{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261028-r03-real-banach-space-0012", "pilot_number": "R0025-13487", "number": 13487, "family_id": "qf-mod28-banach-named-certificate", "title": { "zh-Hant": "第 13487 題：實Banach空間", "en": "Question 13487: A Named Finite-Dimensional Banach Certificate Binds One Space" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261028-r03-real-banach-space-0012", "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-named-certificate；資料={\"dimension\":12,\"norm_name\":\"linfinity\"}。請依 [dimension,norm_name,carrier,metric_binding,completeness_certificate,Banach,boundary] 欄序回答。", "answer": { "kind": "expression", "value": "(12, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, 是, not_Hilbert_claim)", "explanation_zh": "Wave4第二層核對=(12, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, 是, not_Hilbert_claim)；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-named-certificate", "seed": 202610283012, "parameters": { "dimension": 12, "norm_name": "linfinity", "domain": "Named_finite_dimensional_real_Banach_space", "operation": "banach-named-certificate", "semantic_template": "qf-mod28-banach-named-certificate/{\"dimension\":12,\"norm_name\":\"linfinity\"}" } }, "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-0012", "batch_id": "batch-20261014-advance-20261028-r03-real-banach-space-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "2a41cf8825e653f97f41531ac652c6a4bbaf4212fd258bb943e4a7203d75db18", "family_id": "qf-mod28-banach-named-certificate", "seed": 202610283012, "semantic_template": "qf-mod28-banach-named-certificate/{\"dimension\":12,\"norm_name\":\"linfinity\"}" }, "fields": { "title": "A Named Finite-Dimensional Banach Certificate Binds One Space", "stem": "Use the explicit real coordinate space R^n of dimension=12, with the named norm=linfinity, 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. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).", "answer": { "kind": "expression", "value": "(12, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)", "explanation": "Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (12, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)." }, "skills": [ "A Named Finite-Dimensional Banach Certificate Binds One Space" ], "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-named-certificate；資料={\"dimension\":12,\"norm_name\":\"linfinity\"}。請依 [dimension,norm_name,carrier,metric_binding,completeness_certificate,Banach,boundary] 欄序回答。", "en": "Use the explicit real coordinate space R^n of dimension=12, with the named norm=linfinity, 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. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "same real carrier R^12;standard coordinate vector operations;named norm=linfinity", "after": "metric is d(x,y)=this_same_norm(x-y);norm axioms supplied by named formula", "relation": "implies", "rule_id": "Banach_same_space_norm_metric_binding", "conditions": [], "explanation": { "zh-Hant": "Banach結構必須讓載體、向量運算、範數、誘導metric與完備性描述同一空間；不能把別的metric的完備性搬進來。", "en": "Carrier,vector operations,norm,induced metric and completeness must describe the same space.A completeness result for another metric cannot be transplanted." }, "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": "arbitrary Cauchy sequence x_k in R^12 under linfinity;every coordinate obeys abs((x_k-x_m)_i)<=norm(x_k-x_m)", "after": "each real coordinate sequence is Cauchy;real-number completeness supplies a real limit L_i", "relation": "implies", "rule_id": "Banach_coordinate_Cauchy_and_real_completeness_premise", "conditions": [], "explanation": { "zh-Hant": "向量Cauchy對每個正eta的尾段界，同時控制每個座標差，因此各座標Cauchy。此處明示使用實數完備性作數學前提，不稱為本輪新Lean驗證。", "en": "For every positive eta,the vector Cauchy tail bound controls every coordinate difference,so each coordinate is Cauchy.Real-number completeness is an explicit mathematical premise here,not a new Lean verification." }, "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": "coordinate limits L_i form L in R^12;epsilon>0;each coordinate error eventually 0", "relation": "implies", "rule_id": "Banach_finite_coordinate_limits_same_norm_convergence", "conditions": [], "explanation": { "zh-Hant": "維度有限且正，取所有座標門檻最大值可同時控制全部座標。l1用epsilon/n分配誤差，linfinity用有限最大值；得到同一範數收斂到原空間內L，而非只有逐座標或外部極限。", "en": "Finite positive dimension permits one maximum cutoff for all coordinates.Use epsilon/n for l1 and the finite maximum for linfinity.This gives convergence in the same norm to L in the original space,not merely coordinatewise or ambient convergence." }, "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": "S004", "uses": [ "G001", "S003" ], "before": "every Cauchy sequence in this named real R^12 normed space has a same-space norm limit", "after": "Banach certificate=是;no Hilbert or arbitrary infinite-dimensional conclusion", "relation": "implies", "rule_id": "named_finite_dimensional_Banach_certificate_scope", "conditions": [], "explanation": { "zh-Hant": "前述全稱論證閉合此命名有限維空間的完備性；沒有從Banach性推內積、Hilbert、緊性或任意無限維結論。", "en": "The universal argument closes completeness of this named finite-dimensional space.It does not infer an inner product,Hilbert structure,compactness or arbitrary infinite-dimensional conclusions." }, "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": [ 12, "linfinity", "R^n", "same_norm_induced_metric", "finite_dimensional_real_normed_space_complete", true, "not_Hilbert_claim" ] }, "conclusion": { "canonical_value": "(12, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, 是, not_Hilbert_claim)", "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": 12, "original_line_sha256": "6086119e7c3404b5e706c0aaf723b30f882e6ffd79ff96ea9e09a1ec2aa88fa6", "question_compact_sha256": "2a41cf8825e653f97f41531ac652c6a4bbaf4212fd258bb943e4a7203d75db18", "english_row_compact_sha256": "2c701309f88c71b99a630abdba0496882c8418ee5e58c164f38c8cc949d2c168", "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": "9cdcbed94aaca2bf90bbca535cebee627404f9708720260508c4260f0e6a63fb", "original_candidate_line_sha256": "9cdcbed94aaca2bf90bbca535cebee627404f9708720260508c4260f0e6a63fb" }, "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": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "ef70fc3ef5be43f0df9930cf0134103d7076db32cbcceb17d9c826b0bab28bdd", "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." } }