{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261106-r07-quotient-vector-space-0021",
  "pilot_number": "R0038-19921",
  "number": 19921,
  "family_id": "qf-mod37-quotient-coset-equivalence",
  "title": {
    "zh-Hant": "第 19921 題：商向量空間",
    "en": "Question 19921: Equal Cosets Need Not Have Equal Representatives"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261106-r07-quotient-vector-space-0021",
    "batch_id": "batch-20261014-advance-20261106-r07-quotient-vector-space-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-quotient-vector-space"
    ],
    "title_zh": "商向量空間",
    "stem_zh": "有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,21,0]],\"v\":[22,2,3],\"u\":[24,44,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([-2, -42, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)",
      "explanation_zh": "Wave5封閉層核對=([-2, -42, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)；依完整明示資料獨立重算；quotient元素是coset且代表元不唯一；[v]=[u] iff v-u在W；加法與scalar multiplication須與代表元無關；dim公式限有限維且零coset是W。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "商向量空間"
      ]
    },
    "variant": {
      "family_id": "qf-mod37-quotient-coset-equivalence",
      "seed": 202611067021,
      "parameters": {
        "ambient_dimension": 3,
        "basis_w": [
          [
            1,
            21,
            0
          ]
        ],
        "v": [
          22,
          2,
          3
        ],
        "u": [
          24,
          44,
          3
        ],
        "domain": "Finite_dimensional_real_quotient_space",
        "operation": "quotient-coset-equivalence",
        "semantic_template": "qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,21,0]],\"u\":[24,44,3],\"v\":[22,2,3]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261106-r07-quotient-vector-space-0021",
    "batch_id": "batch-20261014-advance-20261106-r07-quotient-vector-space-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "8c732b9fe01015313af90605481d6a094a1c7268ea56fc172d5f71ffcac09758",
      "family_id": "qf-mod37-quotient-coset-equivalence",
      "seed": 202611067021,
      "semantic_template": "qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,21,0]],\"u\":[24,44,3],\"v\":[22,2,3]}"
    },
    "fields": {
      "title": "Equal Cosets Need Not Have Equal Representatives",
      "stem": "Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 21, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[22, 2, 3], u=[24, 44, 3]. Give (difference, equivalent, criterion, boundary).",
      "answer": {
        "kind": "expression",
        "value": "([-2, -42, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)",
        "explanation": "Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -42, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."
      },
      "skills": [
        "Equal Cosets Need Not Have Equal Representatives"
      ],
      "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": "有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={\"ambient_dimension\":3,\"basis_w\":[[1,21,0]],\"v\":[22,2,3],\"u\":[24,44,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 21, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[22, 2, 3], u=[24, 44, 3]. Give (difference, equivalent, criterion, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=quotient-coset-equivalence;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[difference, equivalent, criterion, boundary]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。欄位標籤不自動成為數學證據，No依各題聲稱範圍解讀。",
        "en": "Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "quotient ORIGINAL ordered W basis:COMPLETE exact original ordered matrix=[[1], [21], [0]];rows=3;columns=1",
      "after": "perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed",
      "relation": "implies",
      "rule_id": "quotient_full_matrix_real_linear_equations_not_grid",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐格可逆列運算保整個實解集合，不對係數先設有限整數盒。basis是原column順序，不能以semantic排序調換係數或right side。",
        "en": "Entrywise invertible row operations preserve the whole real solution set,without bounding coefficients to a finite integer box.Keep original basis-column order and right-side roles rather than semantic sorting."
      },
      "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": "quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 := div(1, 1)",
      "after": "quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0，不以小數近似替代。",
          "en": "Recompute quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "quotient ORIGINAL ordered W basis:row=0;column=0;nonzero exact pivot=1;old FULL row=[1]",
      "after": "normalized FULL row=[1]",
      "relation": "implies",
      "rule_id": "quotient_nonzero_pivot_all_entry_scaling",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整列同除非零pivot，不只修改左半或只改一個格。",
        "en": "Divide every entry by the same nonzero pivot,not only the left block or one entry."
