{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261114-r02-finite-quotient-group-0062",
  "pilot_number": "R0049-25387",
  "number": 25387,
  "family_id": "qf-mod45-quotient-representative-well-definedness",
  "title": {
    "zh-Hant": "第 25387 題：有限商群",
    "en": "Question 25387: Every Representative Pair Must Land in the Same Declared Product Coset"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261114-r02-finite-quotient-group-0062",
    "batch_id": "batch-20261015-advance-20261114-r02-finite-quotient-group-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-quotient-group"
    ],
    "title_zh": "有限商群",
    "stem_zh": "完整有限群、subgroup與全部coset partition及代表元責任。操作=quotient-representative-well-definedness；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"identity\":0,\"subset\":[0,2,4,6,8,10,12,14,16,18,20,22,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]], [[0, 0, [0], 是], [0, 1, [1], 是], [1, 0, [1], 是], [1, 1, [0], 是]], 是, every_representative_pair_lands_in_one_declared_product_coset)",
      "explanation_zh": "Wave6第八層核對=([[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]], [[0, 0, [0], 是], [0, 1, [1], 是], [1, 0, [1], 是], [1, 1, [0], 是]], 是, every_representative_pair_lands_in_one_declared_product_coset)；依完整明示資料獨立重算；quotient元素是coset sets；乘法對所有代表元替換不變並保存identity與inverse；nonnormal代表元依賴會阻止商群；不自動commutative或宣稱第一同構定理。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "有限商群"
      ]
    },
    "variant": {
      "family_id": "qf-mod45-quotient-representative-well-definedness",
      "seed": 202611142062,
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      "warning_zh": "候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"
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      "family_id": "qf-mod45-quotient-representative-well-definedness",
      "seed": 202611142062,
      "semantic_template": "qf-mod45-quotient-representative-well-definedness/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"identity\":0,\"subset\":[0,2,4,6,8,10,12,14,16,18,20,22,24]}"
    },
    "fields": {
      "title": "Every Representative Pair Must Land in the Same Declared Product Coset",
      "stem": "Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], identity=0, explicit subset=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24] and the entire left-coset partition. Subgroup qualification precedes normality. Quotient elements are coset sets, not representatives; coset indices follow the source canonical element-list order. Give (cosets, representative_rows, well_defined, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]], [[0, 0, [0], Yes], [0, 1, [1], Yes], [1, 0, [1], Yes], [1, 1, [0], Yes]], Yes, every_representative_pair_lands_in_one_declared_product_coset)",
        "explanation": "For every ordered coset pair, list [left index,right index,all possible product-coset indices,unique]. Exhaust every representative combination. A table produced from only the least representatives is not a substitute for well-definedness. The exact result, retaining the requested order, is ([[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]], [[0, 0, [0], Yes], [0, 1, [1], Yes], [1, 0, [1], Yes], [1, 1, [0], Yes]], Yes, every_representative_pair_lands_in_one_declared_product_coset)."
      },
      "skills": [
        "Every Representative Pair Must Land in the Same Declared Product Coset"
      ],
      "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、subgroup與全部coset partition及代表元責任。操作=quotient-representative-well-definedness；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"identity\":0,\"subset\":[0,2,4,6,8,10,12,14,16,18,20,22,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], identity=0, explicit subset=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24] and the entire left-coset partition. Subgroup qualification precedes normality. Quotient elements are coset sets, not representatives; coset indices follow the source canonical element-list order. Give (cosets, representative_rows, well_defined, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [quotient-representative-well-definedness]",
      "after": "derived = [[cosets, representative_rows, well_defined, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本頁列完整來源欄序，不需JavaScript；LP三種mode雖同為6欄，角色仍由原mode選，不由答案形狀猜。",
        "en": "List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape."
      },
      "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": "source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], 0]",
      "after": "derived = [26, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11], [12, 12, 12, 12], [13, 13, 13, 13], [14, 14, 14, 14], [15, 15, 15, 15], [16, 16, 16, 16], [17, 17, 17, 17], [18, 18, 18, 18], [19, 19, 19, 19], [20, 20, 20, 20], [21, 21, 21, 21], [22, 22, 22, 22], [23, 23, 23, 23], [24, 24, 24, 24], [25, 25, 25, 25]], 是]",
      "relation": "implies",
      "rule_id": "Quotient_complete_group_table_actual_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核整個closed Cayley table與實際identity標籤，不假設0或套舊上限32。",
        "en": "Validate the complete closed Cayley table and actual identity label,without assuming0 or the old32-element cap."
