{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261031-r04-finite-quotient-ring-0079",
  "pilot_number": "R0030-15979",
  "number": 15979,
  "family_id": "qf-mod31-quotring-well-defined",
  "title": {
    "zh-Hant": "第 15979 題：有限商環",
    "en": "Question 15979: Changing Both Representatives Must Preserve Both Operations"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261031-r04-finite-quotient-ring-0079",
    "batch_id": "batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-quotient-ring"
    ],
    "title_zh": "有限商環",
    "stem_zh": "Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":10,\"d\":5,\"a\":3,\"b\":2,\"a_prime\":8,\"b_prime\":7}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([3, 2, 8, 7], [0, 0], [1, 1], 是, 是, 是)",
      "explanation_zh": "Wave4收束核對=([3, 2, 8, 7], [0, 0], [1, 1], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有限商環"
      ]
    },
    "variant": {
      "family_id": "qf-mod31-quotring-well-defined",
      "seed": 202610314079,
      "parameters": {
        "n": 10,
        "d": 5,
        "a": 3,
        "b": 2,
        "a_prime": 8,
        "b_prime": 7,
        "domain": "Exact_quotient_of_Zn_by_proper_ideal",
        "operation": "quotring-well-defined",
        "semantic_template": "qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":8,\"b\":2,\"b_prime\":7,\"d\":5,\"n\":10}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261031-r04-finite-quotient-ring-0079",
    "batch_id": "batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "68f80ae29c68ae01f1789320998978cc1b8b51f03b1fd9d32d17d6d9bdba3880",
      "family_id": "qf-mod31-quotring-well-defined",
      "seed": 202610314079,
      "semantic_template": "qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":8,\"b\":2,\"b_prime\":7,\"d\":5,\"n\":10}"
    },
    "fields": {
      "title": "Changing Both Representatives Must Preserve Both Operations",
      "stem": "Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).",
      "answer": {
        "kind": "expression",
        "value": "([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)",
        "explanation": "Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)."
      },
      "skills": [
        "Changing Both Representatives Must Preserve Both Operations"
      ],
      "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": "Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":10,\"d\":5,\"a\":3,\"b\":2,\"a_prime\":8,\"b_prime\":7}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。",
        "en": "Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "quotient ring parent modulus divided by ideal generator := div(10, 5)",
      "after": "quotient ring parent modulus divided by ideal generator = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient ring parent modulus divided by ideal generator，不以小數近似替代。",
          "en": "Recompute quotient ring parent modulus divided by ideal generator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "10",
          "5"
        ],
        "output": "2",
        "label": "quotient ring parent modulus divided by ideal generator"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "same declared parent Z_10;generator d=5;n/d=2;ideal multiples=[0, 5]",
      "after": "I=kernel of residue map Z_10 to Z_5;proper two-sided ideal=是;zero ideal allowed when d=n",
      "relation": "implies",
      "rule_id": "quotient_ring_actual_parent_proper_two_sided_ideal",
      "conditions": [],
      "explanation": {
        "zh-Hant": "因d整除n，整數代表改加n仍不改mod d，π:Z_n→Z_d良定且保加乘。核I由d倍數組成，加法差封閉且左右乘任意母環元素仍在I。d≥2使1不在I，故proper；d=n可得零理想，不把它排除。",
        "en": "Since d divides n,changing an integer representative by n does not change mod d.The map π:Z_n→Z_d is well-defined and preserves addition and multiplication.Its kernel consists of multiples of d,closed under differences and multiplication on either side by parent elements.With d≥2 it excludes1 and is proper;d=n gives the allowed zero ideal."
      },
      "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": "complete parent residues=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];same ideal I=[0, 5]",
      "after": "all additive cosets in representative order0..d-1=[[0, 5], [1, 6], [2, 7], [3, 8], [4, 9]];quotient size=5;no field conclusion",
      "relation": "implies",
      "rule_id": "quotient_ring_full_cosets_not_representative_carrier",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個加法coset列全體同餘mod d的母環元素，按代表0..d−1排序；它們互斥且覆蓋Z_n。商元素是整類，輸出class標籤只是索引，不因有限商環就宣稱域。",
        "en": "List every parent element congruent mod d in each additive coset,ordered by representatives0 through d−1.They are disjoint and cover Z_n.Quotient elements are whole cosets;class labels are indices only,and a finite quotient ring is not automatically a field."
