{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261101-r01-vector-subspace-0086", "pilot_number": "R0031-16411", "number": 16411, "family_id": "qf-mod32-subspace-classification", "title": { "zh-Hant": "第 16411 題：實向量子空間", "en": "Question 16411: 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-0086", "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\":4,\"c\":-2}。請依 [equation,classification,subspace,certificate] 欄序回答。", "answer": { "kind": "expression", "value": "([0, 4, -2], affine_line_not_through_zero, 否, affine_line_not_through_zero)", "explanation_zh": "Wave5根節點核對=([0, 4, -2], affine_line_not_through_zero, 否, affine_line_not_through_zero)；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": 2, "skills": [ "實向量子空間" ] }, "variant": { "family_id": "qf-mod32-subspace-classification", "seed": 202611011086, "parameters": { "a": 0, "b": 4, "c": -2, "domain": "Exact_R2_subspace_certificate", "operation": "subspace-classification", "semantic_template": "qf-mod32-subspace-classification/{\"a\":0,\"b\":4,\"c\":-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-0086", "batch_id": "batch-20261014-advance-20261101-r01-vector-subspace-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "4497ff9fc3ad07e3d7841e4700f6bac0ba671bd3ee4fe519fc0e89ae4023382a", "family_id": "qf-mod32-subspace-classification", "seed": 202611011086, "semantic_template": "qf-mod32-subspace-classification/{\"a\":0,\"b\":4,\"c\":-2}" }, "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, 4, -2]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.", "answer": { "kind": "expression", "value": "([0, 4, -2], affine_line_not_through_zero, No, affine_line_not_through_zero)", "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, 4, -2], affine_line_not_through_zero, No, affine_line_not_through_zero)." }, "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\":4,\"c\":-2}。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, 4, -2]. 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": "affine nonempty particular y coordinate := div(-2, 4)", "after": "affine nonempty particular y coordinate = -1/2", "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 affine nonempty particular y coordinate，不以小數近似替代。", "en": "Recompute affine nonempty particular y coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-2", "4" ], "output": "-1/2", "label": "affine nonempty particular y coordinate" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "affine particular first term := mul(0, 0)", "after": "affine particular first term = 0", "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 affine particular first term，不以小數近似替代。", "en": "Recompute affine particular first term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "affine particular first term" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "affine particular second term := mul(4, -1/2)", "after": "affine particular second term = -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 affine particular second term，不以小數近似替代。", "en": "Recompute affine particular second term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "-1/2" ], "output": "-2", "label": "affine particular second term" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "affine particular equation value := sum(0, -2)", "after": "affine particular equation value = -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 affine particular equation value，不以小數近似替代。", "en": "Recompute affine particular equation value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "-2" ], "output": "-2", "label": "affine particular equation value" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "nonzero original normal=[0, 4];nonzero right side=-2;particular real solution=[0, -1/2];checked equation=-2", "after": "nonempty affine line;zero gives0 rather than nonzero right side;affine_line_not_through_zero;subspace=否", "relation": "implies", "rule_id": "subspace_affine_nonempty_but_zero_missing", "conditions": [], "explanation": { "zh-Hant": "非零normal保有直線解集，已核一個實解排除空集；但零向量左端為0不等於c，因此不含零，不是子空間。不可把平移後的仿射直線悄悄當過原點直線。", "en": "The nonzero normal gives a line,and the checked real solution rules out emptiness.But the zero vector has left side0 rather than nonzero c,so this affine line omits zero and is not a subspace.Do not silently translate it into a homogeneous 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, 4, -2 ], "affine_line_not_through_zero", false, "affine_line_not_through_zero" ] }, "conclusion": { "canonical_value": "([0, 4, -2], affine_line_not_through_zero, 否, affine_line_not_through_zero)", "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": 86, "original_line_sha256": "2e9096bb850b89698afbb115d985318fa7fc721c756054ab57f87679db58e205", "question_compact_sha256": "4497ff9fc3ad07e3d7841e4700f6bac0ba671bd3ee4fe519fc0e89ae4023382a", "english_row_compact_sha256": "7c0696e661a452ba26c7db9825cdea6ddb43c76098003965cd08595fa2e9e8b5", "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": "5de034e8a13f603b37031b1f7596b7e52381eafc3fd970dc7c752941ccd65dd5", "original_candidate_line_sha256": "5de034e8a13f603b37031b1f7596b7e52381eafc3fd970dc7c752941ccd65dd5" }, "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": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "549204022a870551b156f446ae2fb4e90a558e520743007959eace513727e306", "arithmetic_steps": 4 }, "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." } }