      },
      "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": "source roles = [26, 0]",
      "after": "derived = [[[是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], 17576, [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]]",
      "relation": "implies",
      "rule_id": "Quotient_ALL_group_associativity_triples_and_two_sided_inverses",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個(x,y)矩陣格核全部z的(xy)z=x(yz)，共n³責任；每個inverse同時左右乘回同一identity。完整查核不靠抽樣。",
        "en": "Each pair-matrix entry checks associativity for every z,covering n^3 obligations;each inverse returns the same identity from both sides.This is exhaustive,not sampling."
      },
      "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": "source roles = [0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]]",
      "after": "derived = [[[0, 0, 0, 是], [0, 2, 2, 是], [0, 4, 4, 是], [0, 6, 6, 是], [0, 8, 8, 是], [0, 10, 10, 是], [0, 12, 12, 是], [0, 14, 14, 是], [0, 16, 16, 是], [0, 18, 18, 是], [0, 20, 20, 是], [0, 22, 22, 是], [0, 24, 24, 是], [2, 0, 2, 是], [2, 2, 4, 是], [2, 4, 6, 是], [2, 6, 8, 是], [2, 8, 10, 是], [2, 10, 12, 是], [2, 12, 14, 是], [2, 14, 16, 是], [2, 16, 18, 是], [2, 18, 20, 是], [2, 20, 22, 是], [2, 22, 24, 是], [2, 24, 0, 是], [4, 0, 4, 是], [4, 2, 6, 是], [4, 4, 8, 是], [4, 6, 10, 是], [4, 8, 12, 是], [4, 10, 14, 是], [4, 12, 16, 是], [4, 14, 18, 是], [4, 16, 20, 是], [4, 18, 22, 是], [4, 20, 24, 是], [4, 22, 0, 是], [4, 24, 2, 是], [6, 0, 6, 是], [6, 2, 8, 是], [6, 4, 10, 是], [6, 6, 12, 是], [6, 8, 14, 是], [6, 10, 16, 是], [6, 12, 18, 是], [6, 14, 20, 是], [6, 16, 22, 是], [6, 18, 24, 是], [6, 20, 0, 是], [6, 22, 2, 是], [6, 24, 4, 是], [8, 0, 8, 是], [8, 2, 10, 是], [8, 4, 12, 是], [8, 6, 14, 是], [8, 8, 16, 是], [8, 10, 18, 是], [8, 12, 20, 是], [8, 14, 22, 是], [8, 16, 24, 是], [8, 18, 0, 是], [8, 20, 2, 是], [8, 22, 4, 是], [8, 24, 6, 是], [10, 0, 10, 是], [10, 2, 12, 是], [10, 4, 14, 是], [10, 6, 16, 是], [10, 8, 18, 是], [10, 10, 20, 是], [10, 12, 22, 是], [10, 14, 24, 是], [10, 16, 0, 是], [10, 18, 2, 是], [10, 20, 4, 是], [10, 22, 6, 是], [10, 24, 8, 是], [12, 0, 12, 是], [12, 2, 14, 是], [12, 4, 16, 是], [12, 6, 18, 是], [12, 8, 20, 是], [12, 10, 22, 是], [12, 12, 24, 是], [12, 14, 0, 是], [12, 16, 2, 是], [12, 18, 4, 是], [12, 20, 6, 是], [12, 22, 8, 是], [12, 24, 10, 是], [14, 0, 14, 是], [14, 2, 16, 是], [14, 4, 18, 是], [14, 6, 20, 是], [14, 8, 22, 是], [14, 10, 24, 是], [14, 12, 0, 是], [14, 14, 2, 是], [14, 16, 4, 是], [14, 18, 6, 是], [14, 20, 8, 是], [14, 22, 10, 是], [14, 24, 12, 是], [16, 0, 16, 是], [16, 2, 18, 是], [16, 4, 20, 是], [16, 6, 22, 是], [16, 8, 24, 是], [16, 10, 0, 是], [16, 12, 2, 是], [16, 14, 4, 是], [16, 16, 6, 是], [16, 18, 8, 是], [16, 20, 10, 是], [16, 22, 12, 是], [16, 24, 14, 是], [18, 0, 18, 是], [18, 2, 20, 是], [18, 4, 22, 是], [18, 6, 24, 是], [18, 8, 0, 是], [18, 10, 2, 是], [18, 12, 4, 是], [18, 14, 6, 是], [18, 