      },
      "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": "representative slot0 input class exact quotient fraction := div(3, 5)",
      "after": "representative slot0 input class exact quotient fraction = 3/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative slot0 input class exact quotient fraction，不以小數近似替代。",
          "en": "Recompute representative slot0 input class exact quotient fraction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "3",
          "5"
        ],
        "output": "3/5",
        "label": "representative slot0 input class exact quotient fraction"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "representative slot0 input class:integer value=3;positive modulus=5;exact ratio=3/5",
      "after": "floor quotient=0;quotient<=ratio<quotient+1",
      "relation": "implies",
      "rule_id": "modular_Euclidean_floor_not_rational_division_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取有理商再取向下整數商q，不能把有理除法本身當餘數；正模數下q≤value/m<q+1，故後續減去qm得到0至m−1。",
        "en": "Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1."
      },
      "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": "representative slot0 input class quotient multiple := mul(0, 5)",
      "after": "representative slot0 input class quotient multiple = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative slot0 input class quotient multiple，不以小數近似替代。",
          "en": "Recompute representative slot0 input class quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "5"
        ],
        "output": "0",
        "label": "representative slot0 input class quotient multiple"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "representative slot0 input class canonical nonnegative residue := sub(3, 0)",
      "after": "representative slot0 input class canonical nonnegative residue = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative slot0 input class canonical nonnegative residue，不以小數近似替代。",
          "en": "Recompute representative slot0 input class canonical nonnegative residue with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "0"
        ],
        "output": "3",
        "label": "representative slot0 input class canonical nonnegative residue"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "representative slot1 input class exact quotient fraction := div(2, 5)",
      "after": "representative slot1 input class exact quotient fraction = 2/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative slot1 input class exact quotient fraction，不以小數近似替代。",
          "en": "Recompute representative slot1 input class exact quotient fraction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "5"
        ],
        "output": "2/5",
        "label": "representative slot1 input class exact quotient fraction"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "representative slot1 input class:integer value=2;positive modulus=5;exact ratio=2/5",
      "after": "floor quotient=0;quotient<=ratio<quotient+1",
      "relation": "implies",
      "rule_id": "modular_Euclidean_floor_not_rational_division_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取有理商再取向下整數商q，不能把有理除法本身當餘數；正模數下q≤value/m<q+1，故後續減去qm得到0至m−1。",
        "en": "Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1."
      },
      "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": "representative slot1 input class quotient multiple := mul(0, 5)",
      "after": "representative slot1 input class quotient multiple = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative slot1 input class quotient multiple，不以小數近似替代。",
          "en": "Recompute representative slot1 input class quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "5"
        ],
        "output": "0",
        "label": "representative slot1 input class quotient multiple"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "representative slot1 input class canonical nonnegative residue := sub(2, 0)",
      "after": "representative slot1 input class canonical nonnegative residue = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative slot1 input class canonical nonnegative residue，不以小數近似替代。",
          "en": "Recompute representative slot1 input class canonical nonnegative residue with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "0"
        ],
        "output": "2",
        "label": "representative slot1 input class canonical nonnegative residue"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "representative slot2 input class exact quotient fraction := div(8, 5)",
      "after": "representative slot2 input class exact quotient fraction = 8/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative slot2 input class exact quotient fraction，不以小數近似替代。",
          "en": "Recompute representative slot2 input class exact quotient fraction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "8",
          "5"
        ],
        "output": "8/5",
        "label": "representative slot2 input class exact quotient fraction"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "representative slot2 input class:integer value=8;positive modulus=5;exact ratio=8/5",
      "after": "floor quotient=1;quotient<=ratio<quotient+1",
      "relation": "implies",
      "rule_id": "modular_Euclidean_floor_not_rational_division_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取有理商再取向下整數商q，不能把有理除法本身當餘數；正模數下q≤value/m<q+1，故後續減去qm得到0至m−1。",
        "en": "Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "representative slot2 input class quotient multiple := mul(1, 5)",
