{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261022-r04-real-vector-space-0047",
  "pilot_number": "R0058-29697",
  "number": 29697,
  "family_id": "qf-str22-vectorspace-scalar-multiplication",
  "title": {
    "zh-Hant": "第 29697 題：實座標向量空間",
    "en": "Question 29697: A Declared Real-Scalar Instance on Integer Coordinates"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261022-r04-real-vector-space-0047",
    "batch_id": "batch-20261013-advance-20261022-r04-real-vector-space-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-real-vector-space"
    ],
    "title_zh": "實座標向量空間",
    "stem_zh": "固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-3,\"vector\":[9,-1,-18]}。請依 [scalar,vector,result] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(-3, [9, -1, -18], [-27, 3, 54])",
      "explanation_zh": "結構基礎核對=(-3, [9, -1, -18], [-27, 3, 54])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；只核指定座標公理實例；不冒充任意向量空間完整證明。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "實座標向量空間"
      ]
    },
    "variant": {
      "family_id": "qf-str22-vectorspace-scalar-multiplication",
      "seed": 202610224047,
      "parameters": {
        "scalar": -3,
        "vector": [
          9,
          -1,
          -18
        ],
        "domain": "Finite_coordinate_real_vector_space",
        "operation": "vectorspace-scalar-multiplication",
        "semantic_template": "qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-3,\"vector\":[9,-1,-18]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261022-r04-real-vector-space-0047",
    "batch_id": "batch-20261013-advance-20261022-r04-real-vector-space-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "43a2f24e66617a100ed59cb12a2d20814bbb8c1b6807dbe55cbb9299108024a6",
      "family_id": "qf-str22-vectorspace-scalar-multiplication",
      "seed": 202610224047,
      "semantic_template": "qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-3,\"vector\":[9,-1,-18]}"
    },
    "fields": {
      "title": "A Declared Real-Scalar Instance on Integer Coordinates",
      "stem": "Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[9, -1, -18]. Give (scalar,vector,scaled vector).",
      "answer": {
        "kind": "expression",
        "value": "(-3, [9, -1, -18], [-27, 3, 54])",
        "explanation": "Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, [9, -1, -18], [-27, 3, 54])."
      },
      "skills": [
        "A Declared Real-Scalar Instance on Integer Coordinates"
      ],
      "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": "固定維度實座標空間。操作=vectorspace-scalar-multiplication；資料={\"scalar\":-3,\"vector\":[9,-1,-18]}。請依 [scalar,vector,result] 欄序回答。",
        "en": "Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[9, -1, -18]. Give (scalar,vector,scaled vector)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-str22-vectorspace-scalar-multiplication, vectorspace-scalar-multiplication]",
      "after": "derived = [[scalar, vector, result]]",
      "relation": "implies",
      "rule_id": "R58_EXACT_source_interface_and_answer_ROLE",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題銀行的keys、domain或frame、operation與輸出角色。STR22/App20 tuple與legacy scalar不同；分數是否保/1按原契約，不能用共同外觀改題。",
        "en": "Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。",
          "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Vector ORIGINAL scalar product coordinate0 := mul(-3, 9)",
      "after": "Vector ORIGINAL scalar product coordinate0 = -27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。",
        "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Vector ORIGINAL scalar product coordinate0，不以小數近似替代。",
          "en": "Recompute Vector ORIGINAL scalar product coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "9"
        ],
        "output": "-27",
        "label": "Vector ORIGINAL scalar product coordinate0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Vector ORIGINAL scalar product coordinate1 := mul(-3, -1)",
      "after": "Vector ORIGINAL scalar product coordinate1 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。",
        "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Vector ORIGINAL scalar product coordinate1，不以小數近似替代。",
          "en": "Recompute Vector ORIGINAL scalar product coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "-1"
        ],
        "output": "3",
        "label": "Vector ORIGINAL scalar product coordinate1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Vector ORIGINAL scalar product coordinate2 := mul(-3, -18)",
      "after": "Vector ORIGINAL scalar product coordinate2 = 54",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。",
        "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Vector ORIGINAL scalar product coordinate2，不以小數近似替代。",
          "en": "Recompute Vector ORIGINAL scalar product coordinate2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "-18"
        ],
        "output": "54",
        "label": "Vector ORIGINAL scalar product coordinate2"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [-3, [9, -1, -18]]",
      "after": "derived = [[-27, 3, 54]]",
      "relation": "implies",
      "rule_id": "Vector_scalar_APPLIED_to_ALL_coordinates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一scalar乘全部coordinates，scalar0仍保同一維度zero vector；derived值不重套raw bounds。",
        "en": "Apply the SAME scalar to EVERY coordinate.Scalar0 gives the zero vector in the SAME dimension;derived values do not inherit raw bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。",
          "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      -3,
      [
        9,
        -1,
        -18
      ],
      [
        -27,
        3,
        54
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(-3, [9, -1, -18], [-27, 3, 54])",
    "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-20261013-advance-20261022-r04-real-vector-space-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261022-r04-real-vector-space-foundations/questions.jsonl",
      "batch_sha256": "2e1a8d7fc8d62329f409c52b55d8384e14dbd1965c738725baf612fc9b16df14",
      "original_line": 47,
      "original_line_sha256": "135221c975a69b560d4144d673120202d13d1611cb458499160833dc060964b3",
      "question_compact_sha256": "43a2f24e66617a100ed59cb12a2d20814bbb8c1b6807dbe55cbb9299108024a6",
      "english_row_compact_sha256": "334c06aeeeabe5e211fcf7afb0769b8b3af747abe51995d2dd6722f634c937d2",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261022-r04-real-vector-space-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261022-r04-real-vector-space-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-real-vector-space": {
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        "english_sha256": "b41b524022de7b68103686353c9d8b1f730143e3cb9c5de0be7d936a81343cd8",
        "original_path": "public/data/mko/real-vector-space.json",
        "english_path": "public/data/i18n/editions/mko-real-vector-space.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "623c23f45586b08629c5a347ffbc994cf8dbff2133c4f4cdb227b851d74ea6b6",
    "original_candidate_sha256": "564e0544c13fc28eebe870bb1396c78565e9a35887700f803f6c84dd85380cae",
    "original_candidate_line_sha256": "564e0544c13fc28eebe870bb1396c78565e9a35887700f803f6c84dd85380cae"
  },
  "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-20261026-011-p59",
    "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": "66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df",
      "question_first_witness_sha256": "b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 5,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "a2da72de002cc69fe25ffd5cd8fc6bec070b38d0805d347e761832355309243a",
      "arithmetic_steps": 3
    },
    "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."
  }
}