16, 8, 是], [18, 18, 10, 是], [18, 20, 12, 是], [18, 22, 14, 是], [18, 24, 16, 是], [20, 0, 20, 是], [20, 2, 22, 是], [20, 4, 24, 是], [20, 6, 0, 是], [20, 8, 2, 是], [20, 10, 4, 是], [20, 12, 6, 是], [20, 14, 8, 是], [20, 16, 10, 是], [20, 18, 12, 是], [20, 20, 14, 是], [20, 22, 16, 是], [20, 24, 18, 是], [22, 0, 22, 是], [22, 2, 24, 是], [22, 4, 0, 是], [22, 6, 2, 是], [22, 8, 4, 是], [22, 10, 6, 是], [22, 12, 8, 是], [22, 14, 10, 是], [22, 16, 12, 是], [22, 18, 14, 是], [22, 20, 16, 是], [22, 22, 18, 是], [22, 24, 20, 是], [24, 0, 24, 是], [24, 2, 0, 是], [24, 4, 2, 是], [24, 6, 4, 是], [24, 8, 6, 是], [24, 10, 8, 是], [24, 12, 10, 是], [24, 14, 12, 是], [24, 16, 14, 是], [24, 18, 16, 是], [24, 20, 18, 是], [24, 22, 20, 是], [24, 24, 22, 是]], [[0, 0, 是], [2, 24, 是], [4, 22, 是], [6, 20, 是], [8, 18, 是], [10, 16, 是], [12, 14, 是], [14, 12, 是], [16, 10, 是], [18, 8, 是], [20, 6, 是], [22, 4, 是], [24, 2, 是]], 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]]], 是]",
      "relation": "implies",
      "rule_id": "Quotient_genuine_subgroup_before_actual_normality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先含identity、全product與inverse closure成真subgroup；再對所有g核共軛H。positive branches須normal，boundary須真正nonnormal，不混同未成subgroup。",
        "en": "First qualify identity,all products and inverse closure as a genuine subgroup,then test conjugation for every g.Positive branches require normality and the boundary requires actual nonnormality,not failed subgroup 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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]]",
      "after": "derived = [[[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], 是]",
      "relation": "implies",
      "rule_id": "Quotient_ALL_LEFT_coset_SETS_complete_disjoint_partition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個元素g在gH中；若aH交bH，ah1=bh2給b^-1a∈H，從而兩完整coset相同。列全g後去重且依來源排序，元素是coset SETS，不是最小代表元本身。",
        "en": "Every g belongs to gH.If aH meets bH,ah1=bh2 implies b^-1a in H and the complete cosets coincide.Enumerate every g,deduplicate and retain source order.Quotient elements are coset SETS,not least representatives."
      },
      "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": "source roles = [0, 0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]]",
      "after": "derived = [[[0, 0, 0, 0], [0, 2, 2, 0], [0, 4, 4, 0], [0, 6, 6, 0], [0, 8, 8, 0], [0, 10, 10, 0], [0, 12, 12, 0], [0, 14, 14, 0], [0, 16, 16, 0], [0, 18, 18, 0], [0, 20, 20, 0], [0, 22, 22, 0], [0, 24, 24, 0], [2, 0, 2, 0], [2, 2, 4, 0], [2, 4, 6, 0], [2, 6, 8, 0], [2, 8, 10, 0], [2, 10, 12, 0], [2, 12, 14, 0], [2, 14, 16, 0], [2, 16, 18, 0], [2, 18, 20, 0], [2, 20, 22, 0], [2, 22, 24, 0], [2, 24, 0, 0], [4, 0, 4, 0], [4, 2, 6, 0], [4, 4, 8, 0], [4, 6, 10, 0], [4, 8, 12, 0], [4, 10, 14, 0], [4, 12, 16, 0], [4, 14, 18, 0], [4, 16, 20, 