      "after": "representative slot2 input class quotient multiple = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative slot2 input class quotient multiple，不以小數近似替代。",
          "en": "Recompute representative slot2 input class quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "5"
        ],
        "output": "5",
        "label": "representative slot2 input class quotient multiple"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "representative slot2 input class canonical nonnegative residue := sub(8, 5)",
      "after": "representative slot2 input class canonical nonnegative residue = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative slot2 input class canonical nonnegative residue，不以小數近似替代。",
          "en": "Recompute representative slot2 input class canonical nonnegative residue with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "8",
          "5"
        ],
        "output": "3",
        "label": "representative slot2 input class canonical nonnegative residue"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "representative slot3 input class exact quotient fraction := div(7, 5)",
      "after": "representative slot3 input class exact quotient fraction = 7/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative slot3 input class exact quotient fraction，不以小數近似替代。",
          "en": "Recompute representative slot3 input class exact quotient fraction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "7",
          "5"
        ],
        "output": "7/5",
        "label": "representative slot3 input class exact quotient fraction"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "representative slot3 input class:integer value=7;positive modulus=5;exact ratio=7/5",
      "after": "floor quotient=1;quotient<=ratio<quotient+1",
      "relation": "implies",
      "rule_id": "modular_Euclidean_floor_not_rational_division_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取有理商再取向下整數商q，不能把有理除法本身當餘數；正模數下q≤value/m<q+1，故後續減去qm得到0至m−1。",
        "en": "Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "representative slot3 input class quotient multiple := mul(1, 5)",
      "after": "representative slot3 input class quotient multiple = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative slot3 input class quotient multiple，不以小數近似替代。",
          "en": "Recompute representative slot3 input class quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "5"
        ],
        "output": "5",
        "label": "representative slot3 input class quotient multiple"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "representative slot3 input class canonical nonnegative residue := sub(7, 5)",
      "after": "representative slot3 input class canonical nonnegative residue = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative slot3 input class canonical nonnegative residue，不以小數近似替代。",
          "en": "Recompute representative slot3 input class canonical nonnegative residue with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "7",
          "5"
        ],
        "output": "2",
        "label": "representative slot3 input class canonical nonnegative residue"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "first representative change := sub(3, 8)",
      "after": "first representative change = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first representative change，不以小數近似替代。",
          "en": "Recompute first representative change with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "8"
        ],
        "output": "-5",
        "label": "first representative change"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "second representative change := sub(2, 7)",
      "after": "second representative change = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second representative change，不以小數近似替代。",
          "en": "Recompute second representative change with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "7"
        ],
        "output": "-5",
        "label": "second representative change"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "first representative change divided by d := div(-5, 5)",
      "after": "first representative change divided by d = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first representative change divided by d，不以小數近似替代。",
          "en": "Recompute first representative change divided by d with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-5",
          "5"
        ],
        "output": "-1",
        "label": "first representative change divided by d"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "second representative change divided by d := div(-5, 5)",
      "after": "second representative change divided by d = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second representative change divided by d，不以小數近似替代。",
          "en": "Recompute second representative change divided by d with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-5",
          "5"
        ],
        "output": "-1",
        "label": "second representative change divided by d"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "original representative choices=[3, 2, 8, 7];class labels in slot order=[3, 2, 3, 2];changes a-a_prime=-5;b-b_prime=-5",
      "after": "integer ideal multiples=[-1, -1];corresponding INPUT cosets agree",
      "relation": "implies",
      "rule_id": "quotient_well_defined_input_cosets_are_premise",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先確定兩組代表本來表示相同兩個輸入coset；不符就是問題前提不合，不能把它稱商運算不良定的反例。差除d為整數，明示同理想成員資格。",
