{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261022-r05-finite-group-0034",
  "pilot_number": "R0057-29209",
  "number": 29209,
  "family_id": "qf-str22-group-associativity-audit",
  "title": {
    "zh-Hant": "第 29209 題：完整有限群表",
    "en": "Question 29209: Exhaustive Associativity over All Ordered Triples"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261022-r05-finite-group-0034",
    "batch_id": "batch-20261013-advance-20261022-r05-finite-group-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-group"
    ],
    "title_zh": "完整有限群表",
    "stem_zh": "完整有限Cayley表。操作=group-associativity-audit；資料={\"n\":5,\"table\":[[4,2,3,0,1],[2,0,4,1,3],[3,4,1,2,0],[0,1,2,3,4],[1,3,0,4,2]]}。請依 [n,table,bad_triples,associative] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], [], 是)",
      "explanation_zh": "結構基礎核對=(5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], [], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；結合律逐三元組核對；抽樣不證群公理。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "完整有限群表"
      ]
    },
    "variant": {
      "family_id": "qf-str22-group-associativity-audit",
      "seed": 202610225034,
      "parameters": {
        "n": 5,
        "table": [
          [
            4,
            2,
            3,
            0,
            1
          ],
          [
            2,
            0,
            4,
            1,
            3
          ],
          [
            3,
            4,
            1,
            2,
            0
          ],
          [
            0,
            1,
            2,
            3,
            4
          ],
          [
            1,
            3,
            0,
            4,
            2
          ]
        ],
        "domain": "Complete_finite_operation_table",
        "operation": "group-associativity-audit",
        "semantic_template": "qf-str22-group-associativity-audit/{\"n\":5,\"table\":[[4,2,3,0,1],[2,0,4,1,3],[3,4,1,2,0],[0,1,2,3,4],[1,3,0,4,2]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261022-r05-finite-group-0034",
    "batch_id": "batch-20261013-advance-20261022-r05-finite-group-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "04b15d234f4b8cdac8beaa614fafc69fd6773342dde9568c23c3d389c08cbbba",
      "family_id": "qf-str22-group-associativity-audit",
      "seed": 202610225034,
      "semantic_template": "qf-str22-group-associativity-audit/{\"n\":5,\"table\":[[4,2,3,0,1],[2,0,4,1,3],[3,4,1,2,0],[0,1,2,3,4],[1,3,0,4,2]]}"
    },
    "fields": {
      "title": "Exhaustive Associativity over All Ordered Triples",
      "stem": "The complete finite Cayley table=[[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.",
      "answer": {
        "kind": "expression",
        "value": "(5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], [], Yes)",
        "explanation": "Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], [], Yes)."
      },
      "skills": [
        "Exhaustive Associativity over All Ordered Triples"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "完整有限Cayley表。操作=group-associativity-audit；資料={\"n\":5,\"table\":[[4,2,3,0,1],[2,0,4,1,3],[3,4,1,2,0],[0,1,2,3,4],[1,3,0,4,2]]}。請依 [n,table,bad_triples,associative] 欄序回答。",
        "en": "The complete finite Cayley table=[[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-str22-group-associativity-audit, group-associativity-audit]",
      "after": "derived = [[n, table, bad_triples, associative]]",
      "relation": "implies",
      "rule_id": "STR22_exact_interface_and_source_ORDERED_tuple_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此銀行依原tuple欄序與hyphen operation。fraction約分、正分母且整數省略/1，與前輪舊契約不同；不改原中英快照或題意。",
        "en": "Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [5, 0, [4, 2, 3, 0, 1]]",
      "after": "derived = [[是, 是, 是, 是, 是]]",
      "relation": "implies",
      "rule_id": "Group_EVERY_original_row_closure",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對這個完整Cayley row每一格檢查結果仍在Fin(n)；保row/column標籤。closure只是可作有限binary operation，未證其他群公理。",
        "en": "Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [5, 1, [2, 0, 4, 1, 3]]",
      "after": "derived = [[是, 是, 是, 是, 是]]",
      "relation": "implies",
      "rule_id": "Group_EVERY_original_row_closure",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對這個完整Cayley row每一格檢查結果仍在Fin(n)；保row/column標籤。closure只是可作有限binary operation，未證其他群公理。",
        "en": "Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [5, 2, [3, 4, 1, 2, 0]]",
      "after": "derived = [[是, 是, 是, 是, 是]]",
      "relation": "implies",
      "rule_id": "Group_EVERY_original_row_closure",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對這個完整Cayley row每一格檢查結果仍在Fin(n)；保row/column標籤。closure只是可作有限binary operation，未證其他群公理。",
        "en": "Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [5, 3, [0, 1, 2, 3, 4]]",
      "after": "derived = [[是, 是, 是, 是, 是]]",
      "relation": "implies",
      "rule_id": "Group_EVERY_original_row_closure",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對這個完整Cayley row每一格檢查結果仍在Fin(n)；保row/column標籤。closure只是可作有限binary operation，未證其他群公理。",
        "en": "Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [5, 4, [1, 3, 0, 4, 2]]",
      "after": "derived = [[是, 是, 是, 是, 是]]",
      "relation": "implies",
      "rule_id": "Group_EVERY_original_row_closure",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對這個完整Cayley row每一格檢查結果仍在Fin(n)；保row/column標籤。closure只是可作有限binary operation，未證其他群公理。",
        "en": "Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 0, 0]",
      "after": "derived = [[1, 3, 0, 4, 2], [1, 3, 0, 4, 2], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 0, 1]",
      "after": "derived = [[3, 4, 1, 2, 0], [3, 4, 1, 2, 0], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 0, 2]",
      "after": "derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 0, 3]",