0], [4, 18, 22, 0], [4, 20, 24, 0], [4, 22, 0, 0], [4, 24, 2, 0], [6, 0, 6, 0], [6, 2, 8, 0], [6, 4, 10, 0], [6, 6, 12, 0], [6, 8, 14, 0], [6, 10, 16, 0], [6, 12, 18, 0], [6, 14, 20, 0], [6, 16, 22, 0], [6, 18, 24, 0], [6, 20, 0, 0], [6, 22, 2, 0], [6, 24, 4, 0], [8, 0, 8, 0], [8, 2, 10, 0], [8, 4, 12, 0], [8, 6, 14, 0], [8, 8, 16, 0], [8, 10, 18, 0], [8, 12, 20, 0], [8, 14, 22, 0], [8, 16, 24, 0], [8, 18, 0, 0], [8, 20, 2, 0], [8, 22, 4, 0], [8, 24, 6, 0], [10, 0, 10, 0], [10, 2, 12, 0], [10, 4, 14, 0], [10, 6, 16, 0], [10, 8, 18, 0], [10, 10, 20, 0], [10, 12, 22, 0], [10, 14, 24, 0], [10, 16, 0, 0], [10, 18, 2, 0], [10, 20, 4, 0], [10, 22, 6, 0], [10, 24, 8, 0], [12, 0, 12, 0], [12, 2, 14, 0], [12, 4, 16, 0], [12, 6, 18, 0], [12, 8, 20, 0], [12, 10, 22, 0], [12, 12, 24, 0], [12, 14, 0, 0], [12, 16, 2, 0], [12, 18, 4, 0], [12, 20, 6, 0], [12, 22, 8, 0], [12, 24, 10, 0], [14, 0, 14, 0], [14, 2, 16, 0], [14, 4, 18, 0], [14, 6, 20, 0], [14, 8, 22, 0], [14, 10, 24, 0], [14, 12, 0, 0], [14, 14, 2, 0], [14, 16, 4, 0], [14, 18, 6, 0], [14, 20, 8, 0], [14, 22, 10, 0], [14, 24, 12, 0], [16, 0, 16, 0], [16, 2, 18, 0], [16, 4, 20, 0], [16, 6, 22, 0], [16, 8, 24, 0], [16, 10, 0, 0], [16, 12, 2, 0], [16, 14, 4, 0], [16, 16, 6, 0], [16, 18, 8, 0], [16, 20, 10, 0], [16, 22, 12, 0], [16, 24, 14, 0], [18, 0, 18, 0], [18, 2, 20, 0], [18, 4, 22, 0], [18, 6, 24, 0], [18, 8, 0, 0], [18, 10, 2, 0], [18, 12, 4, 0], [18, 14, 6, 0], [18, 16, 8, 0], [18, 18, 10, 0], [18, 20, 12, 0], [18, 22, 14, 0], [18, 24, 16, 0], [20, 0, 20, 0], [20, 2, 22, 0], [20, 4, 24, 0], [20, 6, 0, 0], [20, 8, 2, 0], [20, 10, 4, 0], [20, 12, 6, 0], [20, 14, 8, 0], [20, 16, 10, 0], [20, 18, 12, 0], [20, 20, 14, 0], [20, 22, 16, 0], [20, 24, 18, 0], [22, 0, 22, 0], [22, 2, 24, 0], [22, 4, 0, 0], [22, 6, 2, 0], [22, 8, 4, 0], [22, 10, 6, 0], [22, 12, 8, 0], [22, 14, 10, 0], [22, 16, 12, 0], [22, 18, 14, 0], [22, 20, 16, 0], [22, 22, 18, 0], [22, 24, 20, 0], [24, 0, 24, 0], [24, 2, 0, 0], [24, 4, 2, 0], [24, 6, 4, 0], [24, 8, 6, 0], [24, 10, 8, 0], [24, 12, 10, 0], [24, 14, 12, 0], [24, 16, 14, 0], [24, 18, 16, 0], [24, 20, 18, 0], [24, 22, 20, 0], [24, 24, 22, 0]], [0], 是]",
      "relation": "implies",
      "rule_id": "Quotient_EVERY_coset_pair_ALL_representative_products",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每組coset的每對代表元都要落在同一product coset，僅用首代表元造表還不足以證良定；保完整可能index集合。",
        "en": "For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices."
      },
      "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": "source roles = [0, 1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]",