        "en": "First require the corresponding representatives to denote the same two input cosets.A mismatch is a failed premise,not a counterexample to well-definedness.Integer differences divided by d explicitly witness membership in the same ideal."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "representative choice0 addition integer addition := sum(3, 2)",
      "after": "representative choice0 addition integer addition = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 addition integer addition，不以小數近似替代。",
          "en": "Recompute representative choice0 addition integer addition with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "2"
        ],
        "output": "5",
        "label": "representative choice0 addition integer addition"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "representative choice0 addition parent mod n exact quotient fraction := div(5, 10)",
      "after": "representative choice0 addition parent mod n exact quotient fraction = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 addition parent mod n exact quotient fraction，不以小數近似替代。",
          "en": "Recompute representative choice0 addition parent mod n exact quotient fraction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "5",
          "10"
        ],
        "output": "1/2",
        "label": "representative choice0 addition parent mod n exact quotient fraction"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "representative choice0 addition parent mod n:integer value=5;positive modulus=10;exact ratio=1/2",
      "after": "floor quotient=0;quotient<=ratio<quotient+1",
      "relation": "implies",
      "rule_id": "modular_Euclidean_floor_not_rational_division_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取有理商再取向下整數商q，不能把有理除法本身當餘數；正模數下q≤value/m<q+1，故後續減去qm得到0至m−1。",
        "en": "Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "representative choice0 addition parent mod n quotient multiple := mul(0, 10)",
      "after": "representative choice0 addition parent mod n quotient multiple = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 addition parent mod n quotient multiple，不以小數近似替代。",
          "en": "Recompute representative choice0 addition parent mod n quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "10"
        ],
        "output": "0",
        "label": "representative choice0 addition parent mod n quotient multiple"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "representative choice0 addition parent mod n canonical nonnegative residue := sub(5, 0)",
      "after": "representative choice0 addition parent mod n canonical nonnegative residue = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 addition parent mod n canonical nonnegative residue，不以小數近似替代。",
          "en": "Recompute representative choice0 addition parent mod n canonical nonnegative residue with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "0"
        ],
        "output": "5",
        "label": "representative choice0 addition parent mod n canonical nonnegative residue"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "representative choice0 addition quotient class mod d exact quotient fraction := div(5, 5)",
      "after": "representative choice0 addition quotient class mod d exact quotient fraction = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 addition quotient class mod d exact quotient fraction，不以小數近似替代。",
          "en": "Recompute representative choice0 addition quotient class mod d exact quotient fraction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "5",
          "5"
        ],
        "output": "1",
        "label": "representative choice0 addition quotient class mod d exact quotient fraction"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "representative choice0 addition quotient class mod d:integer value=5;positive modulus=5;exact ratio=1",
      "after": "floor quotient=1;quotient<=ratio<quotient+1",
      "relation": "implies",
      "rule_id": "modular_Euclidean_floor_not_rational_division_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取有理商再取向下整數商q，不能把有理除法本身當餘數；正模數下q≤value/m<q+1，故後續減去qm得到0至m−1。",
        "en": "Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "representative choice0 addition quotient class mod d quotient multiple := mul(1, 5)",
      "after": "representative choice0 addition quotient class mod d quotient multiple = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 addition quotient class mod d quotient multiple，不以小數近似替代。",
          "en": "Recompute representative choice0 addition quotient class mod d quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "5"
        ],
        "output": "5",
        "label": "representative choice0 addition quotient class mod d quotient multiple"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "representative choice0 addition quotient class mod d canonical nonnegative residue := sub(5, 5)",
      "after": "representative choice0 addition quotient class mod d canonical nonnegative residue = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 addition quotient class mod d canonical nonnegative residue，不以小數近似替代。",
          "en": "Recompute representative choice0 addition quotient class mod d canonical nonnegative residue with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "5"
        ],
        "output": "0",
