{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261105-r02-dual-vector-space-0072",
  "pilot_number": "R0036-18797",
  "number": 18797,
  "family_id": "qf-mod36-dual-linearity-certificate",
  "title": {
    "zh-Hant": "第 18797 題：對偶向量空間",
    "en": "Question 18797: One Row Formula Supplies a Symbolic Linearity Identity"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261105-r02-dual-vector-space-0072",
    "batch_id": "batch-20261014-advance-20261105-r02-dual-vector-space-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-dual-vector-space"
    ],
    "title_zh": "對偶向量空間",
    "stem_zh": "有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[22,-2],\"u\":[1,23],\"v\":[24,-1],\"alpha\":22,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(-1588, -1588, 是, row_covector_gives_universal_linearity_certificate)",
      "explanation_zh": "Wave5第五層核對=(-1588, -1588, 是, row_covector_gives_universal_linearity_certificate)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "對偶向量空間"
      ]
    },
    "variant": {
      "family_id": "qf-mod36-dual-linearity-certificate",
      "seed": 202611052072,
      "parameters": {
        "coefficients": [
          22,
          -2
        ],
        "u": [
          1,
          23
        ],
        "v": [
          24,
          -1
        ],
        "alpha": 22,
        "beta": -2,
        "domain": "Finite_dimensional_real_dual_coordinates",
        "operation": "dual-linearity-certificate",
        "semantic_template": "qf-mod36-dual-linearity-certificate/{\"alpha\":22,\"beta\":-2,\"coefficients\":[22,-2],\"u\":[1,23],\"v\":[24,-1]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r02-dual-vector-space-0072",
    "batch_id": "batch-20261014-advance-20261105-r02-dual-vector-space-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "444c718e777f3cef4d291057311504e84ab91eade4fba6e21512a4c901304341",
      "family_id": "qf-mod36-dual-linearity-certificate",
      "seed": 202611052072,
      "semantic_template": "qf-mod36-dual-linearity-certificate/{\"alpha\":22,\"beta\":-2,\"coefficients\":[22,-2],\"u\":[1,23],\"v\":[24,-1]}"
    },
    "fields": {
      "title": "One Row Formula Supplies a Symbolic Linearity Identity",
      "stem": "Use covector row=[22, -2], u=[1, 23], v=[24, -1], alpha=22, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(-1588, -1588, Yes, row_covector_gives_universal_linearity_certificate)",
        "explanation": "Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1588, -1588, Yes, row_covector_gives_universal_linearity_certificate)."
      },
      "skills": [
        "One Row Formula Supplies a Symbolic Linearity Identity"
      ],
      "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R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={\"coefficients\":[22,-2],\"u\":[1,23],\"v\":[24,-1],\"alpha\":22,\"beta\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use covector row=[22, -2], u=[1, 23], v=[24, -1], alpha=22, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=dual-linearity-certificate;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[left, right, equal, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記仍在快照；這裡直接給原始欄位順序，主文不需JavaScript或外部表。描述欄位不自動證成任何數學聲稱。",
        "en": "Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence."
      },
      "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": "fixed original covector ROW=[22, -2];coordinate dimension=2",
      "after": "define phi(x)=sum_j row[j]*x[j] for EVERY real coordinate vector x;finite distributivity gives phi(alpha*u+beta*v)=alpha*phi(u)+beta*phi(v)",
      "relation": "implies",
      "rule_id": "dual_row_symbolic_universal_linearity_not_one_numeric_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定row每個實座標的有限加乘分配律，對任意u,v,alpha,beta同時成立，才給全稱線性證書。後續代值只是核本題數值，不是用一個成功例子證未知函數線性。row表示實值函數，不是primal向量本身。",
        "en": "Finite distributivity for a fixed row holds for arbitrary real u,v,alpha,beta and gives the universal linearity certificate.The subsequent substitution checks this instance,not linearity of an unspecified function from one passing test.The row represents a real-valued function,not the primal vector itself."
