{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r01-vector-0018",
  "pilot_number": "R0014-07993",
  "number": 7993,
  "family_id": "qf-density16-vector-weighted-combination",
  "title": {
    "zh-Hant": "第 7993 題：兩個加權向量與總和",
    "en": "Question 7993: Ordered Weighted Vector Combination"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r01-vector-0018",
    "batch_id": "batch-20261012-advance-20261016-r01-vector-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-vector"
    ],
    "title_zh": "兩個加權向量與總和",
    "stem_zh": "固定標準基底的整數二分量向量。u=[5, 1]、v=[1, 15]、a=2、b=-1。依序求 (a*u, b*v, a*u+b*v)。",
    "answer": {
      "kind": "expression",
      "value": "([10, 2], [-1, -15], [9, -13])",
      "explanation_zh": "有限結構核對=([10, 2], [-1, -15], [9, -13])。分別把純量乘到每個分量，再逐分量相加。有限精確核對不是一般形式證明。逐分量不當內積或長度。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "兩個加權向量與總和"
      ]
    },
    "variant": {
      "family_id": "qf-density16-vector-weighted-combination",
      "seed": 202610161018,
      "parameters": {
        "u": [
          5,
          1
        ],
        "v": [
          1,
          15
        ],
        "a": 2,
        "b": -1,
        "domain": "Z2_fixed_basis",
        "operation": "vector-weighted-combination",
        "semantic_template": "qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[5,1],\"v\":[1,15]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261016-r01-vector-0018",
    "batch_id": "batch-20261012-advance-20261016-r01-vector-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f2337997cb4e53cf7dd2383a232ae84e12ffd9896b9fd2f09d7727efab11a98e",
      "family_id": "qf-density16-vector-weighted-combination",
      "seed": 202610161018,
      "semantic_template": "qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[5,1],\"v\":[1,15]}"
    },
    "fields": {
      "title": "Ordered Weighted Vector Combination",
      "stem": "Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[5, 1], v=[1, 15], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.",
      "answer": {
        "kind": "expression",
        "value": "([10, 2], [-1, -15], [9, -13])",
        "explanation": "Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10, 2], [-1, -15], [9, -13])."
      },
      "skills": [
        "Ordered Weighted Vector Combination"
      ],
      "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": "固定標準基底的整數二分量向量。u=[5, 1]、v=[1, 15]、a=2、b=-1。依序求 (a*u, b*v, a*u+b*v)。",
        "en": "Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[5, 1], v=[1, 15], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "fixed standard basis;two ordered integer coordinates",
      "after": "componentwise operations;not length or inner product",
      "relation": "implies",
      "rule_id": "ordered_vector_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一固定基底下逐分量運算；交換分量、取長度或算內積都不是此題的向量答案。",
        "en": "Work coordinate by coordinate in the same fixed basis.Swapping coordinates,taking a norm or computing an inner product is not the requested vector answer."
      },
      "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": "au[0] := mul(2, 5)",
      "after": "au[0] = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 au[0]，不以小數近似替代。",
          "en": "Recompute au[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "5"
        ],
        "output": "10",
        "label": "au[0]"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "au[1] := mul(2, 1)",
      "after": "au[1] = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 au[1]，不以小數近似替代。",
          "en": "Recompute au[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "au[1]"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "bv[0] := mul(-1, 1)",
      "after": "bv[0] = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 bv[0]，不以小數近似替代。",
          "en": "Recompute bv[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "bv[0]"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "bv[1] := mul(-1, 15)",
      "after": "bv[1] = -15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 bv[1]，不以小數近似替代。",
          "en": "Recompute bv[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "15"
        ],
        "output": "-15",
        "label": "bv[1]"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "au+bv[0] := sum(10, -1)",
      "after": "au+bv[0] = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 au+bv[0]，不以小數近似替代。",
          "en": "Recompute au+bv[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "-1"
        ],
        "output": "9",
        "label": "au+bv[0]"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "au+bv[1] := sum(2, -15)",
      "after": "au+bv[1] = -13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 au+bv[1]，不以小數近似替代。",
          "en": "Recompute au+bv[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "-15"
        ],
        "output": "-13",
        "label": "au+bv[1]"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "au=[10, 2];bv=[-1, -15]",
      "after": "value=[9, -13]",
      "relation": "implies",
      "rule_id": "weighted_vector_components",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個權重各乘自己的向量，再加同位置分量；負權重與零向量都依同一定義。",
        "en": "Each weight multiplies its own vector before aligned coordinates are added.Negative weights and the zero vector use the same definition."
      },
      "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": [
      [
        "10",
        "2"
      ],
      [
        "-1",
        "-15"
      ],
      [
        "9",
        "-13"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([10, 2], [-1, -15], [9, -13])",
    "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": {
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    "question_binding": {
      "batch_id": "batch-20261012-advance-20261016-r01-vector-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261016-r01-vector-foundations/questions.jsonl",
      "batch_sha256": "3bfc998f6a115fe8a36b505fbe6d5ba211ef040180b9b8fa8684df4ece6c7261",
      "original_line": 18,
      "original_line_sha256": "edb63c336119e98b00e66e0b2e29dfe0a7672b331d8635bb3b75ba3889eb4cf9",
      "question_compact_sha256": "f2337997cb4e53cf7dd2383a232ae84e12ffd9896b9fd2f09d7727efab11a98e",
      "english_row_compact_sha256": "f5a9d3e4e5b55009da516504b2613742d35b3a735c351213bc3cb7653c60e4b9",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261016-r01-vector-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261016-r01-vector-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-vector": {
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        "original_path": "public/data/mko/vector.json",
        "english_path": "public/data/i18n/editions/mko-vector.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d7860245c70062705fa4858c13821db6ff286678dec63f22464364f5d24cfcc1",
    "original_candidate_sha256": "f563aa30960960524a839fcd4a398b3158771e7cf9f8b2bf481c4a6e8adba38d",
    "original_candidate_line_sha256": "f563aa30960960524a839fcd4a398b3158771e7cf9f8b2bf481c4a6e8adba38d"
  },
  "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-20261026-005-p15",
    "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": "23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77",
      "question_first_witness_sha256": "17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 8,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "93b52b0c9507430cc4cecd47497a4576c569b5b79d1529d06a9b54f28611f4d4",
      "arithmetic_steps": 6
    },
    "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."
  }
}