        "label": "representative choice0 addition quotient class mod d canonical nonnegative residue"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "representative choice0 addition:original parent representatives=[3, 2];integer result=5;parent modulus=10",
      "after": "parent result=5;quotient class label=0;class modulus=5",
      "relation": "implies",
      "rule_id": "quotient_operation_parent_result_separate_from_class",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先在指定母環mod n運算，再取mod d的商類；兩層欄位可能不同，不可跳成同一個餘數，也不能把標籤當商元素全體內容。",
        "en": "Operate in the specified parent modulo n,then take the quotient class modulo d.These fields can differ;do not conflate their residues or confuse a class label with its full coset."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "representative choice0 multiplication integer multiplication := mul(3, 2)",
      "after": "representative choice0 multiplication integer multiplication = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 multiplication integer multiplication，不以小數近似替代。",
          "en": "Recompute representative choice0 multiplication integer multiplication with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "2"
        ],
        "output": "6",
        "label": "representative choice0 multiplication integer multiplication"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "representative choice0 multiplication parent mod n exact quotient fraction := div(6, 10)",
      "after": "representative choice0 multiplication parent mod n exact quotient fraction = 3/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 multiplication parent mod n exact quotient fraction，不以小數近似替代。",
          "en": "Recompute representative choice0 multiplication parent mod n exact quotient fraction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6",
          "10"
        ],
        "output": "3/5",
        "label": "representative choice0 multiplication parent mod n exact quotient fraction"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "representative choice0 multiplication parent mod n:integer value=6;positive modulus=10;exact ratio=3/5",
      "after": "floor quotient=0;quotient<=ratio<quotient+1",
      "relation": "implies",
      "rule_id": "modular_Euclidean_floor_not_rational_division_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取有理商再取向下整數商q，不能把有理除法本身當餘數；正模數下q≤value/m<q+1，故後續減去qm得到0至m−1。",
        "en": "Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "representative choice0 multiplication parent mod n quotient multiple := mul(0, 10)",
      "after": "representative choice0 multiplication parent mod n quotient multiple = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 multiplication parent mod n quotient multiple，不以小數近似替代。",
          "en": "Recompute representative choice0 multiplication parent mod n quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "10"
        ],
        "output": "0",
        "label": "representative choice0 multiplication parent mod n quotient multiple"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "representative choice0 multiplication parent mod n canonical nonnegative residue := sub(6, 0)",
      "after": "representative choice0 multiplication parent mod n canonical nonnegative residue = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 multiplication parent mod n canonical nonnegative residue，不以小數近似替代。",
          "en": "Recompute representative choice0 multiplication parent mod n canonical nonnegative residue with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "6",
          "0"
        ],
        "output": "6",
        "label": "representative choice0 multiplication parent mod n canonical nonnegative residue"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "representative choice0 multiplication quotient class mod d exact quotient fraction := div(6, 5)",
      "after": "representative choice0 multiplication quotient class mod d exact quotient fraction = 6/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 multiplication quotient class mod d exact quotient fraction，不以小數近似替代。",
          "en": "Recompute representative choice0 multiplication quotient class mod d exact quotient fraction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6",
          "5"
        ],
        "output": "6/5",
        "label": "representative choice0 multiplication quotient class mod d exact quotient fraction"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "representative choice0 multiplication quotient class mod d:integer value=6;positive modulus=5;exact ratio=6/5",
      "after": "floor quotient=1;quotient<=ratio<quotient+1",
      "relation": "implies",
      "rule_id": "modular_Euclidean_floor_not_rational_division_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取有理商再取向下整數商q，不能把有理除法本身當餘數；正模數下q≤value/m<q+1，故後續減去qm得到0至m−1。",
        "en": "Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "representative choice0 multiplication quotient class mod d quotient multiple := mul(1, 5)",
      "after": "representative choice0 multiplication quotient class mod d quotient multiple = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 multiplication quotient class mod d quotient multiple，不以小數近似替代。",
          "en": "Recompute representative choice0 multiplication quotient class mod d quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "5"
        ],
        "output": "5",
        "label": "representative choice0 multiplication quotient class mod d quotient multiple"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "representative choice0 multiplication quotient class mod d canonical nonnegative residue := sub(6, 5)",