      "after": "derived = [[[0, 1, 1, 1], [0, 3, 3, 1], [0, 5, 5, 1], [0, 7, 7, 1], [0, 9, 9, 1], [0, 11, 11, 1], [0, 13, 13, 1], [0, 15, 15, 1], [0, 17, 17, 1], [0, 19, 19, 1], [0, 21, 21, 1], [0, 23, 23, 1], [0, 25, 25, 1], [2, 1, 3, 1], [2, 3, 5, 1], [2, 5, 7, 1], [2, 7, 9, 1], [2, 9, 11, 1], [2, 11, 13, 1], [2, 13, 15, 1], [2, 15, 17, 1], [2, 17, 19, 1], [2, 19, 21, 1], [2, 21, 23, 1], [2, 23, 25, 1], [2, 25, 1, 1], [4, 1, 5, 1], [4, 3, 7, 1], [4, 5, 9, 1], [4, 7, 11, 1], [4, 9, 13, 1], [4, 11, 15, 1], [4, 13, 17, 1], [4, 15, 19, 1], [4, 17, 21, 1], [4, 19, 23, 1], [4, 21, 25, 1], [4, 23, 1, 1], [4, 25, 3, 1], [6, 1, 7, 1], [6, 3, 9, 1], [6, 5, 11, 1], [6, 7, 13, 1], [6, 9, 15, 1], [6, 11, 17, 1], [6, 13, 19, 1], [6, 15, 21, 1], [6, 17, 23, 1], [6, 19, 25, 1], [6, 21, 1, 1], [6, 23, 3, 1], [6, 25, 5, 1], [8, 1, 9, 1], [8, 3, 11, 1], [8, 5, 13, 1], [8, 7, 15, 1], [8, 9, 17, 1], [8, 11, 19, 1], [8, 13, 21, 1], [8, 15, 23, 1], [8, 17, 25, 1], [8, 19, 1, 1], [8, 21, 3, 1], [8, 23, 5, 1], [8, 25, 7, 1], [10, 1, 11, 1], [10, 3, 13, 1], [10, 5, 15, 1], [10, 7, 17, 1], [10, 9, 19, 1], [10, 11, 21, 1], [10, 13, 23, 1], [10, 15, 25, 1], [10, 17, 1, 1], [10, 19, 3, 1], [10, 21, 5, 1], [10, 23, 7, 1], [10, 25, 9, 1], [12, 1, 13, 1], [12, 3, 15, 1], [12, 5, 17, 1], [12, 7, 19, 1], [12, 9, 21, 1], [12, 11, 23, 1], [12, 13, 25, 1], [12, 15, 1, 1], [12, 17, 3, 1], [12, 19, 5, 1], [12, 21, 7, 1], [12, 23, 9, 1], [12, 25, 11, 1], [14, 1, 15, 1], [14, 3, 17, 1], [14, 5, 19, 1], [14, 7, 21, 1], [14, 9, 23, 1], [14, 11, 25, 1], [14, 13, 1, 1], [14, 15, 3, 1], [14, 17, 5, 1], [14, 19, 7, 1], [14, 21, 9, 1], [14, 23, 11, 1], [14, 25, 13, 1], [16, 1, 17, 1], [16, 3, 19, 1], [16, 5, 21, 1], [16, 7, 23, 1], [16, 9, 25, 1], [16, 11, 1, 1], [16, 13, 3, 1], [16, 15, 5, 1], [16, 17, 7, 1], [16, 19, 9, 1], [16, 21, 11, 1], [16, 23, 13, 1], [16, 25, 15, 1], [18, 1, 19, 1], [18, 3, 21, 1], [18, 5, 23, 1], [18, 7, 25, 1], [18, 9, 1, 1], [18, 11, 3, 1], [18, 13, 5, 1], [18, 15, 7, 1], [18, 17, 9, 1], [18, 19, 11, 1], [18, 21, 13, 1], [18, 23, 15, 1], [18, 25, 17, 1], [20, 1, 21, 1], [20, 3, 23, 1], [20, 5, 25, 1], [20, 7, 1, 1], [20, 9, 3, 1], [20, 11, 5, 1], [20, 13, 7, 1], [20, 15, 9, 1], [20, 17, 11, 1], [20, 19, 13, 1], [20, 21, 15, 1], [20, 23, 17, 1], [20, 25, 19, 1], [22, 1, 23, 1], [22, 3, 25, 1], [22, 5, 1, 1], [22, 7, 3, 1], [22, 9, 5, 1], [22, 11, 7, 1], [22, 13, 9, 1], [22, 15, 11, 1], [22, 17, 13, 1], [22, 19, 15, 1], [22, 21, 17, 1], [22, 23, 19, 1], [22, 25, 21, 1], [24, 1, 25, 1], [24, 3, 1, 1], [24, 5, 3, 1], [24, 7, 5, 1], [24, 9, 7, 1], [24, 11, 9, 1], [24, 13, 11, 1], [24, 15, 13, 1], [24, 17, 15, 1], [24, 19, 17, 1], [24, 21, 19, 1], [24, 23, 21, 1], [24, 25, 23, 1]], [1], 是]",
      "relation": "implies",
      "rule_id": "Quotient_EVERY_coset_pair_ALL_representative_products",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每組coset的每對代表元都要落在同一product coset，僅用首代表元造表還不足以證良定；保完整可能index集合。",
        "en": "For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices."
