{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261023-r02-linear-combination-span-0097",
  "pilot_number": "R0020-10847",
  "number": 10847,
  "family_id": "qf-mod23-span-single-generator-membership",
  "title": {
    "zh-Hant": "第 10847 題：有限線性組合",
    "en": "Question 10847: A Single Generator, Including the Zero-Vector Case"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261023-r02-linear-combination-span-0097",
    "batch_id": "batch-20261013-advance-20261023-r02-linear-combination-span-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-combination-span"
    ],
    "title_zh": "有限線性組合",
    "stem_zh": "固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[22,44]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([2, 4], [22, 44], 11, 是)",
      "explanation_zh": "現代橋梁核對=([2, 4], [22, 44], 11, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限線性組合"
      ]
    },
    "variant": {
      "family_id": "qf-mod23-span-single-generator-membership",
      "seed": 202610232097,
      "parameters": {
        "generator": [
          2,
          4
        ],
        "target": [
          22,
          44
        ],
        "domain": "Finite_coordinate_linear_combinations",
        "operation": "span-single-generator-membership",
        "semantic_template": "qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[22,44]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、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-20261023-r02-linear-combination-span-0097",
    "batch_id": "batch-20261013-advance-20261023-r02-linear-combination-span-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2eb473ee0c7d5c1f87c470d63db19a9fd96ca01aeb20262f81e9888dbba50a93",
      "family_id": "qf-mod23-span-single-generator-membership",
      "seed": 202610232097,
      "semantic_template": "qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[22,44]}"
    },
    "fields": {
      "title": "A Single Generator, Including the Zero-Vector Case",
      "stem": "Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[22, 44]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "([2, 4], [22, 44], 11, Yes)",
        "explanation": "Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [22, 44], 11, Yes)."
      },
      "skills": [
        "A Single Generator, Including the Zero-Vector Case"
      ],
      "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": "固定維度與明示有限生成族。操作=span-single-generator-membership；資料={\"generator\":[2,4],\"target\":[22,44]}。請依 [generator,target,scalar_or_status,in_span] 欄序回答。",
        "en": "Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[22, 44]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "fixed real coordinate space;span uses all finite real linear combinations",
      "after": "given coefficient witness and existence of some coefficients are separate",
      "relation": "implies",
      "rule_id": "span_quantifier_and_witness_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "span量化所有有限實係數組合；本題的整數／有理資料是明示切片，一個指定見證失敗不等於所有係數都失敗。",
        "en": "Span quantifies all finite real linear combinations.These integer/rational instances are declared slices;a failed designated witness does not show every coefficient choice fails."
      },
      "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": "single-generator first nonzero pivot scalar := div(22, 2)",
      "after": "single-generator first nonzero pivot scalar = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 single-generator first nonzero pivot scalar，不以小數近似替代。",
          "en": "Recompute single-generator first nonzero pivot scalar with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "22",
          "2"
        ],
        "output": "11",
        "label": "single-generator first nonzero pivot scalar"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "generator=[2, 4];target=[22, 44];first nonzero coordinate=0",
      "after": "necessary scalar=11",
      "relation": "implies",
      "rule_id": "single_generator_pivot_necessary_scalar",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非零首樞軸方程強迫任何可行純量等於此比值，負分母照精確有理數規約。",
        "en": "The first nonzero pivot equation forces every possible scalar to equal this ratio,with a negative denominator normalized exactly."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "single-generator coordinate0 scaled := mul(11, 2)",
      "after": "single-generator coordinate0 scaled = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 single-generator coordinate0 scaled，不以小數近似替代。",
          "en": "Recompute single-generator coordinate0 scaled with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "2"
        ],
        "output": "22",
        "label": "single-generator coordinate0 scaled"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "single-generator coordinate0 residual := sub(22, 22)",
      "after": "single-generator coordinate0 residual = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 single-generator coordinate0 residual，不以小數近似替代。",
          "en": "Recompute single-generator coordinate0 residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "22",
          "22"
        ],
        "output": "0",
        "label": "single-generator coordinate0 residual"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "single-generator coordinate1 scaled := mul(11, 4)",
      "after": "single-generator coordinate1 scaled = 44",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 single-generator coordinate1 scaled，不以小數近似替代。",
          "en": "Recompute single-generator coordinate1 scaled with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "4"
        ],
        "output": "44",
        "label": "single-generator coordinate1 scaled"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "single-generator coordinate1 residual := sub(44, 44)",
      "after": "single-generator coordinate1 residual = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 single-generator coordinate1 residual，不以小數近似替代。",
          "en": "Recompute single-generator coordinate1 residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "44",
          "44"
        ],
        "output": "0",
        "label": "single-generator coordinate1 residual"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "pivot scalar=11;all coordinate residuals=[0, 0]",
      "after": "in single-generator span=是;returned scalar remains=11",
      "relation": "implies",
      "rule_id": "single_generator_all_coordinates_after_pivot",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩條方向都核：全相等給實際見證；任一不等，因樞軸已強迫唯一純量，無其他純量能補救。即使失敗也按協定保留原候選比值。",
        "en": "Both directions matter:all equalities give a witness;if any fails,the pivot already forces the unique scalar,so no other scalar repairs it.Retain the candidate ratio even on failure as specified."
      },
      "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,
        4
      ],
      [
        22,
        44
      ],
      "11",
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([2, 4], [22, 44], 11, 是)",
    "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-20261013-advance-20261023-r02-linear-combination-span-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261023-r02-linear-combination-span-foundations/questions.jsonl",
      "batch_sha256": "da062918e67f92a171ca3b576c4e8fd07cc820916a8047a29a22a0da6e0df6d4",
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      "original_line_sha256": "999347fc788c4eebe96a7d4383b383b94f76c08e09ac336636d65e9a9d85ca69",
      "question_compact_sha256": "2eb473ee0c7d5c1f87c470d63db19a9fd96ca01aeb20262f81e9888dbba50a93",
      "english_row_compact_sha256": "89f6cead0c036708d684b6acaf20752e63e269368d2d033ba64110326fc67309",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261023-r02-linear-combination-span-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261023-r02-linear-combination-span-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-linear-combination-span": {
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        "english_sha256": "3d16797b83376915e08d16c68eee0e87ae7d9b28ca5d5ba76a9e2466ce8195b3",
        "original_path": "public/data/mko/linear-combination-span.json",
        "english_path": "public/data/i18n/editions/mko-linear-combination-span.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "bd67681fffd9e29972c322d497cd945a668a794dd25e0c8cd26f1a6ce22e4dfb",
    "original_candidate_sha256": "cdc8a22e11e8db05c35cc3c270f549f40fd411526226c04b4a41173637a7e73c",
    "original_candidate_line_sha256": "cdc8a22e11e8db05c35cc3c270f549f40fd411526226c04b4a41173637a7e73c"
  },
  "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-20261026-006-p21",
    "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": "b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f",
      "question_first_witness_sha256": "3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4"
    },
    "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": "e59ac10e2e0bb2d6f4cf54ecf3fa7873d788568608b2ab756e79512e94e47241",
      "arithmetic_steps": 5
    },
    "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."
  }
}