      "after": "representative choice0 multiplication quotient class mod d canonical nonnegative residue = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice0 multiplication quotient class mod d canonical nonnegative residue，不以小數近似替代。",
          "en": "Recompute representative choice0 multiplication quotient class mod d canonical nonnegative residue with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "6",
          "5"
        ],
        "output": "1",
        "label": "representative choice0 multiplication quotient class mod d canonical nonnegative residue"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "representative choice0 multiplication:original parent representatives=[3, 2];integer result=6;parent modulus=10",
      "after": "parent result=6;quotient class label=1;class modulus=5",
      "relation": "implies",
      "rule_id": "quotient_operation_parent_result_separate_from_class",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先在指定母環mod n運算，再取mod d的商類；兩層欄位可能不同，不可跳成同一個餘數，也不能把標籤當商元素全體內容。",
        "en": "Operate in the specified parent modulo n,then take the quotient class modulo d.These fields can differ;do not conflate their residues or confuse a class label with its full coset."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "representative choice1 addition integer addition := sum(8, 7)",
      "after": "representative choice1 addition integer addition = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 addition integer addition，不以小數近似替代。",
          "en": "Recompute representative choice1 addition integer addition with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "8",
          "7"
        ],
        "output": "15",
        "label": "representative choice1 addition integer addition"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "representative choice1 addition parent mod n exact quotient fraction := div(15, 10)",
      "after": "representative choice1 addition parent mod n exact quotient fraction = 3/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 addition parent mod n exact quotient fraction，不以小數近似替代。",
          "en": "Recompute representative choice1 addition parent mod n exact quotient fraction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "15",
          "10"
        ],
        "output": "3/2",
        "label": "representative choice1 addition parent mod n exact quotient fraction"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "representative choice1 addition parent mod n:integer value=15;positive modulus=10;exact ratio=3/2",
      "after": "floor quotient=1;quotient<=ratio<quotient+1",
      "relation": "implies",
      "rule_id": "modular_Euclidean_floor_not_rational_division_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取有理商再取向下整數商q，不能把有理除法本身當餘數；正模數下q≤value/m<q+1，故後續減去qm得到0至m−1。",
        "en": "Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "representative choice1 addition parent mod n quotient multiple := mul(1, 10)",
      "after": "representative choice1 addition parent mod n quotient multiple = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 addition parent mod n quotient multiple，不以小數近似替代。",
          "en": "Recompute representative choice1 addition parent mod n quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "10"
        ],
        "output": "10",
        "label": "representative choice1 addition parent mod n quotient multiple"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "representative choice1 addition parent mod n canonical nonnegative residue := sub(15, 10)",
      "after": "representative choice1 addition parent mod n canonical nonnegative residue = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 addition parent mod n canonical nonnegative residue，不以小數近似替代。",
          "en": "Recompute representative choice1 addition parent mod n canonical nonnegative residue with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "15",
          "10"
        ],
        "output": "5",
        "label": "representative choice1 addition parent mod n canonical nonnegative residue"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "representative choice1 addition quotient class mod d exact quotient fraction := div(5, 5)",
      "after": "representative choice1 addition quotient class mod d exact quotient fraction = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 addition quotient class mod d exact quotient fraction，不以小數近似替代。",
          "en": "Recompute representative choice1 addition quotient class mod d exact quotient fraction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "5",
          "5"
        ],
        "output": "1",
        "label": "representative choice1 addition quotient class mod d exact quotient fraction"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "representative choice1 addition quotient class mod d:integer value=5;positive modulus=5;exact ratio=1",
      "after": "floor quotient=1;quotient<=ratio<quotient+1",
      "relation": "implies",
      "rule_id": "modular_Euclidean_floor_not_rational_division_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取有理商再取向下整數商q，不能把有理除法本身當餘數；正模數下q≤value/m<q+1，故後續減去qm得到0至m−1。",
        "en": "Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "representative choice1 addition quotient class mod d quotient multiple := mul(1, 5)",