      },
      "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": "quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product := mul(21, 1)",
      "after": "quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product，不以小數近似替代。",
          "en": "Recompute quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "1"
        ],
        "output": "21",
        "label": "quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference := sub(21, 21)",
      "after": "quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference，不以小數近似替代。",
          "en": "Recompute quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "21",
          "21"
        ],
        "output": "0",
        "label": "quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "quotient ORIGINAL ordered W basis:target row=1;old row=[21];subtract factor=21 times FULL pivot row=[1]",
      "after": "new exact FULL row=[0]",
      "relation": "implies",
      "rule_id": "quotient_invertible_complete_row_elimination",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一倍數逐每格相減，反向加回可逆；所有方程同時保留，不用抽樣解測試代替。",
        "en": "Subtract the same multiple in every entry.The inverse adds it back,preserving every equation rather than replacing the full system with sampled solutions."
      },
      "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": "quotient ORIGINAL ordered W basis:complete exact reduced matrix=[[1], [0], [0]];ORIGINAL pivot-column indices=[0]",
      "after": "rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved",
      "relation": "implies",
      "rule_id": "quotient_full_rank_and_solution_set_certificate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "主元identity座標給獨立性，其餘欄由主元列座標表出，故rank可由pivot數讀出；可逆列運算把這些關係帶回原矩陣。",
        "en": "Pivot identity coordinates prove independence and express other columns by their pivot-row coordinates,so pivot count gives rank.Invertible operations carry these relations back to the original matrix."
      },
      "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": "ACTUAL ambient real space R^3;complete ORIGINAL independent basis columns of W=[[1, 21, 0]]",
      "after": "W is the FULL real span of these columns;dim W=1;zero coset is the WHOLE subspace W,not the vector0 alone",
      "relation": "implies",
      "rule_id": "quotient_declared_subspace_full_real_span_zero_coset",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整basis的獨立性逐精確rank核過，所以數量是dimW；列表不是W所有元素。空basis代表W={0}，零coset仍是集合W，不把商元素與某一代表向量混型。",
        "en": "Exact rank verifies independence of the declared complete basis,whose count is dim W.The list is not the set of all W elements.An empty basis gives W={0};the zero coset is still the set W,not a representative vector."
      },
      "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": "quotient ORIGINAL v minus u coordinate0 := sub(22, 24)",
      "after": "quotient ORIGINAL v minus u coordinate0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient ORIGINAL v minus u coordinate0，不以小數近似替代。",
          "en": "Recompute quotient ORIGINAL v minus u coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "22",
          "24"
        ],
        "output": "-2",
        "label": "quotient ORIGINAL v minus u coordinate0"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "quotient ORIGINAL v minus u coordinate1 := sub(2, 44)",
      "after": "quotient ORIGINAL v minus u coordinate1 = -42",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient ORIGINAL v minus u coordinate1，不以小數近似替代。",
          "en": "Recompute quotient ORIGINAL v minus u coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "44"
        ],
        "output": "-42",
        "label": "quotient ORIGINAL v minus u coordinate1"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "quotient ORIGINAL v minus u coordinate2 := sub(3, 3)",
      "after": "quotient ORIGINAL v minus u coordinate2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient ORIGINAL v minus u coordinate2，不以小數近似替代。",
          "en": "Recompute quotient ORIGINAL v minus u coordinate2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "3"
        ],
        "output": "0",
        "label": "quotient ORIGINAL v minus u coordinate2"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "quotient coset original difference membership:COMPLETE exact original ordered matrix=[[1, -2], [21, -42], [0, 0]];rows=3;columns=2",
      "after": "perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed",
      "relation": "implies",
      "rule_id": "quotient_full_matrix_real_linear_equations_not_grid",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐格可逆列運算保整個實解集合，不對係數先設有限整數盒。basis是原column順序，不能以semantic排序調換係數或right side。",
        "en": "Entrywise invertible row operations preserve the whole real solution set,without bounding coefficients to a finite integer box.Keep original basis-column order and right-side roles rather than semantic sorting."
