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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r01-direct-sum-0025","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-internal-audit；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,25]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 2, 0, 是, 否, 否)","explanation_zh":"Wave5第五層核對=(1, 1, 2, 0, 是, 否, 否)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-internal-audit","seed":202611051025,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,25]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-internal-audit","semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0026","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,1]],\"vector\":[1,2,2],\"coefficients_u\":[1],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 0], [0, 2, 2], [1, 2, 2], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([1, 0, 0], [0, 2, 2], [1, 2, 2], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051026,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,1]],"vector":[1,2,2],"coefficients_u":[1],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,1]],\"coefficients_u\":[1],\"coefficients_w\":[2],\"vector\":[1,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0027","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,2]],\"vector\":[2,2,4],\"coefficients_u\":[2],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 0, 0], [0, 2, 4], [2, 2, 4], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([2, 0, 0], [0, 2, 4], [2, 2, 4], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051027,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,2]],"vector":[2,2,4],"coefficients_u":[2],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,2]],\"coefficients_u\":[2],\"coefficients_w\":[2],\"vector\":[2,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0028","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,3]],\"vector\":[3,2,6],\"coefficients_u\":[3],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 0], [0, 2, 6], [3, 2, 6], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([3, 0, 0], [0, 2, 6], [3, 2, 6], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051028,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,3]],"vector":[3,2,6],"coefficients_u":[3],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,3]],\"coefficients_u\":[3],\"coefficients_w\":[2],\"vector\":[3,2,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0029","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,4]],\"vector\":[4,2,8],\"coefficients_u\":[4],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 0, 0], [0, 2, 8], [4, 2, 8], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([4, 0, 0], [0, 2, 8], [4, 2, 8], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051029,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,4]],"vector":[4,2,8],"coefficients_u":[4],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,4]],\"coefficients_u\":[4],\"coefficients_w\":[2],\"vector\":[4,2,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0030","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,5]],\"vector\":[5,2,10],\"coefficients_u\":[5],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0], [0, 2, 10], [5, 2, 10], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([5, 0, 0], [0, 2, 10], [5, 2, 10], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051030,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,5]],"vector":[5,2,10],"coefficients_u":[5],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,5]],\"coefficients_u\":[5],\"coefficients_w\":[2],\"vector\":[5,2,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0031","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,6]],\"vector\":[6,2,12],\"coefficients_u\":[6],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, 0, 0], [0, 2, 12], [6, 2, 12], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([6, 0, 0], [0, 2, 12], [6, 2, 12], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051031,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,6]],"vector":[6,2,12],"coefficients_u":[6],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,6]],\"coefficients_u\":[6],\"coefficients_w\":[2],\"vector\":[6,2,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0032","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,7]],\"vector\":[7,2,14],\"coefficients_u\":[7],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, 0, 0], [0, 2, 14], [7, 2, 14], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([7, 0, 0], [0, 2, 14], [7, 2, 14], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051032,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,7]],"vector":[7,2,14],"coefficients_u":[7],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,7]],\"coefficients_u\":[7],\"coefficients_w\":[2],\"vector\":[7,2,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0033","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,8]],\"vector\":[8,2,16],\"coefficients_u\":[8],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, 0, 0], [0, 2, 16], [8, 2, 16], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([8, 0, 0], [0, 2, 16], [8, 2, 16], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051033,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,8]],"vector":[8,2,16],"coefficients_u":[8],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,8]],\"coefficients_u\":[8],\"coefficients_w\":[2],\"vector\":[8,2,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0034","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,9]],\"vector\":[9,2,18],\"coefficients_u\":[9],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, 0, 0], [0, 2, 18], [9, 2, 18], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([9, 0, 0], [0, 2, 18], [9, 2, 18], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051034,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,9]],"vector":[9,2,18],"coefficients_u":[9],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,9]],\"coefficients_u\":[9],\"coefficients_w\":[2],\"vector\":[9,2,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0035","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,10]],\"vector\":[10,2,20],\"coefficients_u\":[10],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, 0, 0], [0, 2, 20], [10, 2, 20], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([10, 0, 0], [0, 2, 20], [10, 2, 20], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051035,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,10]],"vector":[10,2,20],"coefficients_u":[10],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,10]],\"coefficients_u\":[10],\"coefficients_w\":[2],\"vector\":[10,2,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0036","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,11]],\"vector\":[11,2,22],\"coefficients_u\":[11],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11, 