      "after": "derived = [[4, 2, 3, 0, 1], [4, 2, 3, 0, 1], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 0, 4]",
      "after": "derived = [[2, 0, 4, 1, 3], [2, 0, 4, 1, 3], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 1, 0]",
      "after": "derived = [[3, 4, 1, 2, 0], [3, 4, 1, 2, 0], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 1, 1]",
      "after": "derived = [[4, 2, 3, 0, 1], [4, 2, 3, 0, 1], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 1, 2]",
      "after": "derived = [[1, 3, 0, 4, 2], [1, 3, 0, 4, 2], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 1, 3]",
      "after": "derived = [[2, 0, 4, 1, 3], [2, 0, 4, 1, 3], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 1, 4]",
      "after": "derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 2, 0]",
      "after": "derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 2, 1]",
      "after": "derived = [[1, 3, 0, 4, 2], [1, 3, 0, 4, 2], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 2, 2]",
      "after": "derived = [[2, 0, 4, 1, 3], [2, 0, 4, 1, 3], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 2, 3]",
      "after": "derived = [[3, 4, 1, 2, 0], [3, 4, 1, 2, 0], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 2, 4]",
      "after": "derived = [[4, 2, 3, 0, 1], [4, 2, 3, 0, 1], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 3, 0]",
      "after": "derived = [[4, 2, 3, 0, 1], [4, 2, 3, 0, 1], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 3, 1]",
      "after": "derived = [[2, 0, 4, 1, 3], [2, 0, 4, 1, 3], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 3, 2]",
      "after": "derived = [[3, 4, 1, 2, 0], [3, 4, 1, 2, 0], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 3, 3]",
      "after": "derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 3, 4]",
      "after": "derived = [[1, 3, 0, 4, 2], [1, 3, 0, 4, 2], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 4, 0]",
      "after": "derived = [[2, 0, 4, 1, 3], [2, 0, 4, 1, 3], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 4, 1]",
      "after": "derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 4, 2]",
      "after": "derived = [[4, 2, 3, 0, 1], [4, 2, 3, 0, 1], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 4, 3]",
      "after": "derived = [[1, 3, 0, 4, 2], [1, 3, 0, 4, 2], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], 4, 4]",
      "after": "derived = [[3, 4, 1, 2, 0], [3, 4, 1, 2, 0], []]",
      "relation": "implies",
      "rule_id": "Group_ALL_third_coordinates_for_EACH_ordered_first_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定原(a,b)，完整列c=0..n-1兩種括號(a*b)*c與a*(b*c)及全部失敗c；包含重複元素，不抽樣、不只列首反例。",
        "en": "For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "source roles = [5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]]]",
      "after": "derived = [[], 是]",
      "relation": "implies",
      "rule_id": "Group_ALL_bad_triples_nested_source_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "雙層(a,b)配每個c已涵蓋n³ ordered triples；依nested升序保全部bad list，空list才是完整associative。有限完整表的結論不外推未知structure。",
        "en": "The(a,b) loops and ALL c cover n³ ordered triples.Retain ALL bad triples in nested increasing order;only an empty list certifies associativity of THIS complete finite table,not an unknown structure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。",
          "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      5,
      [
        [
          4,
          2,
          3,
          0,
          1
        ],
        [
          2,
          0,
          4,
          1,
          3
        ],
        [
          3,
          4,
          1,
          2,
          0
        ],
        [
          0,
          1,
          2,
          3,
          4
        ],
        [
          1,
          3,
          0,
          4,
          2
        ]
      ],
      [],
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], [], 是)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261013-advance-20261022-r05-finite-group-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261022-r05-finite-group-foundations/questions.jsonl",
      "batch_sha256": "77b65df8e9a0d3e23c7ae294869da93a08c0d74f9ac38b442deaa7b2a3673f58",
      "original_line": 34,
      "original_line_sha256": "4e6aa14b9c14299bac2a92d2f2cfe6ebaa8e49d08905d14cd4a8e7bde3515cde",
      "question_compact_sha256": "04b15d234f4b8cdac8beaa614fafc69fd6773342dde9568c23c3d389c08cbbba",
      "english_row_compact_sha256": "7a8d12265f82f93d9d4029ca6e004aea034628090ac5a0799dc1d575611fa3c0",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261022-r05-finite-group-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261022-r05-finite-group-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-group": {
        "sha256": "491d28b1773ac105187d1f17664e8e55b1ab4831613c354693901e40fae75f01",
        "english_sha256": "dc483579d967de45b346797aa7d8b882baaf395c1c61b932fcf9072babf13bc2",
        "original_path": "public/data/mko/finite-group.json",
        "english_path": "public/data/i18n/editions/mko-finite-group.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "cd714b09e116fbc0088007895c4d956bc0f4fad5a61507e4a16f09649d05a3b0",
    "original_candidate_sha256": "f2c874a742e576f87b35903f2ba8cf6937f29541f85447ab1a828cf96395cf06",
    "original_candidate_line_sha256": "f2c874a742e576f87b35903f2ba8cf6937f29541f85447ab1a828cf96395cf06"
  },
  "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-20261026-011-p58",
    "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": "db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6",
      "question_first_witness_sha256": "58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 32,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "386ea26d733b46e6110c98b4c0f7c3cc97330e5ff3456e20fc9af8b8a594ce82",
      "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."
  }
}