      },
      "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": "dual linear combination coordinate0 alpha u := mul(22, 1)",
      "after": "dual linear combination coordinate0 alpha u = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linear combination coordinate0 alpha u，不以小數近似替代。",
          "en": "Recompute dual linear combination coordinate0 alpha u with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "1"
        ],
        "output": "22",
        "label": "dual linear combination coordinate0 alpha u"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "dual linear combination coordinate0 beta v := mul(-2, 24)",
      "after": "dual linear combination coordinate0 beta v = -48",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linear combination coordinate0 beta v，不以小數近似替代。",
          "en": "Recompute dual linear combination coordinate0 beta v with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "24"
        ],
        "output": "-48",
        "label": "dual linear combination coordinate0 beta v"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "dual linear combination coordinate0 sum := sum(22, -48)",
      "after": "dual linear combination coordinate0 sum = -26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linear combination coordinate0 sum，不以小數近似替代。",
          "en": "Recompute dual linear combination coordinate0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "-48"
        ],
        "output": "-26",
        "label": "dual linear combination coordinate0 sum"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "dual linear combination coordinate1 alpha u := mul(22, 23)",
      "after": "dual linear combination coordinate1 alpha u = 506",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linear combination coordinate1 alpha u，不以小數近似替代。",
          "en": "Recompute dual linear combination coordinate1 alpha u with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "23"
        ],
        "output": "506",
        "label": "dual linear combination coordinate1 alpha u"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "dual linear combination coordinate1 beta v := mul(-2, -1)",
      "after": "dual linear combination coordinate1 beta v = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linear combination coordinate1 beta v，不以小數近似替代。",
          "en": "Recompute dual linear combination coordinate1 beta v with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-1"
        ],
        "output": "2",
        "label": "dual linear combination coordinate1 beta v"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "dual linear combination coordinate1 sum := sum(506, 2)",
      "after": "dual linear combination coordinate1 sum = 508",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linear combination coordinate1 sum，不以小數近似替代。",
          "en": "Recompute dual linear combination coordinate1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "506",
          "2"
        ],
        "output": "508",
        "label": "dual linear combination coordinate1 sum"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "dual linearity left evaluation coordinate0 product := mul(22, -26)",
      "after": "dual linearity left evaluation coordinate0 product = -572",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linearity left evaluation coordinate0 product，不以小數近似替代。",
          "en": "Recompute dual linearity left evaluation coordinate0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "-26"
        ],
        "output": "-572",
        "label": "dual linearity left evaluation coordinate0 product"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "dual linearity left evaluation coordinate1 product := mul(-2, 508)",
      "after": "dual linearity left evaluation coordinate1 product = -1016",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linearity left evaluation coordinate1 product，不以小數近似替代。",
          "en": "Recompute dual linearity left evaluation coordinate1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "508"
        ],
        "output": "-1016",
        "label": "dual linearity left evaluation coordinate1 product"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "dual linearity left evaluation complete sum := sum(-572, -1016)",
      "after": "dual linearity left evaluation complete sum = -1588",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linearity left evaluation complete sum，不以小數近似替代。",
          "en": "Recompute dual linearity left evaluation complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-572",
          "-1016"
        ],
        "output": "-1588",
        "label": "dual linearity left evaluation complete sum"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "dual linearity phi u coordinate0 product := mul(22, 1)",
      "after": "dual linearity phi u coordinate0 product = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linearity phi u coordinate0 product，不以小數近似替代。",
          "en": "Recompute dual linearity phi u coordinate0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "1"
        ],
        "output": "22",
        "label": "dual linearity phi u coordinate0 product"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "dual linearity phi u coordinate1 product := mul(-2, 23)",
      "after": "dual linearity phi u coordinate1 product = -46",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linearity phi u coordinate1 product，不以小數近似替代。",
          "en": "Recompute dual linearity phi u coordinate1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "23"
        ],
        "output": "-46",