0, 0], [0, 2, 22], [11, 2, 22], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([11, 0, 0], [0, 2, 22], [11, 2, 22], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051036,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,11]],"vector":[11,2,22],"coefficients_u":[11],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,11]],\"coefficients_u\":[11],\"coefficients_w\":[2],\"vector\":[11,2,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0037","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,12]],\"vector\":[12,2,24],\"coefficients_u\":[12],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, 0, 0], [0, 2, 24], [12, 2, 24], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([12, 0, 0], [0, 2, 24], [12, 2, 24], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051037,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,12]],"vector":[12,2,24],"coefficients_u":[12],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,12]],\"coefficients_u\":[12],\"coefficients_w\":[2],\"vector\":[12,2,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0038","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,13]],\"vector\":[13,2,26],\"coefficients_u\":[13],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([13, 0, 0], [0, 2, 26], [13, 2, 26], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([13, 0, 0], [0, 2, 26], [13, 2, 26], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051038,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,13]],"vector":[13,2,26],"coefficients_u":[13],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,13]],\"coefficients_u\":[13],\"coefficients_w\":[2],\"vector\":[13,2,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0039","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,14]],\"vector\":[14,2,28],\"coefficients_u\":[14],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([14, 0, 0], [0, 2, 28], [14, 2, 28], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([14, 0, 0], [0, 2, 28], [14, 2, 28], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051039,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,14]],"vector":[14,2,28],"coefficients_u":[14],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,14]],\"coefficients_u\":[14],\"coefficients_w\":[2],\"vector\":[14,2,28]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0040","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,15]],\"vector\":[15,2,30],\"coefficients_u\":[15],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([15, 0, 0], [0, 2, 30], [15, 2, 30], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([15, 0, 0], [0, 2, 30], [15, 2, 30], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051040,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,15]],"vector":[15,2,30],"coefficients_u":[15],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,15]],\"coefficients_u\":[15],\"coefficients_w\":[2],\"vector\":[15,2,30]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0041","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,16]],\"vector\":[16,2,32],\"coefficients_u\":[16],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([16, 0, 0], [0, 2, 32], [16, 2, 32], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([16, 0, 0], [0, 2, 32], [16, 2, 32], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051041,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,16]],"vector":[16,2,32],"coefficients_u":[16],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,16]],\"coefficients_u\":[16],\"coefficients_w\":[2],\"vector\":[16,2,32]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0042","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,17]],\"vector\":[17,2,34],\"coefficients_u\":[17],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([17, 0, 0], [0, 2, 34], [17, 2, 34], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([17, 0, 0], [0, 2, 34], [17, 2, 34], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051042,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,17]],"vector":[17,2,34],"coefficients_u":[17],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,17]],\"coefficients_u\":[17],\"coefficients_w\":[2],\"vector\":[17,2,34]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0043","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,18]],\"vector\":[18,2,36],\"coefficients_u\":[18],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([18, 0, 0], [0, 2, 36], [18, 2, 36], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([18, 0, 0], [0, 2, 36], [18, 2, 36], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051043,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,18]],"vector":[18,2,36],"coefficients_u":[18],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,18]],\"coefficients_u\":[18],\"coefficients_w\":[2],\"vector\":[18,2,36]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0044","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,19]],\"vector\":[19,2,38],\"coefficients_u\":[19],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([19, 0, 0], [0, 2, 38], [19, 2, 38], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([19, 0, 0], [0, 2, 38], [19, 2, 38], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051044,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,19]],"vector":[19,2,38],"coefficients_u":[19],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,19]],\"coefficients_u\":[19],\"coefficients_w\":[2],\"vector\":[19,2,38]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0045","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,20]],\"vector\":[20,2,40],\"coefficients_u\":[20],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([20, 0, 0], [0, 2, 40], [20, 2, 40], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([20, 0, 0], [0, 2, 40], [20, 2, 40], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051045,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,20]],"vector":[20,2,40],"coefficients_u":[20],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,20]],\"coefficients_u\":[20],\"coefficients_w\":[2],\"vector\":[20,2,40]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0046","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,21]],\"vector\":[21,2,42],\"coefficients_u\":[21],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([21, 0, 0], [0, 2, 42], [21, 2, 42], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([21, 0, 0], [0, 2, 42], [21, 2, 42], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051046,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,21]],"vector":[21,2,42],"coefficients_u":[21],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,21]],\"coefficients_u\":[21],\"coefficients_w\":[2],\"vector\":[21,2,42]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0047","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,22]],\"vector\":[22,2,44],\"coefficients_u\":[22],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([22, 