{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r01-vector-0061",
  "pilot_number": "R0014-07961",
  "number": 7961,
  "family_id": "qf-density16-vector-two-scalar-additivity",
  "title": {
    "zh-Hant": "第 7961 題：同向量的雙純量兩路",
    "en": "Question 7961: Two Scalar-Addition Routes and Component Defect"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r01-vector-0061",
    "batch_id": "batch-20261012-advance-20261016-r01-vector-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-vector"
    ],
    "title_zh": "同向量的雙純量兩路",
    "stem_zh": "固定標準基底的整數二分量向量。u=[-2, 2]、a=1、b=2。求 (u, (a+b)u, au+bu, 後兩路之差)。",
    "answer": {
      "kind": "expression",
      "value": "([-2, 2], [-6, 6], [-6, 6], [0, 0])",
      "explanation_zh": "有限結構核對=([-2, 2], [-6, 6], [-6, 6], [0, 0])。一條路先加純量，另一條路先乘分量再加，核對實際兩路向量。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": 2,
      "skills": [
        "同向量的雙純量兩路"
      ]
    },
    "variant": {
      "family_id": "qf-density16-vector-two-scalar-additivity",
      "seed": 202610161061,
      "parameters": {
        "u": [
          -2,
          2
        ],
        "a": 1,
        "b": 2,
        "domain": "Z2_fixed_basis",
        "operation": "vector-two-scalar-additivity",
        "semantic_template": "qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[-2,2]}"
      }
    },
    "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-0061",
    "batch_id": "batch-20261012-advance-20261016-r01-vector-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "27e35e572dd3e51308effb641d184771486d78ed852d35a7c07f163a630cb8f1",
      "family_id": "qf-density16-vector-two-scalar-additivity",
      "seed": 202610161061,
      "semantic_template": "qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[-2,2]}"
    },
    "fields": {
      "title": "Two Scalar-Addition Routes and Component Defect",
      "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=[-2, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).",
      "answer": {
        "kind": "expression",
        "value": "([-2, 2], [-6, 6], [-6, 6], [0, 0])",
        "explanation": "One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2, 2], [-6, 6], [-6, 6], [0, 0])."
      },
      "skills": [
        "Two Scalar-Addition Routes and Component Defect"
      ],
      "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=[-2, 2]、a=1、b=2。求 (u, (a+b)u, au+bu, 後兩路之差)。",
        "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=[-2, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes)."
      }
    }
  ],
  "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": "a+b := sum(1, 2)",
      "after": "a+b = 3",
      "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 a+b，不以小數近似替代。",
          "en": "Recompute a+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "2"
        ],
        "output": "3",
        "label": "a+b"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "(a+b)u[0] := mul(3, -2)",
      "after": "(a+b)u[0] = -6",
      "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 (a+b)u[0]，不以小數近似替代。",
          "en": "Recompute (a+b)u[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "-2"
        ],
        "output": "-6",
        "label": "(a+b)u[0]"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "(a+b)u[1] := mul(3, 2)",
      "after": "(a+b)u[1] = 6",
      "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 (a+b)u[1]，不以小數近似替代。",
          "en": "Recompute (a+b)u[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "2"
        ],
        "output": "6",
        "label": "(a+b)u[1]"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "au[0] := mul(1, -2)",
      "after": "au[0] = -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[0]，不以小數近似替代。",
          "en": "Recompute au[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-2"
        ],
        "output": "-2",
        "label": "au[0]"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "au[1] := mul(1, 2)",
      "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": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "au[1]"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "bu[0] := mul(2, -2)",
      "after": "bu[0] = -4",
      "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 bu[0]，不以小數近似替代。",
          "en": "Recompute bu[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-2"
        ],
        "output": "-4",
        "label": "bu[0]"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "bu[1] := mul(2, 2)",
      "after": "bu[1] = 4",
      "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 bu[1]，不以小數近似替代。",
          "en": "Recompute bu[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "bu[1]"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "au+bu[0] := sum(-2, -4)",
      "after": "au+bu[0] = -6",
      "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+bu[0]，不以小數近似替代。",
          "en": "Recompute au+bu[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "-4"
        ],
        "output": "-6",
        "label": "au+bu[0]"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "au+bu[1] := sum(2, 4)",
      "after": "au+bu[1] = 6",
      "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+bu[1]，不以小數近似替代。",
          "en": "Recompute au+bu[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "4"
        ],
        "output": "6",
        "label": "au+bu[1]"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "scalar additivity residual[0] := sub(-6, -6)",
      "after": "scalar additivity residual[0] = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按題目指定的前項減後項，保留殘差的正負。",
        "en": "Subtract the second operand from the first in the requested order,retaining the residual sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 scalar additivity residual[0]，不以小數近似替代。",
          "en": "Recompute scalar additivity residual[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-6",
          "-6"
        ],
        "output": "0",
        "label": "scalar additivity residual[0]"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "scalar additivity residual[1] := sub(6, 6)",
      "after": "scalar additivity residual[1] = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按題目指定的前項減後項，保留殘差的正負。",
        "en": "Subtract the second operand from the first in the requested order,retaining the residual sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 scalar additivity residual[1]，不以小數近似替代。",
          "en": "Recompute scalar additivity residual[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "6",
          "6"
        ],
        "output": "0",
        "label": "scalar additivity residual[1]"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "a=1;b=2;u=[-2, 2]",
      "after": "left=[-6, 6];right=[-6, 6];residual=[0, 0]",
      "relation": "implies",
      "rule_id": "two_scalar_routes_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加純量與先各自縮放是兩條路；每個殘差分量為0才核對到這一組給定值。",
        "en": "Adding scalars first and scaling separately are two routes.Check that each residual component is zero for these supplied values."
      },
      "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,
        2
      ],
      [
        "-6",
        "6"
      ],
      [
        "-6",
        "6"
      ],
      [
        "0",
        "0"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([-2, 2], [-6, 6], [-6, 6], [0, 0])",
    "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-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": 61,
      "original_line_sha256": "1fe83fc37514df2492978d92b6e371cfa3c212c53c7887f477a2ccfae32a1dbe",
      "question_compact_sha256": "27e35e572dd3e51308effb641d184771486d78ed852d35a7c07f163a630cb8f1",
      "english_row_compact_sha256": "7208f3a3c32795fd4f8feb2fba0982098471b1b1687af9805dd43ebe64e8efa8",
      "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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        "english_sha256": "8b23c6e9698bb13ffb12ef863fc1ba12d88c96f880c720e6504681e634fcd99d",
        "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": "a719c1c719a4889766f0378b457e8c0b844399a7d61b9987b9666d76bc7c3a09",
    "original_candidate_line_sha256": "a719c1c719a4889766f0378b457e8c0b844399a7d61b9987b9666d76bc7c3a09"
  },
  "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": 13,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "7dcbca10c4175b3525979b9164787ae4a8cfea53c134890f9e90b18b121d9da0",
      "arithmetic_steps": 11
    },
    "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."
  }
}