{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261101-r01-vector-subspace-0096",
  "pilot_number": "R0031-16421",
  "number": 16421,
  "family_id": "qf-mod32-subspace-classification",
  "title": {
    "zh-Hant": "第 16421 題：實向量子空間",
    "en": "Question 16421: Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261101-r01-vector-subspace-0096",
    "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-classification；資料={\"a\":0,\"b\":0,\"c\":8}。請依 [equation,classification,subspace,certificate] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([0, 0, 8], empty_set, 否, empty_set)",
      "explanation_zh": "Wave5根節點核對=([0, 0, 8], empty_set, 否, empty_set)；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": 3,
      "skills": [
        "實向量子空間"
      ]
    },
    "variant": {
      "family_id": "qf-mod32-subspace-classification",
      "seed": 202611011096,
      "parameters": {
        "a": 0,
        "b": 0,
        "c": 8,
        "domain": "Exact_R2_subspace_certificate",
        "operation": "subspace-classification",
        "semantic_template": "qf-mod32-subspace-classification/{\"a\":0,\"b\":0,\"c\":8}"
      }
    },
    "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-0096",
    "batch_id": "batch-20261014-advance-20261101-r01-vector-subspace-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "7e8c9efb6d33875f476f93c352dbf4fa9aa312141f4da95dade33070e7a4417d",
      "family_id": "qf-mod32-subspace-classification",
      "seed": 202611011096,
      "semantic_template": "qf-mod32-subspace-classification/{\"a\":0,\"b\":0,\"c\":8}"
    },
    "fields": {
      "title": "Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set",
      "stem": "In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 0, 8]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.",
      "answer": {
        "kind": "expression",
        "value": "([0, 0, 8], empty_set, No, empty_set)",
        "explanation": "If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([0, 0, 8], empty_set, No, empty_set)."
      },
      "skills": [
        "Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set"
      ],
      "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-classification；資料={\"a\":0,\"b\":0,\"c\":8}。請依 [equation,classification,subspace,certificate] 欄序回答。",
        "en": "In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 0, 8]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "original equation coefficients=[0, 0, 8];left side is identically0",
      "after": "equation0=8;classification=empty_set;subspace=否",
      "relation": "implies",
      "rule_id": "subspace_zero_normal_whole_or_empty_not_line",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩係數同零必先分支：c=0則所有實向量都滿足，c非0則無解且缺零，不把這兩種情況硬叫一條直線。",
        "en": "When both coefficients vanish,split first:c=0 gives every real vector,while nonzero c gives no solution and omits zero.Neither case is a nondegenerate line."
      },
      "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,
        8
      ],
      "empty_set",
      false,
      "empty_set"
    ]
  },
  "conclusion": {
    "canonical_value": "([0, 0, 8], empty_set, 否, empty_set)",
    "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-20261014-advance-20261101-r01-vector-subspace-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261101-r01-vector-subspace-foundations/questions.jsonl",
      "batch_sha256": "be86d8792c3336b601ef1d82acef63d9c51e5095f202e938fc2873f72209ba19",
      "original_line": 96,
      "original_line_sha256": "e6b2159d62175d420e55c7b643caa79e3b803023367a000793017338ae09dd3c",
      "question_compact_sha256": "7e8c9efb6d33875f476f93c352dbf4fa9aa312141f4da95dade33070e7a4417d",
      "english_row_compact_sha256": "f3ec4a2a3a405a41ba4bb69734b678efa3ad766f607f7a839af1f330e8c0c26c",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261101-r01-vector-subspace-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261101-r01-vector-subspace-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-vector-subspace": {
        "sha256": "a92727827e6ab175bea8b4da7021fe6b063af40e68f09beb9ad7437cd72c1abc",
        "english_sha256": "23093cbd91680008b49cf03eba212164f763c1dccecc9b30a56bccbe278919ca",
        "original_path": "public/data/mko/vector-subspace.json",
        "english_path": "public/data/i18n/editions/mko-vector-subspace.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "633793a5f242dece0a240bd92f77a33c587d12463e0beaedd8f70a14c70cf9e7",
    "original_candidate_sha256": "444202076f82d3a36007247c1894cf772bf1ce086a5f256179cfae1274701bbd",
    "original_candidate_line_sha256": "444202076f82d3a36007247c1894cf772bf1ce086a5f256179cfae1274701bbd"
  },
  "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-20261026-007-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.",
    "review_receipts": {
      "semantic_review_sha256": "55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369a",
      "question_first_witness_sha256": "754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bb"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 1,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "d230a6a8b7611af9d23419833d86c1457caa45e8da139629820428776b8cae02",
      "arithmetic_steps": 0
    },
    "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."
  }
}