0, 0], [0, 2, 44], [22, 2, 44], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([22, 0, 0], [0, 2, 44], [22, 2, 44], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051047,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,22]],"vector":[22,2,44],"coefficients_u":[22],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,22]],\"coefficients_u\":[22],\"coefficients_w\":[2],\"vector\":[22,2,44]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0048","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,23]],\"vector\":[23,2,46],\"coefficients_u\":[23],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([23, 0, 0], [0, 2, 46], [23, 2, 46], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([23, 0, 0], [0, 2, 46], [23, 2, 46], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051048,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,23]],"vector":[23,2,46],"coefficients_u":[23],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,23]],\"coefficients_u\":[23],\"coefficients_w\":[2],\"vector\":[23,2,46]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0049","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,24]],\"vector\":[24,2,48],\"coefficients_u\":[24],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([24, 0, 0], [0, 2, 48], [24, 2, 48], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([24, 0, 0], [0, 2, 48], [24, 2, 48], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051049,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,24]],"vector":[24,2,48],"coefficients_u":[24],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,24]],\"coefficients_u\":[24],\"coefficients_w\":[2],\"vector\":[24,2,48]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0050","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,25]],\"vector\":[25,2,50],\"coefficients_u\":[25],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([25, 0, 0], [0, 2, 50], [25, 2, 50], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation_zh":"Wave5第五層核對=([25, 0, 0], [0, 2, 50], [25, 2, 50], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051050,"parameters":{"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,25]],"vector":[25,2,50],"coefficients_u":[25],"coefficients_w":[2],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-ambient-decomposition","semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,25]],\"coefficients_u\":[25],\"coefficients_w\":[2],\"vector\":[25,2,50]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0051","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,1]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051051,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,1]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[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":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0052","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,2]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051052,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,2]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0053","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,3]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051053,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,3]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,3]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0054","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,4]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051054,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,4]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0055","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,5]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051055,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,5]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,5]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0056","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,6]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051056,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,6]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,6]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0057","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,7]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051057,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,7]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,7]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0058","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,8]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051058,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,8]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,8]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0059","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,9]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051059,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,9]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,9]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0060","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,10]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051060,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,10]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,10]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0061","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,11]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051061,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,11]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,11]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0062","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,12]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051062,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,12]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,12]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0063","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,13]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051063,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,13]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,13]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0064","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,14]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051064,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,14]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,14]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0065","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,15]