      },
      "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": "quotient coset original difference membership pivot0 column0 normalized entry0 := div(1, 1)",
      "after": "quotient coset original difference membership pivot0 column0 normalized entry0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient coset original difference membership pivot0 column0 normalized entry0，不以小數近似替代。",
          "en": "Recompute quotient coset original difference membership pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "quotient coset original difference membership pivot0 column0 normalized entry0"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "quotient coset original difference membership pivot0 column0 normalized entry1 := div(-2, 1)",
      "after": "quotient coset original difference membership pivot0 column0 normalized entry1 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient coset original difference membership pivot0 column0 normalized entry1，不以小數近似替代。",
          "en": "Recompute quotient coset original difference membership pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-2",
        "label": "quotient coset original difference membership pivot0 column0 normalized entry1"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "quotient coset original difference membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, -2]",
      "after": "normalized FULL row=[1, -2]",
      "relation": "implies",
      "rule_id": "quotient_nonzero_pivot_all_entry_scaling",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整列同除非零pivot，不只修改左半或只改一個格。",
        "en": "Divide every entry by the same nonzero pivot,not only the left block or one entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "quotient coset original difference membership eliminate row1 pivot0 entry0 product := mul(21, 1)",
      "after": "quotient coset original difference membership eliminate row1 pivot0 entry0 product = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient coset original difference membership eliminate row1 pivot0 entry0 product，不以小數近似替代。",
          "en": "Recompute quotient coset original difference membership eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "1"
        ],
        "output": "21",
        "label": "quotient coset original difference membership eliminate row1 pivot0 entry0 product"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "quotient coset original difference membership eliminate row1 pivot0 entry0 difference := sub(21, 21)",
      "after": "quotient coset original difference membership eliminate row1 pivot0 entry0 difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient coset original difference membership eliminate row1 pivot0 entry0 difference，不以小數近似替代。",
          "en": "Recompute quotient coset original difference membership eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "21",
          "21"
        ],
        "output": "0",
        "label": "quotient coset original difference membership eliminate row1 pivot0 entry0 difference"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "quotient coset original difference membership eliminate row1 pivot0 entry1 product := mul(21, -2)",
      "after": "quotient coset original difference membership eliminate row1 pivot0 entry1 product = -42",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient coset original difference membership eliminate row1 pivot0 entry1 product，不以小數近似替代。",
          "en": "Recompute quotient coset original difference membership eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "-2"
        ],
        "output": "-42",
        "label": "quotient coset original difference membership eliminate row1 pivot0 entry1 product"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "quotient coset original difference membership eliminate row1 pivot0 entry1 difference := sub(-42, -42)",
      "after": "quotient coset original difference membership eliminate row1 pivot0 entry1 difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient coset original difference membership eliminate row1 pivot0 entry1 difference，不以小數近似替代。",
          "en": "Recompute quotient coset original difference membership eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-42",
          "-42"
        ],
        "output": "0",
        "label": "quotient coset original difference membership eliminate row1 pivot0 entry1 difference"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "quotient coset original difference membership:target row=1;old row=[21, -42];subtract factor=21 times FULL pivot row=[1, -2]",
      "after": "new exact FULL row=[0, 0]",
      "relation": "implies",
      "rule_id": "quotient_invertible_complete_row_elimination",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一倍數逐每格相減，反向加回可逆；所有方程同時保留，不用抽樣解測試代替。",