      },
      "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": "source roles = [1, 0, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]]",
      "after": "derived = [[[1, 0, 1, 1], [1, 2, 3, 1], [1, 4, 5, 1], [1, 6, 7, 1], [1, 8, 9, 1], [1, 10, 11, 1], [1, 12, 13, 1], [1, 14, 15, 1], [1, 16, 17, 1], [1, 18, 19, 1], [1, 20, 21, 1], [1, 22, 23, 1], [1, 24, 25, 1], [3, 0, 3, 1], [3, 2, 5, 1], [3, 4, 7, 1], [3, 6, 9, 1], [3, 8, 11, 1], [3, 10, 13, 1], [3, 12, 15, 1], [3, 14, 17, 1], [3, 16, 19, 1], [3, 18, 21, 1], [3, 20, 23, 1], [3, 22, 25, 1], [3, 24, 1, 1], [5, 0, 5, 1], [5, 2, 7, 1], [5, 4, 9, 1], [5, 6, 11, 1], [5, 8, 13, 1], [5, 10, 15, 1], [5, 12, 17, 1], [5, 14, 19, 1], [5, 16, 21, 1], [5, 18, 23, 1], [5, 20, 25, 1], [5, 22, 1, 1], [5, 24, 3, 1], [7, 0, 7, 1], [7, 2, 9, 1], [7, 4, 11, 1], [7, 6, 13, 1], [7, 8, 15, 1], [7, 10, 17, 1], [7, 12, 19, 1], [7, 14, 21, 1], [7, 16, 23, 1], [7, 18, 25, 1], [7, 20, 1, 1], [7, 22, 3, 1], [7, 24, 5, 1], [9, 0, 9, 1], [9, 2, 11, 1], [9, 4, 13, 1], [9, 6, 15, 1], [9, 8, 17, 1], [9, 10, 19, 1], [9, 12, 21, 1], [9, 14, 23, 1], [9, 16, 25, 1], [9, 18, 1, 1], [9, 20, 3, 1], [9, 22, 5, 1], [9, 24, 7, 1], [11, 0, 11, 1], [11, 2, 13, 1], [11, 4, 15, 1], [11, 6, 17, 1], [11, 8, 19, 1], [11, 10, 21, 1], [11, 12, 23, 1], [11, 14, 25, 1], [11, 16, 1, 1], [11, 18, 3, 1], [11, 20, 5, 1], [11, 22, 7, 1], [11, 24, 9, 1], [13, 0, 13, 1], [13, 2, 15, 1], [13, 4, 17, 1], [13, 6, 19, 1], [13, 8, 21, 1], [13, 10, 23, 1], [13, 12, 25, 1], [13, 14, 1, 1], [13, 16, 3, 1], [13, 18, 5, 1], [13, 20, 7, 1], [13, 22, 9, 1], [13, 24, 11, 1], [15, 0, 15, 1], [15, 2, 17, 1], [15, 4, 19, 1], [15, 6, 21, 1], [15, 8, 23, 1], [15, 10, 25, 1], [15, 12, 1, 1], [15, 14, 3, 1], [15, 16, 5, 1], [15, 18, 7, 1], [15, 20, 9, 1], [15, 22, 11, 1], [15, 24, 13, 1], [17, 0, 17, 1], [17, 2, 19, 1], [17, 4, 21, 1], [17, 6, 23, 1], [17, 8, 25, 1], [17, 10, 1, 1], [17, 12, 3, 1], [17, 14, 5, 1], [17, 16, 7, 1], [17, 18, 9, 1], [17, 20, 11, 1], [17, 22, 13, 1], [17, 24, 15, 1], [19, 0, 19, 1], [19, 2, 21, 1], [19, 4, 23, 1], [19, 6, 25, 1], [19, 8, 1, 1], [19, 10, 3, 1], [19, 12, 5, 1], [19, 14, 7, 1], [19, 16, 9, 1], [19, 18, 11, 1], [19, 20, 13, 1], [19, 22, 15, 1], [19, 24, 17, 1], [21, 0, 21, 1], [21, 2, 23, 1], [21, 4, 25, 1], [21, 6, 1, 1], [21, 8, 3, 1], [21, 10, 5, 1], [21, 12, 7, 1], [21, 14, 9, 1], [21, 16, 11, 1], [21, 18, 13, 1], [21, 20, 15, 1], [21, 22, 17, 1], [21, 24, 19, 1], [23, 0, 23, 1], [23, 2, 25, 1], [23, 4, 1, 1], [23, 6, 3, 1], [23, 8, 5, 1], [23, 10, 7, 1], [23, 12, 9, 1], [23, 14, 11, 1], [23, 16, 13, 1], [23, 18, 15, 1], [23, 20, 17, 1], [23, 22, 19, 1], [23, 24, 21, 1], [25, 0, 25, 1], [25, 2, 1, 1], [25, 4, 3, 1], [25, 6, 5, 1], [25, 8, 7, 1], [25, 10, 9, 1], [25, 12, 11, 1], [25, 14, 13, 1], [25, 16, 15, 1], [25, 18, 17, 1], [25, 20, 19, 1], [25, 22, 21, 1], [25, 24, 23, 1]], [1], 是]",
      "relation": "implies",
      "rule_id": "Quotient_EVERY_coset_pair_ALL_representative_products",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每組coset的每對代表元都要落在同一product coset，僅用首代表元造表還不足以證良定；保完整可能index集合。",
        "en": "For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices."
