{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261101-r01-vector-subspace-0020", "pilot_number": "R0031-16470", "number": 16470, "family_id": "qf-mod32-subspace-line-membership", "title": { "zh-Hant": "第 16470 題：實向量子空間", "en": "Question 16470: Membership in a Homogeneous Real Line Is an Exact Equation Check" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261101-r01-vector-subspace-0020", "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-line-membership；資料={\"a\":20,\"b\":-5,\"x\":7,\"y\":2}。請依 [point,normal,equation_value,in_line] 欄序回答。", "answer": { "kind": "expression", "value": "([7, 2], [20, -5], 130, 否)", "explanation_zh": "Wave5根節點核對=([7, 2], [20, -5], 130, 否)；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-line-membership", "seed": 202611011020, "parameters": { "a": 20, "b": -5, "x": 7, "y": 2, "domain": "Exact_R2_subspace_certificate", "operation": "subspace-line-membership", "semantic_template": "qf-mod32-subspace-line-membership/{\"a\":20,\"b\":-5,\"x\":7,\"y\":2}" } }, "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-0020", "batch_id": "batch-20261014-advance-20261101-r01-vector-subspace-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "4274bae80ad0380653f8dcf4c863136819e736431f818a33a103f394e581fa6b", "family_id": "qf-mod32-subspace-line-membership", "seed": 202611011020, "semantic_template": "qf-mod32-subspace-line-membership/{\"a\":20,\"b\":-5,\"x\":7,\"y\":2}" }, "fields": { "title": "Membership in a Homogeneous Real Line Is an Exact Equation Check", "stem": "Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[20, -5]. Selected point=[7, 2]. Give (point,normal,equation_value,in_line?).", "answer": { "kind": "expression", "value": "([7, 2], [20, -5], 130, No)", "explanation": "Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([7, 2], [20, -5], 130, No)." }, "skills": [ "Membership in a Homogeneous Real Line Is an Exact Equation Check" ], "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-line-membership；資料={\"a\":20,\"b\":-5,\"x\":7,\"y\":2}。請依 [point,normal,equation_value,in_line] 欄序回答。", "en": "Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[20, -5]. Selected point=[7, 2]. Give (point,normal,equation_value,in_line?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "homogeneous selected first coordinate term := mul(20, 7)", "after": "homogeneous selected first coordinate term = 140", "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 homogeneous selected first coordinate term，不以小數近似替代。", "en": "Recompute homogeneous selected first coordinate term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "20", "7" ], "output": "140", "label": "homogeneous selected first coordinate term" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "homogeneous selected second coordinate term := mul(-5, 2)", "after": "homogeneous selected second coordinate term = -10", "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 homogeneous selected second coordinate term，不以小數近似替代。", "en": "Recompute homogeneous selected second coordinate term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-5", "2" ], "output": "-10", "label": "homogeneous selected second coordinate term" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "homogeneous selected equation value := sum(140, -10)", "after": "homogeneous selected equation value = 130", "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 homogeneous selected equation value，不以小數近似替代。", "en": "Recompute homogeneous selected equation value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "140", "-10" ], "output": "130", "label": "homogeneous selected equation value" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "same homogeneous line normal=[20, -5];selected point=[7, 2];equation value=130", "after": "selected point belongs to line=否;one membership check is not universal subspace closure", "relation": "implies", "rule_id": "subspace_point_equation_not_universal_certificate", "conditions": [], "explanation": { "zh-Hant": "只由這個點的方程值判本點是否在直線，不把一個成功點變成全稱加法／實數倍封閉證書。", "en": "The selected equation value decides only this point's membership.A successful point is not a universal addition/scalar-closure certificate." }, "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": [ [ 7, 2 ], [ 20, -5 ], 130, false ] }, "conclusion": { "canonical_value": "([7, 2], [20, -5], 130, 否)", "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": 20, "original_line_sha256": "562d825fc67608644cad3d97151aa346868067c0e9007ddaeffca0527fa972c7", "question_compact_sha256": "4274bae80ad0380653f8dcf4c863136819e736431f818a33a103f394e581fa6b", "english_row_compact_sha256": "6207d5adb9146981aa6ea8e42ebf1e52cc26583a8acb8944e35d7aa922987e4f", "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": "3f0c9e8973caded8c19b85c588973a19be5500a713c54d145f92c50e0409f0e1", "original_candidate_line_sha256": "3f0c9e8973caded8c19b85c588973a19be5500a713c54d145f92c50e0409f0e1" }, "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": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "40009bbdca4570880918f2d96bfbae3a43844fe351bef4f53db95df894a8a1d7", "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." } }