{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261028-r03-real-banach-space-0084", "pilot_number": "R0025-13459", "number": 13459, "family_id": "qf-mod28-banach-evidence-mode-audit", "title": { "zh-Hant": "第 13459 題：實Banach空間", "en": "Question 13459: Distinguish a Completeness Theorem from Sampled Cauchy Sequences" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261028-r03-real-banach-space-0084", "batch_id": "batch-20261014-advance-20261028-r03-real-banach-space-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-real-banach-space" ], "title_zh": "實Banach空間", "stem_zh": "載體norm誘導metric與完備證書必須描述同一空間。操作=banach-evidence-mode-audit；資料={\"dimension\":9,\"norm_name\":\"l1\",\"evidence_mode\":\"sampled_cauchy_sequences\"}。請依 [dimension,norm_name,evidence_mode,sufficient,boundary] 欄序回答。", "answer": { "kind": "expression", "value": "(9, l1, sampled_cauchy_sequences, 否, samples_insufficient_for_completeness)", "explanation_zh": "Wave4第二層核對=(9, l1, sampled_cauchy_sequences, 否, samples_insufficient_for_completeness)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；有限維實賦範空間使用命名全稱完備定理；抽樣Cauchy序列不足且不推出Hilbert。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "實Banach空間" ] }, "variant": { "family_id": "qf-mod28-banach-evidence-mode-audit", "seed": 202610283084, "parameters": { "dimension": 9, "norm_name": "l1", "evidence_mode": "sampled_cauchy_sequences", "domain": "Named_finite_dimensional_real_Banach_space", "operation": "banach-evidence-mode-audit", "semantic_template": "qf-mod28-banach-evidence-mode-audit/{\"dimension\":9,\"evidence_mode\":\"sampled_cauchy_sequences\",\"norm_name\":\"l1\"}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式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-20261028-r03-real-banach-space-0084", "batch_id": "batch-20261014-advance-20261028-r03-real-banach-space-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "d165cf152a7bd35a868c2ffa86b2d5c2e1d1bc4af697a4fd5c631437982d1a13", "family_id": "qf-mod28-banach-evidence-mode-audit", "seed": 202610283084, "semantic_template": "qf-mod28-banach-evidence-mode-audit/{\"dimension\":9,\"evidence_mode\":\"sampled_cauchy_sequences\",\"norm_name\":\"l1\"}" }, "fields": { "title": "Distinguish a Completeness Theorem from Sampled Cauchy Sequences", "stem": "Use the explicit real coordinate space R^n of dimension=9, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=sampled_cauchy_sequences. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.", "answer": { "kind": "expression", "value": "(9, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)", "explanation": "The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (9, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)." }, "skills": [ "Distinguish a Completeness Theorem from Sampled Cauchy Sequences" ], "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": "載體norm誘導metric與完備證書必須描述同一空間。操作=banach-evidence-mode-audit；資料={\"dimension\":9,\"norm_name\":\"l1\",\"evidence_mode\":\"sampled_cauchy_sequences\"}。請依 [dimension,norm_name,evidence_mode,sufficient,boundary] 欄序回答。", "en": "Use the explicit real coordinate space R^n of dimension=9, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=sampled_cauchy_sequences. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "named real space R^9;norm=l1;evidence=sampled_cauchy_sequences", "after": "sufficient evidence=否;sample insufficiency is not incompleteness", "relation": "implies", "rule_id": "Banach_evidence_quantifier_scope", "conditions": [], "explanation": { "zh-Hant": "命名全稱定理／上述座標論證覆蓋每個Cauchy序列，有限樣本不行；樣本模式No不是說此已知空間不完備，也不是新形式核驗收據。", "en": "The named universal theorem and coordinate argument cover every Cauchy sequence;finite samples do not.No in sample mode neither declares this known space incomplete nor supplies a formal-verification receipt." }, "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": [ 9, "l1", "sampled_cauchy_sequences", false, "samples_insufficient_for_completeness" ] }, "conclusion": { "canonical_value": "(9, l1, sampled_cauchy_sequences, 否, samples_insufficient_for_completeness)", "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-20261028-r03-real-banach-space-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261028-r03-real-banach-space-foundations/questions.jsonl", "batch_sha256": "0c7e534de890d481ab1182c0a29ecdd37e1106a7f1ab84e240ef036eee9a6808", "original_line": 84, "original_line_sha256": "b73de843a838d9bea15fc46c35446f13bf4c6fbd271a619ee224612e6410e08e", "question_compact_sha256": "d165cf152a7bd35a868c2ffa86b2d5c2e1d1bc4af697a4fd5c631437982d1a13", "english_row_compact_sha256": "61255bbd7cb4400b2898f2870c3e915b8db190f46e44753f62ec7a278b23ed46", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261028-r03-real-banach-space-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261028-r03-real-banach-space-foundations.en.jsonl" }, "mko_bindings": { "mko-real-banach-space": { "sha256": "1c66f08e81b1ae1e47f887e3890cc179fd0aa8071970f275641c4c720de626ca", "english_sha256": "2a5bb4c0c274231ae79955cc73e08fe30d45fea796fa77c3d8ca5213dc75c4f4", "original_path": "public/data/mko/real-banach-space.json", "english_path": "public/data/i18n/editions/mko-real-banach-space.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a02aa2dccf790869cf864b821d0e60119686214561153daba232d921e07e7924", "original_candidate_sha256": "38c48474c2b8c44a89e92b79d2e0bf8adb26a0ddae9a757515fad4354781a35d", "original_candidate_line_sha256": "38c48474c2b8c44a89e92b79d2e0bf8adb26a0ddae9a757515fad4354781a35d" }, "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-p26", "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": "0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5", "question_first_witness_sha256": "ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a" }, "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": "c1ef97cc3fbead065cb78bb3d46f6f46e1f2f0a411263e189ee74fd41e12d327", "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." } }