      },
      "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": "source roles = [1, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]",
      "after": "derived = [[[1, 1, 2, 0], [1, 3, 4, 0], [1, 5, 6, 0], [1, 7, 8, 0], [1, 9, 10, 0], [1, 11, 12, 0], [1, 13, 14, 0], [1, 15, 16, 0], [1, 17, 18, 0], [1, 19, 20, 0], [1, 21, 22, 0], [1, 23, 24, 0], [1, 25, 0, 0], [3, 1, 4, 0], [3, 3, 6, 0], [3, 5, 8, 0], [3, 7, 10, 0], [3, 9, 12, 0], [3, 11, 14, 0], [3, 13, 16, 0], [3, 15, 18, 0], [3, 17, 20, 0], [3, 19, 22, 0], [3, 21, 24, 0], [3, 23, 0, 0], [3, 25, 2, 0], [5, 1, 6, 0], [5, 3, 8, 0], [5, 5, 10, 0], [5, 7, 12, 0], [5, 9, 14, 0], [5, 11, 16, 0], [5, 13, 18, 0], [5, 15, 20, 0], [5, 17, 22, 0], [5, 19, 24, 0], [5, 21, 0, 0], [5, 23, 2, 0], [5, 25, 4, 0], [7, 1, 8, 0], [7, 3, 10, 0], [7, 5, 12, 0], [7, 7, 14, 0], [7, 9, 16, 0], [7, 11, 18, 0], [7, 13, 20, 0], [7, 15, 22, 0], [7, 17, 24, 0], [7, 19, 0, 0], [7, 21, 2, 0], [7, 23, 4, 0], [7, 25, 6, 0], [9, 1, 10, 0], [9, 3, 12, 0], [9, 5, 14, 0], [9, 7, 16, 0], [9, 9, 18, 0], [9, 11, 20, 0], [9, 13, 22, 0], [9, 15, 24, 0], [9, 17, 0, 0], [9, 19, 2, 0], [9, 21, 4, 0], [9, 23, 6, 0], [9, 25, 8, 0], [11, 1, 12, 0], [11, 3, 14, 0], [11, 5, 16, 0], [11, 7, 18, 0], [11, 9, 20, 0], [11, 11, 22, 0], [11, 13, 24, 0], [11, 15, 0, 0], [11, 17, 2, 0], [11, 19, 4, 0], [11, 21, 6, 0], [11, 23, 8, 0], [11, 25, 10, 0], [13, 1, 14, 0], [13, 3, 16, 0], [13, 5, 18, 0], [13, 7, 20, 0], [13, 9, 22, 0], [13, 11, 24, 0], [13, 13, 0, 0], [13, 15, 2, 0], [13, 17, 4, 0], [13, 19, 6, 0], [13, 21, 8, 0], [13, 23, 10, 0], [13, 25, 12, 0], [15, 1, 16, 0], [15, 3, 18, 0], [15, 5, 20, 0], [15, 7, 22, 0], [15, 9, 24, 0], [15, 11, 0, 0], [15, 13, 2, 0], [15, 15, 4, 0], [15, 17, 6, 0], [15, 19, 8, 0], [15, 21, 10, 0], [15, 23, 12, 0], [15, 25, 14, 0], [17, 1, 18, 0], [17, 3, 20, 0], [17, 5, 22, 0], [17, 7, 24, 0], [17, 9, 0, 0], [17, 11, 2, 0], [17, 13, 4, 0], [17, 15, 6, 0], [17, 17, 8, 0], [17, 19, 10, 0], [17, 21, 12, 0], [17, 23, 14, 0], [17, 25, 16, 0], [19, 1, 20, 0], [19, 3, 22, 0], [19, 5, 24, 0], [19, 7, 0, 0], [19, 9, 2, 0], [19, 11, 4, 0], [19, 13, 6, 0], [19, 15, 8, 0], [19, 17, 10, 0], [19, 19, 12, 0], [19, 21, 14, 0], [19, 23, 16, 0], [19, 25, 18, 0], [21, 1, 22, 0], [21, 3, 24, 0], [21, 5, 0, 0], [21, 7, 2, 0], [21, 9, 4, 0], [21, 11, 6, 0], [21, 13, 8, 0], [21, 15, 10, 0], [21, 17, 12, 0], [21, 19, 14, 0], [21, 21, 16, 0], [21, 23, 18, 0], [21, 25, 20, 0], [23, 1, 24, 0], [23, 3, 0, 0], [23, 5, 2, 0], [23, 7, 4, 0], [23, 9, 6, 0], [23, 11, 8, 0], [23, 13, 10, 0], [23, 15, 12, 0], [23, 17, 14, 0], [23, 19, 16, 0], [23, 21, 18, 0], [23, 23, 20, 0], [23, 25, 22, 0], [25, 1, 0, 0], [25, 3, 2, 0], [25, 5, 4, 0], [25, 7, 6, 0], [25, 9, 8, 0], [25, 11, 10, 0], [25, 13, 12, 0], [25, 15, 14, 0], [25, 17, 16, 0], [25, 19, 18, 0], [25, 21, 20, 0], [25, 23, 22, 0], [25, 25, 24, 0]], [0], 是]",
      "relation": "implies",
      "rule_id": "Quotient_EVERY_coset_pair_ALL_representative_products",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每組coset的每對代表元都要落在同一product coset，僅用首代表元造表還不足以證良定；保完整可能index集合。",
        "en": "For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices."