        "label": "dual linearity phi u coordinate1 product"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "dual linearity phi u complete sum := sum(22, -46)",
      "after": "dual linearity phi u complete sum = -24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linearity phi u complete sum，不以小數近似替代。",
          "en": "Recompute dual linearity phi u complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "-46"
        ],
        "output": "-24",
        "label": "dual linearity phi u complete sum"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "dual linearity phi v coordinate0 product := mul(22, 24)",
      "after": "dual linearity phi v coordinate0 product = 528",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linearity phi v coordinate0 product，不以小數近似替代。",
          "en": "Recompute dual linearity phi v coordinate0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "24"
        ],
        "output": "528",
        "label": "dual linearity phi v coordinate0 product"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "dual linearity phi v coordinate1 product := mul(-2, -1)",
      "after": "dual linearity phi v coordinate1 product = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linearity phi v coordinate1 product，不以小數近似替代。",
          "en": "Recompute dual linearity phi v coordinate1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-1"
        ],
        "output": "2",
        "label": "dual linearity phi v coordinate1 product"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "dual linearity phi v complete sum := sum(528, 2)",
      "after": "dual linearity phi v complete sum = 530",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linearity phi v complete sum，不以小數近似替代。",
          "en": "Recompute dual linearity phi v complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "528",
          "2"
        ],
        "output": "530",
        "label": "dual linearity phi v complete sum"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "dual linearity alpha times phi u := mul(22, -24)",
      "after": "dual linearity alpha times phi u = -528",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linearity alpha times phi u，不以小數近似替代。",
          "en": "Recompute dual linearity alpha times phi u with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "-24"
        ],
        "output": "-528",
        "label": "dual linearity alpha times phi u"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "dual linearity beta times phi v := mul(-2, 530)",
      "after": "dual linearity beta times phi v = -1060",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linearity beta times phi v，不以小數近似替代。",
          "en": "Recompute dual linearity beta times phi v with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "530"
        ],
        "output": "-1060",
        "label": "dual linearity beta times phi v"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "dual linearity right complete sum := sum(-528, -1060)",
      "after": "dual linearity right complete sum = -1588",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dual linearity right complete sum，不以小數近似替代。",
          "en": "Recompute dual linearity right complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-528",
          "-1060"
        ],
        "output": "-1588",
        "label": "dual linearity right complete sum"
      }
    }
  ],
  "math_model": {
    "result": [
      "-1588",
      "-1588",
      true,
      "row_covector_gives_universal_linearity_certificate"
    ]
  },
  "conclusion": {
    "canonical_value": "(-1588, -1588, 是, row_covector_gives_universal_linearity_certificate)",
    "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-20261105-r02-dual-vector-space-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261105-r02-dual-vector-space-foundations/questions.jsonl",
      "batch_sha256": "710a303eff4a61aad547a2ca9501595d14cef5d84a5e44e51d08864c41eb9f14",
      "original_line": 72,
      "original_line_sha256": "9a75905f54c50f22e9bd5026c779a7a75450760e4afa0da07b3ae6d41a024615",
      "question_compact_sha256": "444c718e777f3cef4d291057311504e84ab91eade4fba6e21512a4c901304341",
      "english_row_compact_sha256": "1cf8c2b586f0f13aa5e677db4f7eac425a736a95c048564046f55f9192c560d1",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261105-r02-dual-vector-space-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261105-r02-dual-vector-space-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-dual-vector-space": {
        "sha256": "1994bbbcc73dff239e5c7cfdaba60f5dc5aadb1a0a4603f613e4b1371d59234a",
        "english_sha256": "1e23ee2fa2f740284ff7f2e81139957b6196a81326cb161bae3a56d54fe07153",
        "original_path": "public/data/mko/source13-20261018-dual-vector-space.json",
        "english_path": "public/data/i18n/editions/mko-dual-vector-space.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "49e5d7674cefa01298c2386543f777d2bdfc99446b4027156b80142f10031994",
    "original_candidate_sha256": "2a5df8b41f17360b40ffaef497df7113ca68699a86f01c58d305c9f21da85804",
    "original_candidate_line_sha256": "2a5df8b41f17360b40ffaef497df7113ca68699a86f01c58d305c9f21da85804"
  },
  "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-p37",
    "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": "efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301",
      "question_first_witness_sha256": "acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 20,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "134e80cb03ad7fd74038b402eff93cd561e2450858ac729c23ef3d1bf26e2832",
      "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."
  }
}