{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0001","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021001,"parameters":{"dimension":2,"vectors":[[1,0],[0,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[0,1],[1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0002","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[1,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021002,"parameters":{"dimension":2,"vectors":[[1,0],[1,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[1,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0003","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[2,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021003,"parameters":{"dimension":2,"vectors":[[1,0],[2,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[2,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0004","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[3,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021004,"parameters":{"dimension":2,"vectors":[[1,0],[3,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[3,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0005","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[4,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021005,"parameters":{"dimension":2,"vectors":[[1,0],[4,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[4,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0006","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[5,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021006,"parameters":{"dimension":2,"vectors":[[1,0],[5,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[5,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0007","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[6,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021007,"parameters":{"dimension":2,"vectors":[[1,0],[6,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[6,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0008","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[7,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021008,"parameters":{"dimension":2,"vectors":[[1,0],[7,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[7,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0009","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[8,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021009,"parameters":{"dimension":2,"vectors":[[1,0],[8,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[8,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0010","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[9,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021010,"parameters":{"dimension":2,"vectors":[[1,0],[9,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[9,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0011","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[10,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021011,"parameters":{"dimension":2,"vectors":[[1,0],[10,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[10,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0012","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[11,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021012,"parameters":{"dimension":2,"vectors":[[1,0],[11,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[11,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0013","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[12,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021013,"parameters":{"dimension":2,"vectors":[[1,0],[12,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[12,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0014","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[13,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021014,"parameters":{"dimension":2,"vectors":[[1,0],[13,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[13,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0015","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[14,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021015,"parameters":{"dimension":2,"vectors":[[1,0],[14,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[14,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0016","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[15,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021016,"parameters":{"dimension":2,"vectors":[[1,0],[15,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[15,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0017","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[16,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021017,"parameters":{"dimension":2,"vectors":[[1,0],[16,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[16,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0018","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[17,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021018,"parameters":{"dimension":2,"vectors":[[1,0],[17,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[17,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0019","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[18,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021019,"parameters":{"dimension":2,"vectors":[[1,0],[18,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[18,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0020","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[19,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021020,"parameters":{"dimension":2,"vectors":[[1,0],[19,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[19,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0021","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[20,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021021,"parameters":{"dimension":2,"vectors":[[1,0],[20,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[20,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0022","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[21,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021022,"parameters":{"dimension":2,"vectors":[[1,0],[21,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[21,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0023","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[22,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021023,"parameters":{"dimension":2,"vectors":[[1,0],[22,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[22,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0024","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[23,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021024,"parameters":{"dimension":2,"vectors":[[1,0],[23,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[23,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0025","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-basis-audit；資料={\"dimension\":2,\"vectors\":[[1,0],[24,1]]}。請依 [ambient_dimension,vector_count,rank,independent,spanning,basis] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, 是, 是)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 是, 是, 是)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-basis-audit","seed":202611021025,"parameters":{"dimension":2,"vectors":[[1,0],[24,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-basis-audit","semantic_template":"qf-mod33-dimension-basis-audit/{\"dimension\":2,\"vectors\":[[1,0],[24,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0026","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021026,"parameters":{"dimension":2,"vectors":[[1,0],[0,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[0,1],[1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0027","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[1,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021027,"parameters":{"dimension":2,"vectors":[[1,0],[1,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[1,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0028","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[2,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021028,"parameters":{"dimension":2,"vectors":[[1,0],[2,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[2,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0029","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[3,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021029,"parameters":{"dimension":2,"vectors":[[1,0],[3,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[3,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0030","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[4,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021030,"parameters":{"dimension":2,"vectors":[[1,0],[4,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[4,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0031","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[5,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021031,"parameters":{"dimension":2,"vectors":[[1,0],[5,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[5,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0032","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[6,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021032,"parameters":{"dimension":2,"vectors":[[1,0],[6,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[6,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0033","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[7,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021033,"parameters":{"dimension":2,"vectors":[[1,0],[7,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[7,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0034","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[8,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021034,"parameters":{"dimension":2,"vectors":[[1,0],[8,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[8,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0035","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[9,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021035,"parameters":{"dimension":2,"vectors":[[1,0],[9,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[9,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0036","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[10,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021036,"parameters":{"dimension":2,"vectors":[[1,0],[10,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[10,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0037","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[11,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021037,"parameters":{"dimension":2,"vectors":[[1,0],[11,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[11,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0038","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[12,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021038,"parameters":{"dimension":2,"vectors":[[1,0],[12,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[12,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0039","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[13,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021039,"parameters":{"dimension":2,"vectors":[[1,0],[13,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[13,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0040","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[14,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021040,"parameters":{"dimension":2,"vectors":[[1,0],[14,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[14,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0041","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[15,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021041,"parameters":{"dimension":2,"vectors":[[1,0],[15,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[15,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0042","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[16,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021042,"parameters":{"dimension":2,"vectors":[[1,0],[16,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[16,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0043","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[17,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021043,"parameters":{"dimension":2,"vectors":[[1,0],[17,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[17,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0044","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[18,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021044,"parameters":{"dimension":2,"vectors":[[1,0],[18,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[18,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0045","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[19,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021045,"parameters":{"dimension":2,"vectors":[[1,0],[19,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[19,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0046","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[20,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021046,"parameters":{"dimension":2,"vectors":[[1,0],[20,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[20,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0047","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[21,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021047,"parameters":{"dimension":2,"vectors":[[1,0],[21,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[21,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0048","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[22,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021048,"parameters":{"dimension":2,"vectors":[[1,0],[22,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[22,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0049","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[23,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021049,"parameters":{"dimension":2,"vectors":[[1,0],[23,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[23,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0050","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[24,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)","explanation_zh":"Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-exact-value","seed":202611021050,"parameters":{"dimension":2,"vectors":[[1,0],[24,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-exact-value","semantic_template":"qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[24,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0051","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[0,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021051,"parameters":{"dimension":3,"vectors":[[1,0,0],[0,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[0,1,0],[1,0,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0052","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[1,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021052,"parameters":{"dimension":3,"vectors":[[1,0,0],[1,