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051065,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,15]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,15]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0066","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,16]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051066,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,16]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,16]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0067","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,17]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051067,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,17]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,17]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0068","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,18]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051068,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,18]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,18]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0069","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,19]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051069,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,19]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,19]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0070","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,20]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051070,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,20]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,20]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0071","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,21]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051071,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,21]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,21]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0072","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,22]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051072,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,22]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,22]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0073","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,23]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051073,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,23]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,23]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0074","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,24]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051074,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,24]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,24]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0075","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,25]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)","explanation_zh":"Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051075,"parameters":{"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,25]]],"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-multi-subspace-global","semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,25]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0076","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[0,1],\"right\":[2],\"other_left\":[3,1],\"other_right\":[4],\"scalar\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2]], [[3, 2], [6]], [[0, 1], [2]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[0, 1], [2]], [[3, 2], [6]], [[0, 1], [2]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051076,"parameters":{"left":[0,1],"right":[2],"other_left":[3,1],"other_right":[4],"scalar":1,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[0,1],\"other_left\":[3,1],\"other_right\":[4],\"right\":[2],\"scalar\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0077","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[1,2],\"right\":[3],\"other_left\":[4,3],\"other_right\":[5],\"scalar\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [3]], [[5, 5], [8]], [[2, 4], [6]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[1, 2], [3]], [[5, 5], [8]], [[2, 4], [6]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051077,"parameters":{"left":[1,2],"right":[3],"other_left":[4,3],"other_right":[5],"scalar":2,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[1,2],\"other_left\":[4,3],\"other_right\":[5],\"right\":[3],\"scalar\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0078","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[2,3],\"right\":[4],\"other_left\":[5,5],\"other_right\":[6],\"scalar\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[2, 3], [4]], [[7, 8], [10]], [[6, 9], [12]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[2, 3], [4]], [[7, 8], [10]], [[6, 9], [12]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051078,"parameters":{"left":[2,3],"right":[4],"other_left":[5,5],"other_right":[6],"scalar":3,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[2,3],\"other_left\":[5,5],\"other_right\":[6],\"right\":[4],\"scalar\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0079","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[3,4],\"right\":[5],\"other_left\":[6,7],\"other_right\":[7],\"scalar\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[3, 4], [5]], [[9, 11], [12]], [[12, 16], [20]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[3, 4], [5]], [[9, 11], [12]], [[12, 16], [20]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051079,"parameters":{"left":[3,4],"right":[5],"other_left":[6,7],"other_right":[7],"scalar":4,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[3,4],\"other_left\":[6,7],\"other_right\":[7],\"right\":[5],\"scalar\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0080","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[4,5],\"right\":[6],\"other_left\":[7,9],\"other_right\":[8],\"scalar\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[4, 5], [6]], [[11, 14], [14]], [[20, 25], [30]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[4, 5], [6]], [[11, 14], [14]], [[20, 25], [30]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051080,"parameters":{"left":[4,5],"right":[6],"other_left":[7,9],"other_right":[8],"scalar":5,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[4,5],\"other_left\":[7,9],\"other_right\":[8],\"right\":[6],\"scalar\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0081","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[5,6],\"right\":[7],\"other_left\":[8,11],\"other_right\":[9],\"scalar\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[5, 6], [7]], [[13, 17], [16]], [[30, 36], [42]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[5, 