{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261030-r02-real-hilbert-space-0034", "pilot_number": "R0029-15209", "number": 15209, "family_id": "qf-mod30-hilbert-component-checklist", "title": { "zh-Hant": "第 15209 題：實Hilbert空間", "en": "Question 15209: A Completeness Checklist Must Use One Carrier, Norm and Metric" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261030-r02-real-hilbert-space-0034", "batch_id": "batch-20261014-advance-20261030-r02-real-hilbert-space-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-real-hilbert-space" ], "title_zh": "實Hilbert空間", "stem_zh": "inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[10,2],[2,5]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。", "answer": { "kind": "expression", "value": "(2, [[10, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)", "explanation_zh": "Wave4第四層核對=(2, [[10, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "實Hilbert空間" ] }, "variant": { "family_id": "qf-mod30-hilbert-component-checklist", "seed": 202610302034, "parameters": { "dimension": 2, "matrix": [ [ 10, 2 ], [ 2, 5 ] ], "norm_source": "same_inner_product", "metric_source": "same_norm", "completeness_source": "finite_dimensional_theorem", "domain": "Named_finite_dimensional_real_Hilbert_space", "operation": "hilbert-component-checklist", "semantic_template": "qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[10,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或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-20261030-r02-real-hilbert-space-0034", "batch_id": "batch-20261014-advance-20261030-r02-real-hilbert-space-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "a4fd8df53955e339aa577fed27eacf44b8fd5b6e309a35241c1c4153ff7ee317", "family_id": "qf-mod30-hilbert-component-checklist", "seed": 202610302034, "semantic_template": "qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[10,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}" }, "fields": { "title": "A Completeness Checklist Must Use One Carrier, Norm and Metric", "stem": "Use the same real plane with symmetric positive-definite matrix G=[[10, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).", "answer": { "kind": "expression", "value": "(2, [[10, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)", "explanation": "The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[10, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)." }, "skills": [ "A Completeness Checklist Must Use One Carrier, Norm and Metric" ], "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": "inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[10,2],[2,5]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。", "en": "Use the same real plane with symmetric positive-definite matrix G=[[10, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "Hilbert Gram diagonal product := mul(10, 5)", "after": "Hilbert Gram diagonal product = 50", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert Gram diagonal product，不以小數近似替代。", "en": "Recompute Hilbert Gram diagonal product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "10", "5" ], "output": "50", "label": "Hilbert Gram diagonal product" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Hilbert Gram off-diagonal product := mul(2, 2)", "after": "Hilbert Gram off-diagonal product = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert Gram off-diagonal product，不以小數近似替代。", "en": "Recompute Hilbert Gram off-diagonal product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "Hilbert Gram off-diagonal product" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Hilbert Gram determinant := sub(50, 4)", "after": "Hilbert Gram determinant = 46", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert Gram determinant，不以小數近似替代。", "en": "Recompute Hilbert Gram determinant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "50", "4" ], "output": "46", "label": "Hilbert Gram determinant" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Hilbert completing-square slope := div(2, 10)", "after": "Hilbert completing-square slope = 1/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert completing-square slope，不以小數近似替代。", "en": "Recompute Hilbert completing-square slope with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2", "10" ], "output": "1/5", "label": "Hilbert completing-square slope" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Hilbert completing-square positive residual := div(46, 10)", "after": "Hilbert completing-square positive residual = 23/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert completing-square positive residual，不以小數近似替代。", "en": "Recompute Hilbert completing-square positive residual with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "46", "10" ], "output": "23/5", "label": "Hilbert completing-square positive residual" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "complete real carrier R^2;Gram matrix=[[10, 2], [2, 5]];a=10>0;det=46>0;dimension declaration=2", "after": "x^T M x=10*(x0+(1/5)*x1)^2+(23/5)*x1^2;zero only at x=0", "relation": "implies", "rule_id": "hilbert_same_complete_matrix_positive_inner_product", "conditions": [], "explanation": { "zh-Hant": "對稱矩陣加正主子式，以配方給所有實向量的正定性，不是只測本題向量。配方的三角變換可逆，故此內積norm與R²座標norm等價；其Cauchy序列座標收斂，極限仍在同一R²。", "en": "Symmetry and positive principal minors give positivity for every real vector by completing the square,not just testing the displayed vectors.The triangular transformation is invertible,so this inner-product norm is equivalent to the coordinate norm on R².A Cauchy sequence has convergent coordinates and its limit remains in this same R²." }, "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": "S007", "uses": [ "G001", "S006" ], "before": "complete matrix=[[10, 2], [2, 5]];dimension=2;norm source=same_inner_product;metric source=same_norm;completeness source=finite_dimensional_theorem", "after": "same inner product norm=是;same norm metric=是;same finite-dimensional completeness source=是;all components qualified=是", "relation": "implies", "rule_id": "hilbert_component_binding_not_name_only", "conditions": [], "explanation": { "zh-Hant": "三個來源須同時對應同一結構；任一不符只表示這份checklist未給完整證書，不等於證明底層空間不可能是Hilbert。", "en": "All three sources must refer to the same structure.A mismatch means this checklist has not supplied the full certificate;it does not prove the underlying space could never be Hilbert." }, "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": [ 2, [ [ 10, 2 ], [ 2, 5 ] ], "same_inner_product", "same_norm", "finite_dimensional_theorem", true ] }, "conclusion": { "canonical_value": "(2, [[10, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)", "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-20261030-r02-real-hilbert-space-foundations", "batch_path": "public/data/questions/revisions/ocme-parameter-revision-20261017/batch-20261014-advance-20261030-r02-real-hilbert-space-foundations/questions.jsonl", "batch_sha256": "4e3813f5a6d26ef66584109baa548d64ea1c9ec1d0158cd3f4cfa19fdb5990e1", "original_line": 34, "original_line_sha256": "96078d012f9df2429cbea3d28e9dc482c80745105a01690d4b10fd4f18d76181", "question_compact_sha256": "a4fd8df53955e339aa577fed27eacf44b8fd5b6e309a35241c1c4153ff7ee317", "english_row_compact_sha256": "a5d888f321fe071a61b9c1b3710abfcad57ed8c43a8873eae7ddd4465175e04b", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261030-r02-real-hilbert-space-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261030-r02-real-hilbert-space-foundations.en.jsonl" }, "mko_bindings": { "mko-real-hilbert-space": { "sha256": "89b1eab58b853963348761bf140ffff91d2903053edbdf9baf157b32a5ac731e", "english_sha256": "3ac4455274e5bba35d48841ef7df59ae0ed9e3c9645b4035ebd5961f396f7052", "original_path": "public/data/mko/real-hilbert-space.json", "english_path": "public/data/i18n/editions/mko-real-hilbert-space.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "54de60dca082018923bf089f2b1eb0a6666cc401916115af792056c8bae935db", "original_candidate_sha256": "2e59b6ade1c1471476252bce06867dcf34191050b47cf48412ed47501f257119", "original_candidate_line_sha256": "2e59b6ade1c1471476252bce06867dcf34191050b47cf48412ed47501f257119" }, "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-p30", "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": "8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725", "question_first_witness_sha256": "c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 7, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "570c1b3886a46d709cc8b43254d21d80c4273e7277592f40bc98a8c39e0896aa", "arithmetic_steps": 5 }, "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." } }