      },
      "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": "source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是, [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]]",
      "after": "derived = [[[0, 1], [1, 0]], 0, [[0, 0], [1, 1]], 是, 否]",
      "relation": "implies",
      "rule_id": "Quotient_GENERAL_normality_representative_replacement_and_group_laws",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若a換ah、b換bk，(ah)(bk)=ab(b^-1hb)k∈abH，normality才保代表元替換不變。associativity由G傳下，eH為identity、a^-1H為inverse。固定No只不自動提供commutativity證書，不說具體cyclic quotient非交換，也不引第一同構定理。",
        "en": "Replacing a by ah and b by bk yields(ah)(bk)=ab(b^-1hb)k in abH,using normality for representative independence.Associativity descends from G,eH is identity and a^-1H inverse.The fixed No withholds automatic commutativity;it does not call a particular cyclic quotient noncommutative or invoke first isomorphism."
      },
      "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": [
      [
        [
          0,
          2,
          4,
          6,
          8,
          10,
          12,
          14,
          16,
          18,
          20,
          22,
          24
        ],
        [
          1,
          3,
          5,
          7,
          9,
          11,
          13,
          15,
          17,
          19,
          21,
          23,
          25
        ]
      ],
      [
        [
          0,
          0,
          [
            0
          ],
          true
        ],
        [
          0,
          1,
          [
            1
          ],
          true
        ],
        [
          1,
          0,
          [
            1
          ],
          true
        ],
        [
          1,
          1,
          [
            0
          ],
          true
        ]
      ],
      true,
      "every_representative_pair_lands_in_one_declared_product_coset"
    ]
  },
  "conclusion": {
    "canonical_value": "([[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]], [[0, 0, [0], 是], [0, 1, [1], 是], [1, 0, [1], 是], [1, 1, [0], 是]], 是, every_representative_pair_lands_in_one_declared_product_coset)",
    "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-20261015-advance-20261114-r02-finite-quotient-group-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261114-r02-finite-quotient-group-foundations/questions.jsonl",
      "batch_sha256": "0398c3196e77c77c239fa72091680c1fa5e961a520dc298756d2d4591c79a2ff",
      "original_line": 62,
      "original_line_sha256": "b37c17843720d31995456bca65a5a88aea7b7acf6947434984fc8850c4cec262",
      "question_compact_sha256": "e0ba1240ad21d11bbab7680ab9f684e8ab693ec8d7085501e42cf2dbd9ec9522",
      "english_row_compact_sha256": "3514f3471bf280175351cb1ff86bd6994f7c3d3d944323007ff2dc0486491aa9",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261114-r02-finite-quotient-group-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261114-r02-finite-quotient-group-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-quotient-group": {
        "sha256": "f3e42a5b51685f805af8727447418b3b158bf4c512da0e78fd2804f7a04bba24",
        "english_sha256": "036ebd3e0c66d661a3e151b97f378a2d60aae4fc019e98ba7d711292436a7f89",
        "original_path": "public/data/mko/finite-quotient-group.json",
        "english_path": "public/data/i18n/editions/mko-finite-quotient-group.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "991c73df066352bfcfb1142a2492e104cd1d20dafe7e9ecea8cff26a4f9d8e50",
    "original_candidate_sha256": "965db751f569aa54a5bfed7f3dd7a4ead0cedfd22802e66381d2f18a33f0b7a4",
    "original_candidate_line_sha256": "965db751f569aa54a5bfed7f3dd7a4ead0cedfd22802e66381d2f18a33f0b7a4"
  },
  "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-20261026-010-p50",
    "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": "6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3",
      "question_first_witness_sha256": "f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 10,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "06edb163ba25c9122ac1a8d34d1991933c6025f14bb1d438d455062c5742140a",
      "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."
  }
}