        "en": "Subtract the same multiple in every entry.The inverse adds it back,preserving every equation rather than replacing the full system with sampled solutions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "quotient coset original difference membership:column=1;remaining exact entries=[0, 0]",
      "after": "no pivot added;current rank=1",
      "relation": "implies",
      "rule_id": "quotient_zero_remaining_column_no_fake_pivot",
      "conditions": [],
      "explanation": {
        "zh-Hant": "剩餘全零欄不補pivot，保其原角色與索引。",
        "en": "Do not add a pivot for an all-zero remaining column;retain its original role and index."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "quotient coset original difference membership:complete exact reduced matrix=[[1, -2], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]",
      "after": "rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved",
      "relation": "implies",
      "rule_id": "quotient_full_rank_and_solution_set_certificate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "主元identity座標給獨立性，其餘欄由主元列座標表出，故rank可由pivot數讀出；可逆列運算把這些關係帶回原矩陣。",
        "en": "Pivot identity coordinates prove independence and express other columns by their pivot-row coordinates,so pivot count gives rank.Invertible operations carry these relations back to the original matrix."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "quotient coset original difference membership original coordinate0 reconstruction coefficient0 product := mul(1, -2)",
      "after": "quotient coset original difference membership original coordinate0 reconstruction coefficient0 product = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient coset original difference membership original coordinate0 reconstruction coefficient0 product，不以小數近似替代。",
          "en": "Recompute quotient coset original difference membership original coordinate0 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-2"
        ],
        "output": "-2",
        "label": "quotient coset original difference membership original coordinate0 reconstruction coefficient0 product"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "quotient coset original difference membership original coordinate0 reconstruction complete sum := sum(-2)",
      "after": "quotient coset original difference membership original coordinate0 reconstruction complete sum = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient coset original difference membership original coordinate0 reconstruction complete sum，不以小數近似替代。",
          "en": "Recompute quotient coset original difference membership original coordinate0 reconstruction complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2"
        ],
        "output": "-2",
        "label": "quotient coset original difference membership original coordinate0 reconstruction complete sum"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "quotient coset original difference membership original coordinate1 reconstruction coefficient0 product := mul(21, -2)",
      "after": "quotient coset original difference membership original coordinate1 reconstruction coefficient0 product = -42",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient coset original difference membership original coordinate1 reconstruction coefficient0 product，不以小數近似替代。",
          "en": "Recompute quotient coset original difference membership original coordinate1 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "-2"
        ],
        "output": "-42",
        "label": "quotient coset original difference membership original coordinate1 reconstruction coefficient0 product"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "quotient coset original difference membership original coordinate1 reconstruction complete sum := sum(-42)",
      "after": "quotient coset original difference membership original coordinate1 reconstruction complete sum = -42",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient coset original difference membership original coordinate1 reconstruction complete sum，不以小數近似替代。",
          "en": "Recompute quotient coset original difference membership original coordinate1 reconstruction complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-42"
        ],
        "output": "-42",
        "label": "quotient coset original difference membership original coordinate1 reconstruction complete sum"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "quotient coset original difference membership original coordinate2 reconstruction coefficient0 product := mul(0, -2)",
      "after": "quotient coset original difference membership original coordinate2 reconstruction coefficient0 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient coset original difference membership original coordinate2 reconstruction coefficient0 product，不以小數近似替代。",
          "en": "Recompute quotient coset original difference membership original coordinate2 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "0",
