{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261101-r01-vector-subspace-0054",
  "pilot_number": "R0031-16429",
  "number": 16429,
  "family_id": "qf-mod32-subspace-finite-set-counterexample",
  "title": {
    "zh-Hant": "第 16429 題：實向量子空間",
    "en": "Question 16429: A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261101-r01-vector-subspace-0054",
    "batch_id": "batch-20261014-advance-20261101-r01-vector-subspace-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-vector-subspace"
    ],
    "title_zh": "實向量子空間",
    "stem_zh": "R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[4,1]]}。請依 [points,witness,subspace,classification] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[0, 0], [4, 1]], [scalar_failure, 2, [4, 1], [8, 2]], 否, not_subspace_exact_counterexample)",
      "explanation_zh": "Wave5根節點核對=([[0, 0], [4, 1]], [scalar_failure, 2, [4, 1], [8, 2]], 否, not_subspace_exact_counterexample)；依完整明示資料重算；子空間須含零並對加法與每個實純量封閉；有限正樣本不足；不過原點仿射直線不是子空間。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "實向量子空間"
      ]
    },
    "variant": {
      "family_id": "qf-mod32-subspace-finite-set-counterexample",
      "seed": 202611011054,
      "parameters": {
        "points": [
          [
            0,
            0
          ],
          [
            4,
            1
          ]
        ],
        "domain": "Exact_R2_subspace_certificate",
        "operation": "subspace-finite-set-counterexample",
        "semantic_template": "qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[4,1]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式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-20261101-r01-vector-subspace-0054",
    "batch_id": "batch-20261014-advance-20261101-r01-vector-subspace-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ebe0dd79ceb9bb0e2fcea53bdec8d48daedb0fa81baf92caad805ef28d6fa30d",
      "family_id": "qf-mod32-subspace-finite-set-counterexample",
      "seed": 202611011054,
      "semantic_template": "qf-mod32-subspace-finite-set-counterexample/{\"points\":[[0,0],[4,1]]}"
    },
    "fields": {
      "title": "A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure",
      "stem": "Use the complete nonempty duplicate-free finite point set=[[0, 0], [4, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.",
      "answer": {
        "kind": "expression",
        "value": "([[0, 0], [4, 1]], [scalar_failure, 2, [4, 1], [8, 2]], No, not_subspace_exact_counterexample)",
        "explanation": "Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [4, 1]], [scalar_failure, 2, [4, 1], [8, 2]], No, not_subspace_exact_counterexample)."
      },
      "skills": [
        "A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure"
      ],
      "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": "R²通常運算與明示全稱代數證書。操作=subspace-finite-set-counterexample；資料={\"points\":[[0,0],[4,1]]}。請依 [points,witness,subspace,classification] 欄序回答。",
        "en": "Use the complete nonempty duplicate-free finite point set=[[0, 0], [4, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "DECLARED complete finite candidate subset of usual real R2=[[0, 0], [4, 1]];original point order retained",
      "after": "zero vector present=是;raw points nonempty and duplicate-free=是",
      "relation": "implies",
      "rule_id": "subspace_complete_finite_set_not_unknown_sampled_universe",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整有限點集是候選集合本身，不是未知子空間的抽樣；保留原點順序，先優先找缺零反例，不能去重或重排改變原見證。",
        "en": "The complete finite list is the candidate set itself,not a sample from an unknown subspace.Keep original point order and prefer a missing-zero counterexample;do not deduplicate or reorder to change the source witness."
      },
      "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": "first original nonzero point index=1;point=[4, 1];complete set size=2",
      "after": "integer multiples k*point for k=2 through4 are 3 DISTINCT vectors;some image must be absent",
      "relation": "implies",
      "rule_id": "subspace_nonzero_real_multiples_finite_pigeonhole_obligation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非零點有某個非零座標，所以兩純量給同倍點可消去該座標得兩純量相同；L+1個不同倍點不能全放進L點集合。此論證保證搜尋會停，不是用有限正例猜封閉。",
        "en": "A nonzero coordinate lets equal scalar multiples imply equal scalars.Thus L+1 distinct multiples cannot all lie in an L-point set.This guarantees a missing multiple rather than guessing closure from positive tests."
      },
      "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": "finite scalar2 image coordinate0 := mul(2, 4)",
      "after": "finite scalar2 image coordinate0 = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始欄位角色與已核中間量精確計算，保留分母、正負號、索引及完整載體。",
        "en": "Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite scalar2 image coordinate0，不以小數近似替代。",
          "en": "Recompute finite scalar2 image coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "4"
        ],
        "output": "8",
        "label": "finite scalar2 image coordinate0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "finite scalar2 image coordinate1 := mul(2, 1)",
      "after": "finite scalar2 image coordinate1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始欄位角色與已核中間量精確計算，保留分母、正負號、索引及完整載體。",
        "en": "Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite scalar2 image coordinate1，不以小數近似替代。",
          "en": "Recompute finite scalar2 image coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "finite scalar2 image coordinate1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "first original nonzero point=[4, 1];candidate scalar=2;exact image=[8, 2]",
      "after": "image belongs to complete candidate=否",
      "relation": "implies",
      "rule_id": "subspace_first_missing_integer_scalar_in_source_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依2、3、…順序核真實集合成員資格，第一個缺失才是原協定見證；推得的像可超過原始座標輸入界限，不能因此刪掉反例。",
        "en": "Test actual membership for scalars2,3,...in order.The first missing image is the source witness.Derived coordinates may exceed raw input bounds;that does not erase the counterexample."
      },
      "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": [
      [
        [
          0,
          0
        ],
        [
          4,
          1
        ]
      ],
      [
        "scalar_failure",
        2,
        [
          4,
          1
        ],
        [
          8,
          2
        ]
      ],
      false,
      "not_subspace_exact_counterexample"
    ]
  },
  "conclusion": {
    "canonical_value": "([[0, 0], [4, 1]], [scalar_failure, 2, [4, 1], [8, 2]], 否, not_subspace_exact_counterexample)",
    "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."
  },
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      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
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    "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."
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  "revision": {
    "id": "review-20261026-007-p32",
    "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.",
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    "bilingual_semantics": true,
    "steps_reviewed": 5,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}