df018ca607ef69362e5a7a349c2cc8b3b8a59164250b811280cc3e19613",
      "question_compact_sha256": "64d262a49a2609930ad12daefc6fd1e7c6080bacbb59943ae1b65661af9249c5",
      "english_row_compact_sha256": "9ce4672c698453e67ca2ac40b2f0b22178850468ec437d44c0dbb9c648853357",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261107-r01-finite-group-action-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261107-r01-finite-group-action-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-group-action": {
        "sha256": "2e31b8a871f6fd32317885aa68f63e2380a6c3a7d97c031b152b653c6a879de0",
        "english_sha256": "d2fb382c41e0c0388963c9f896b8d1cc26ad6fce451783c0262bb3ae5312cc51",
        "original_path": "public/data/mko/finite-group-action.json",
        "english_path": "public/data/i18n/editions/mko-finite-group-action.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "521929dfe9b0919224c38312ad78e5fa56457fc886faaa12b137edd83962251e",
    "original_candidate_sha256": "8df190b5119a1a13d63764869a4d567577dfc74a7604baa99ed4322452a803dd",
    "original_candidate_line_sha256": "8df190b5119a1a13d63764869a4d567577dfc74a7604baa99ed4322452a803dd"
  },
  "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-20261026-008-p40",
    "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": "6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034",
      "question_first_witness_sha256": "cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 22,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "d1caf0a5ae1fab9d730f1a328a53cb33236ebba42e167f7476f57bea0a015a2e",
      "arithmetic_steps": 2
    },
    "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."
  }
}