{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261101-r01-vector-subspace-0030", "pilot_number": "R0031-16480", "number": 16480, "family_id": "qf-mod32-subspace-universal-line-certificate", "title": { "zh-Hant": "第 16480 題：實向量子空間", "en": "Question 16480: A Homogeneous Kernel Has Universal Closure under Real Linear Combinations" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261101-r01-vector-subspace-0030", "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-universal-line-certificate；資料={\"a\":5,\"b\":-5}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。", "answer": { "kind": "expression", "value": "([5, -5], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)", "explanation_zh": "Wave5根節點核對=([5, -5], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)；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-universal-line-certificate", "seed": 202611011030, "parameters": { "a": 5, "b": -5, "domain": "Exact_R2_subspace_certificate", "operation": "subspace-universal-line-certificate", "semantic_template": "qf-mod32-subspace-universal-line-certificate/{\"a\":5,\"b\":-5}" } }, "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-0030", "batch_id": "batch-20261014-advance-20261101-r01-vector-subspace-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "21f71c30af82a00f01e28d2273d73aeef1fff52097d11c522b4d1d3da204d0ed", "family_id": "qf-mod32-subspace-universal-line-certificate", "seed": 202611011030, "semantic_template": "qf-mod32-subspace-universal-line-certificate/{\"a\":5,\"b\":-5}" }, "fields": { "title": "A Homogeneous Kernel Has Universal Closure under Real Linear Combinations", "stem": "Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[5, -5] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).", "answer": { "kind": "expression", "value": "([5, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)", "explanation": "The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([5, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)." }, "skills": [ "A Homogeneous Kernel Has Universal Closure under Real Linear Combinations" ], "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-universal-line-certificate；資料={\"a\":5,\"b\":-5}。請依 [normal,zero_in,addition_closed,scalar_closed,certificate] 欄序回答。", "en": "Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[5, -5] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "same usual real R2;fixed normal=[5, -5];U consists of ALL x with(5)*x0+(-5)*x1=0", "after": "zero vector:( 5)*0+(-5)*0=0;zero belongs to U", "relation": "implies", "rule_id": "subspace_homogeneous_zero_vector_universal_parent", "conditions": [], "explanation": { "zh-Hant": "方程定義完整集合，不是有限樣本。先代入零向量，且往後所有向量與純量都在同一通常實向量空間。", "en": "The equation defines the whole set,not a finite sample.First substitute the zero vector;all later vectors and scalars stay in the same usual real vector space." }, "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": "ANY u,v in this U;(5)*u0+(-5)*u1=0 and(5)*v0+(-5)*v1=0", "after": "(5)*(u0+v0)+(-5)*(u1+v1)=((5)*u0+(-5)*u1)+((5)*v0+(-5)*v1)=0+0=0;u+v in U", "relation": "implies", "rule_id": "subspace_addition_linearity_for_all_solutions", "conditions": [], "explanation": { "zh-Hant": "逐項分配後重排成兩個已知為零的方程，才得到任意兩個解相加仍是解；不是只加本題幾個點。", "en": "Distribute and regroup into the two equations already equal to zero.This proves closure for any two solutions,not just selected points." }, "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": "ANY real scalar r and ANY u in U;(5)*u0+(-5)*u1=0", "after": "(5)*(r*u0)+(-5)*(r*u1)=r*((5)*u0+(-5)*u1)=r*0=0;r*u in U", "relation": "implies", "rule_id": "subspace_every_real_scalar_not_finite_sample_scalars", "conditions": [], "explanation": { "zh-Hant": "把固定方程的係數與任意實數r交換、提出r，得到0；包括0、負數、無理數，不把有限整數測試冒充每個實純量封閉。", "en": "Commute the fixed coefficients with an arbitrary real r and factor it out.The result is zero,including zero,negative and irrational scalars.Finitely many integer tests are not a substitute for every-real-scalar closure." }, "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": [ [ 5, -5 ], true, true, true, "homogeneous_kernel_closed_for_all_real_scalars" ] }, "conclusion": { "canonical_value": "([5, -5], 是, 是, 是, homogeneous_kernel_closed_for_all_real_scalars)", "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": 30, "original_line_sha256": "32da0f95d285adf42a30d37677b4e366389803b65e2505718cdfb5d630567d74", "question_compact_sha256": "21f71c30af82a00f01e28d2273d73aeef1fff52097d11c522b4d1d3da204d0ed", "english_row_compact_sha256": "c7d90e975ee07ce2577c6f0acf93a9fbc8f04bd4a66ff8edf4605d76b36f23c9", "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": "1f73c6327cfad36608b57665a5994989ed5273637ee26274b48dda5f95392cef", "original_candidate_line_sha256": "1f73c6327cfad36608b57665a5994989ed5273637ee26274b48dda5f95392cef" }, "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": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "07517caed7688871d41fc259ea6bde853e91ff622f5ba494d8d52226c02c70c9", "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." } }