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[1,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0053","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[2,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021053,"parameters":{"dimension":3,"vectors":[[1,0,0],[2,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[2,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0054","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[3,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021054,"parameters":{"dimension":3,"vectors":[[1,0,0],[3,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[3,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0055","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[4,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021055,"parameters":{"dimension":3,"vectors":[[1,0,0],[4,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[4,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0056","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[5,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021056,"parameters":{"dimension":3,"vectors":[[1,0,0],[5,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[5,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0057","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[6,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021057,"parameters":{"dimension":3,"vectors":[[1,0,0],[6,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[6,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0058","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[7,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021058,"parameters":{"dimension":3,"vectors":[[1,0,0],[7,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[7,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0059","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[8,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021059,"parameters":{"dimension":3,"vectors":[[1,0,0],[8,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[8,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0060","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[9,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021060,"parameters":{"dimension":3,"vectors":[[1,0,0],[9,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[9,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0061","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[10,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021061,"parameters":{"dimension":3,"vectors":[[1,0,0],[10,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[10,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0062","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[11,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021062,"parameters":{"dimension":3,"vectors":[[1,0,0],[11,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[11,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0063","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[12,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021063,"parameters":{"dimension":3,"vectors":[[1,0,0],[12,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[12,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0064","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[13,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021064,"parameters":{"dimension":3,"vectors":[[1,0,0],[13,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[13,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0065","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[14,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021065,"parameters":{"dimension":3,"vectors":[[1,0,0],[14,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[14,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0066","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[15,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021066,"parameters":{"dimension":3,"vectors":[[1,0,0],[15,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[15,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0067","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[16,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021067,"parameters":{"dimension":3,"vectors":[[1,0,0],[16,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[16,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0068","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[17,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021068,"parameters":{"dimension":3,"vectors":[[1,0,0],[17,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[17,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0069","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[18,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021069,"parameters":{"dimension":3,"vectors":[[1,0,0],[18,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[18,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0070","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[19,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021070,"parameters":{"dimension":3,"vectors":[[1,0,0],[19,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[19,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0071","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[20,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021071,"parameters":{"dimension":3,"vectors":[[1,0,0],[20,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[20,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0072","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[21,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021072,"parameters":{"dimension":3,"vectors":[[1,0,0],[21,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[21,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0073","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[22,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021073,"parameters":{"dimension":3,"vectors":[[1,0,0],[22,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[22,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0074","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[23,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021074,"parameters":{"dimension":3,"vectors":[[1,0,0],[23,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[23,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0075","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-independent-lower-bound；資料={\"dimension\":3,\"vectors\":[[1,0,0],[24,1,0]]}。請依 [vector_count,rank,independent,lower_bound] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, dimension_at_least_2)","explanation_zh":"Wave5第二層核對=(2, 2, 是, dimension_at_least_2)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-independent-lower-bound","seed":202611021075,"parameters":{"dimension":3,"vectors":[[1,0,0],[24,1,0]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-independent-lower-bound","semantic_template":"qf-mod33-dimension-independent-lower-bound/{\"dimension\":3,\"vectors\":[[1,0,0],[24,1,0]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0076","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[1,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021076,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[1,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[1,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0077","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[2,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021077,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[2,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[2,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0078","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[3,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021078,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[3,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[3,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0079","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[4,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021079,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[4,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[4,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0080","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[5,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021080,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[5,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[5,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0081","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[6,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021081,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[6,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[6,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0082","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[7,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021082,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[7,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[7,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0083","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[8,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021083,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[8,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[8,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0084","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[9,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021084,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[9,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[9,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0085","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[10,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021085,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[10,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[10,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0086","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[11,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021086,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[11,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[11,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0087","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[12,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021087,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[12,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[12,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0088","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[13,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021088,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[13,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[13,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0089","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[14,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021089,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[14,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[14,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0090","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[15,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021090,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[15,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[15,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0091","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[16,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021091,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[16,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[16,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0092","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[17,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021092,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[17,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[17,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0093","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[18,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021093,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[18,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[18,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0094","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[19,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021094,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[19,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[19,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0095","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[20,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021095,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[20,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[20,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0096","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[21,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021096,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[21,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[21,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0097","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[22,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021097,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[22,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[22,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0098","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[23,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021098,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[23,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[23,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0099","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[24,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021099,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[24,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[24,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r01-vector-space-dimension-0100","batch_id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","language":"zh-Hant","source_mko_ids":["mko-vector-space-dimension"],"title_zh":"向量空間維度","stem_zh":"R²或R³完整座標族與精確rank證書。操作=dimension-spanning-upper-bound；資料={\"dimension\":2,\"vectors\":[[1,0],[0,1],[25,1]]}。請依 [vector_count,rank,spanning,upper_bound] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, dimension_at_most_3)","explanation_zh":"Wave5第二層核對=(3, 2, 是, dimension_at_most_3)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["向量空間維度"]},"variant":{"family_id":"qf-mod33-dimension-spanning-upper-bound","seed":202611021100,"parameters":{"dimension":2,"vectors":[[1,0],[0,1],[25,1]],"domain":"Exact_R2_R3_basis_dimension","operation":"dimension-spanning-upper-bound","semantic_template":"qf-mod33-dimension-spanning-upper-bound/{\"dimension\":2,\"vectors\":[[0,1],[1,0],[25,1]]}"}},"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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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"}}