        "label": "quotient coset original difference membership original coordinate2 reconstruction coefficient0 product"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "quotient coset original difference membership original coordinate2 reconstruction complete sum := sum(0)",
      "after": "quotient coset original difference membership original coordinate2 reconstruction complete sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 quotient coset original difference membership original coordinate2 reconstruction complete sum，不以小數近似替代。",
          "en": "Recompute quotient coset original difference membership original coordinate2 reconstruction complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0"
        ],
        "output": "0",
        "label": "quotient coset original difference membership original coordinate2 reconstruction complete sum"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "quotient coset original difference membership:original proposed vector=[-2, -42, 0];original ordered independent columns=[[1, 21, 0]];complete reduced augmented system=[[1, -2], [0, 0], [0, 0]]",
      "after": "exact real-span coefficient witness=[-2];FULL original reconstruction=[-2, -42, 0];membership=是",
      "relation": "implies",
      "rule_id": "quotient_full_span_coefficient_witness_not_sampled_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "主元方程給精確係數，再逐原座標完整代回。零basis時空係數只重建零向量；非零basis可有非整數或超出原輸入界限的合法實係數，不把它們排除。",
        "en": "Read exact coefficients from the pivot equations and reconstruct every original coordinate.Empty coefficients reconstruct only the zero vector for an empty basis.Valid coefficients for nonempty bases may be fractional or outside original input bounds and must not be excluded."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "ORIGINAL v=[22, 2, 3],u=[24, 44, 3];ORDERED difference v-u=[-2, -42, 0];full-real-span membership=是",
      "after": "v+W=u+W iff v-u in W;coset equality=是;not a test of representative vector equality",
      "relation": "implies",
      "rule_id": "quotient_coset_equality_both_inclusions_not_representative_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若cosets相等，v=v+0也在u+W，故v−u在W。反之v−u在W時，v+w=u+(v−u+w)落u+W，反向用其負值亦成立，兩集合相等。差可非0，不要求代表向量相等。",
        "en": "If the cosets agree,v=v+0 belongs to u+W,so v−u lies in W.Conversely,if v−u lies in W,then v+w=u+(v−u+w)belongs to u+W,and the negative difference proves the reverse inclusion.The difference need not be0;representative vectors need not agree."
      },
      "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": [
      [
        "-2",
        "-42",
        "0"
      ],
      true,
      "v_plus_W_equals_u_plus_W_iff_difference_in_W",
      "quotient_elements_are_cosets_not_unique_representatives"
    ]
  },
  "conclusion": {
    "canonical_value": "([-2, -42, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)",
    "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-20261106-r07-quotient-vector-space-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261106-r07-quotient-vector-space-foundations/questions.jsonl",
      "batch_sha256": "397a375663db7b70c770ebdf31978bf670a4045d83c4d06d54ad214ef66ac82f",
      "original_line": 21,
      "original_line_sha256": "dae62ba2b3c86d110ac6152f8236f41bbb3d2057b944ef37082d7e120ad8add1",
      "question_compact_sha256": "8c732b9fe01015313af90605481d6a094a1c7268ea56fc172d5f71ffcac09758",
      "english_row_compact_sha256": "3b006f281f5af94871a8609e2b61ef83718a15b7ffa9a93b91ee9ac8d6247e1c",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261106-r07-quotient-vector-space-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261106-r07-quotient-vector-space-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-quotient-vector-space": {
        "sha256": "0f381f20543e1db106973cf749847f066bf4cb237412005cfb871209df548ed0",
        "english_sha256": "2591a61d4bb585ffcd836dbd49d9f2f74f4f3731c5c0b78d89bfbeb40b2729cd",
        "original_path": "public/data/mko/quotient-vector-space.json",
        "english_path": "public/data/i18n/editions/mko-quotient-vector-space.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "9ed24319f1e4df0dd58d2a2dfbf2846def03f020cb5ebf70613fab53c9165f11",
    "original_candidate_sha256": "fe96e0f8d7524890aa07e1c7df9dd98118051e58ca22a5074eb7529e07278fd2",
    "original_candidate_line_sha256": "fe96e0f8d7524890aa07e1c7df9dd98118051e58ca22a5074eb7529e07278fd2"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-008-p39",
    "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": "d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612",
      "question_first_witness_sha256": "c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 31,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "6cea5a39c0f958b932e741761d07886cf8ddc931ed5bb2b598980bf76bd5a229",
      "arithmetic_steps": 18
    },
    "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."
  }
}