      "after": "representative choice1 addition quotient class mod d quotient multiple = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 addition quotient class mod d quotient multiple，不以小數近似替代。",
          "en": "Recompute representative choice1 addition quotient class mod d quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "5"
        ],
        "output": "5",
        "label": "representative choice1 addition quotient class mod d quotient multiple"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "representative choice1 addition quotient class mod d canonical nonnegative residue := sub(5, 5)",
      "after": "representative choice1 addition quotient class mod d canonical nonnegative residue = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 addition quotient class mod d canonical nonnegative residue，不以小數近似替代。",
          "en": "Recompute representative choice1 addition quotient class mod d canonical nonnegative residue with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "5"
        ],
        "output": "0",
        "label": "representative choice1 addition quotient class mod d canonical nonnegative residue"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "representative choice1 addition:original parent representatives=[8, 7];integer result=15;parent modulus=10",
      "after": "parent result=5;quotient class label=0;class modulus=5",
      "relation": "implies",
      "rule_id": "quotient_operation_parent_result_separate_from_class",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先在指定母環mod n運算，再取mod d的商類；兩層欄位可能不同，不可跳成同一個餘數，也不能把標籤當商元素全體內容。",
        "en": "Operate in the specified parent modulo n,then take the quotient class modulo d.These fields can differ;do not conflate their residues or confuse a class label with its full coset."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "representative choice1 multiplication integer multiplication := mul(8, 7)",
      "after": "representative choice1 multiplication integer multiplication = 56",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 multiplication integer multiplication，不以小數近似替代。",
          "en": "Recompute representative choice1 multiplication integer multiplication with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "7"
        ],
        "output": "56",
        "label": "representative choice1 multiplication integer multiplication"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "representative choice1 multiplication parent mod n exact quotient fraction := div(56, 10)",
      "after": "representative choice1 multiplication parent mod n exact quotient fraction = 28/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 multiplication parent mod n exact quotient fraction，不以小數近似替代。",
          "en": "Recompute representative choice1 multiplication parent mod n exact quotient fraction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "56",
          "10"
        ],
        "output": "28/5",
        "label": "representative choice1 multiplication parent mod n exact quotient fraction"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "representative choice1 multiplication parent mod n:integer value=56;positive modulus=10;exact ratio=28/5",
      "after": "floor quotient=5;quotient<=ratio<quotient+1",
      "relation": "implies",
      "rule_id": "modular_Euclidean_floor_not_rational_division_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取有理商再取向下整數商q，不能把有理除法本身當餘數；正模數下q≤value/m<q+1，故後續減去qm得到0至m−1。",
        "en": "Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "representative choice1 multiplication parent mod n quotient multiple := mul(5, 10)",
      "after": "representative choice1 multiplication parent mod n quotient multiple = 50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 multiplication parent mod n quotient multiple，不以小數近似替代。",
          "en": "Recompute representative choice1 multiplication parent mod n quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "10"
        ],
        "output": "50",
        "label": "representative choice1 multiplication parent mod n quotient multiple"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "representative choice1 multiplication parent mod n canonical nonnegative residue := sub(56, 50)",
      "after": "representative choice1 multiplication parent mod n canonical nonnegative residue = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 multiplication parent mod n canonical nonnegative residue，不以小數近似替代。",
          "en": "Recompute representative choice1 multiplication parent mod n canonical nonnegative residue with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "56",
          "50"
        ],
        "output": "6",
        "label": "representative choice1 multiplication parent mod n canonical nonnegative residue"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "representative choice1 multiplication quotient class mod d exact quotient fraction := div(6, 5)",
      "after": "representative choice1 multiplication quotient class mod d exact quotient fraction = 6/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 multiplication quotient class mod d exact quotient fraction，不以小數近似替代。",
          "en": "Recompute representative choice1 multiplication quotient class mod d exact quotient fraction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6",
          "5"
        ],
        "output": "6/5",
        "label": "representative choice1 multiplication quotient class mod d exact quotient fraction"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "representative choice1 multiplication quotient class mod d:integer value=6;positive modulus=5;exact ratio=6/5",
      "after": "floor quotient=1;quotient<=ratio<quotient+1",
      "relation": "implies",
      "rule_id": "modular_Euclidean_floor_not_rational_division_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取有理商再取向下整數商q，不能把有理除法本身當餘數；正模數下q≤value/m<q+1，故後續減去qm得到0至m−1。",
        "en": "Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "representative choice1 multiplication quotient class mod d quotient multiple := mul(1, 5)",
      "after": "representative choice1 multiplication quotient class mod d quotient multiple = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 multiplication quotient class mod d quotient multiple，不以小數近似替代。",
          "en": "Recompute representative choice1 multiplication quotient class mod d quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "5"
        ],
        "output": "5",
        "label": "representative choice1 multiplication quotient class mod d quotient multiple"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "representative choice1 multiplication quotient class mod d canonical nonnegative residue := sub(6, 5)",
      "after": "representative choice1 multiplication quotient class mod d canonical nonnegative residue = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 representative choice1 multiplication quotient class mod d canonical nonnegative residue，不以小數近似替代。",
          "en": "Recompute representative choice1 multiplication quotient class mod d canonical nonnegative residue with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "6",
          "5"
        ],
        "output": "1",
        "label": "representative choice1 multiplication quotient class mod d canonical nonnegative residue"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "representative choice1 multiplication:original parent representatives=[8, 7];integer result=56;parent modulus=10",
      "after": "parent result=6;quotient class label=1;class modulus=5",
      "relation": "implies",
      "rule_id": "quotient_operation_parent_result_separate_from_class",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先在指定母環mod n運算，再取mod d的商類；兩層欄位可能不同，不可跳成同一個餘數，也不能把標籤當商元素全體內容。",
        "en": "Operate in the specified parent modulo n,then take the quotient class modulo d.These fields can differ;do not conflate their residues or confuse a class label with its full coset."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "ANY changes a-a_prime,b-b_prime in same two-sided ideal I;actual addition classes=[0, 0];actual multiplication classes=[1, 1]",
      "after": "(a+b)-(a_prime+b_prime)=(a-a_prime)+(b-b_prime) in I;ab-a_prime*b_prime=a*(b-b_prime)+(a-a_prime)*b_prime in I;both universal operation obligations=是",
      "relation": "implies",
      "rule_id": "quotient_ring_universal_representative_independence_not_two_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法差由I加法封閉，乘法差由左／右吸收再相加，故任意合法代表替換都不改coset。本題兩次重算是實例核對，真正的全稱良定性來自這個理想論證，不只兩樣本同值。",
        "en": "Closure of I handles the additive difference;left/right absorption and addition handle the multiplicative difference.Thus any legal representative changes preserve both cosets.The two recomputations instantiate the result;universal well-definedness comes from this ideal argument,not merely two matching samples."
      },
      "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": [
      [
        3,
        2,
        8,
        7
      ],
      [
        0,
        0
      ],
      [
        1,
        1
      ],
      true,
      true,
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([3, 2, 8, 7], [0, 0], [1, 1], 是, 是, 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-20261014-advance-20261031-r04-finite-quotient-ring-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations/questions.jsonl",
      "batch_sha256": "990159f2eda0c7cd420834e0b7b4b5b70e6cde01e33924c6340b2b0b90b0d4f8",
      "original_line": 79,
      "original_line_sha256": "e806f5ad4e1c80d7fd791bf3898794d99a425153fdbec172979681c18947bae5",
      "question_compact_sha256": "68f80ae29c68ae01f1789320998978cc1b8b51f03b1fd9d32d17d6d9bdba3880",
      "english_row_compact_sha256": "16a92de7a7bfcc745382cf4d194374f6a9031816bfec24afdca5330d50d58ede",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-quotient-ring": {
        "sha256": "8c2b45b38428b422d01b869c06eaca3a0db6e14e6fafc691ef583d4614d39216",
        "english_sha256": "b144945c75f4a276442029de263e7ef036043832247b0b951049af0cc80623fa",
        "original_path": "public/data/mko/finite-quotient-ring.json",
        "english_path": "public/data/i18n/editions/mko-finite-quotient-ring.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d85118d74859ec7e59736714f767732c6ec092eaa0bf5279abeb28e4a628e3c3",
    "original_candidate_sha256": "2aae3883191ddb7b90419a722f2660cc8f3a685d2da051fa4f86d4b9b0d31793",
    "original_candidate_line_sha256": "2aae3883191ddb7b90419a722f2660cc8f3a685d2da051fa4f86d4b9b0d31793"
  },
  "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-20261026-007-p31",
    "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": "73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc",
      "question_first_witness_sha256": "ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 65,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "7be995d9ba997db1240ef5ec5121b556fc8531c5492f3014d3843f0b4e299776",
      "arithmetic_steps": 45
    },
    "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."
  }
}