6], [7]], [[13, 17], [16]], [[30, 36], [42]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051081,"parameters":{"left":[5,6],"right":[7],"other_left":[8,11],"other_right":[9],"scalar":6,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[5,6],\"other_left\":[8,11],\"other_right\":[9],\"right\":[7],\"scalar\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0082","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[6,7],\"right\":[8],\"other_left\":[9,13],\"other_right\":[10],\"scalar\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[6, 7], [8]], [[15, 20], [18]], [[42, 49], [56]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[6, 7], [8]], [[15, 20], [18]], [[42, 49], [56]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051082,"parameters":{"left":[6,7],"right":[8],"other_left":[9,13],"other_right":[10],"scalar":7,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[6,7],\"other_left\":[9,13],\"other_right\":[10],\"right\":[8],\"scalar\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0083","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[7,8],\"right\":[9],\"other_left\":[10,15],\"other_right\":[11],\"scalar\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[7, 8], [9]], [[17, 23], [20]], [[56, 64], [72]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[7, 8], [9]], [[17, 23], [20]], [[56, 64], [72]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051083,"parameters":{"left":[7,8],"right":[9],"other_left":[10,15],"other_right":[11],"scalar":8,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[7,8],\"other_left\":[10,15],\"other_right\":[11],\"right\":[9],\"scalar\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0084","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[8,9],\"right\":[10],\"other_left\":[11,17],\"other_right\":[12],\"scalar\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[8, 9], [10]], [[19, 26], [22]], [[72, 81], [90]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[8, 9], [10]], [[19, 26], [22]], [[72, 81], [90]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051084,"parameters":{"left":[8,9],"right":[10],"other_left":[11,17],"other_right":[12],"scalar":9,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[8,9],\"other_left\":[11,17],\"other_right\":[12],\"right\":[10],\"scalar\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0085","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[9,10],\"right\":[11],\"other_left\":[12,19],\"other_right\":[13],\"scalar\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[9, 10], [11]], [[21, 29], [24]], [[90, 100], [110]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[9, 10], [11]], [[21, 29], [24]], [[90, 100], [110]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051085,"parameters":{"left":[9,10],"right":[11],"other_left":[12,19],"other_right":[13],"scalar":10,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[9,10],\"other_left\":[12,19],\"other_right\":[13],\"right\":[11],\"scalar\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0086","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[10,11],\"right\":[12],\"other_left\":[13,21],\"other_right\":[14],\"scalar\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[10, 11], [12]], [[23, 32], [26]], [[110, 121], [132]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[10, 11], [12]], [[23, 32], [26]], [[110, 121], [132]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051086,"parameters":{"left":[10,11],"right":[12],"other_left":[13,21],"other_right":[14],"scalar":11,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[10,11],\"other_left\":[13,21],\"other_right\":[14],\"right\":[12],\"scalar\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0087","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[11,12],\"right\":[13],\"other_left\":[14,23],\"other_right\":[15],\"scalar\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[11, 12], [13]], [[25, 35], [28]], [[132, 144], [156]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[11, 12], [13]], [[25, 35], [28]], [[132, 144], [156]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051087,"parameters":{"left":[11,12],"right":[13],"other_left":[14,23],"other_right":[15],"scalar":12,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[11,12],\"other_left\":[14,23],\"other_right\":[15],\"right\":[13],\"scalar\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0088","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[12,13],\"right\":[14],\"other_left\":[15,25],\"other_right\":[16],\"scalar\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[12, 13], [14]], [[27, 38], [30]], [[156, 169], [182]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[12, 13], [14]], [[27, 38], [30]], [[156, 169], [182]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051088,"parameters":{"left":[12,13],"right":[14],"other_left":[15,25],"other_right":[16],"scalar":13,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[12,13],\"other_left\":[15,25],\"other_right\":[16],\"right\":[14],\"scalar\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0089","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[13,14],\"right\":[15],\"other_left\":[16,27],\"other_right\":[17],\"scalar\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[13, 14], [15]], [[29, 41], [32]], [[182, 196], [210]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[13, 14], [15]], [[29, 41], [32]], [[182, 196], [210]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051089,"parameters":{"left":[13,14],"right":[15],"other_left":[16,27],"other_right":[17],"scalar":14,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[13,14],\"other_left\":[16,27],\"other_right\":[17],\"right\":[15],\"scalar\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0090","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[14,15],\"right\":[16],\"other_left\":[17,29],\"other_right\":[18],\"scalar\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[14, 15], [16]], [[31, 44], [34]], [[210, 225], [240]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[14, 15], [16]], [[31, 44], [34]], [[210, 225], [240]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051090,"parameters":{"left":[14,15],"right":[16],"other_left":[17,29],"other_right":[18],"scalar":15,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[14,15],\"other_left\":[17,29],\"other_right\":[18],\"right\":[16],\"scalar\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0091","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[15,16],\"right\":[17],\"other_left\":[18,31],\"other_right\":[19],\"scalar\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[15, 16], [17]], [[33, 47], [36]], [[240, 256], [272]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[15, 16], [17]], [[33, 47], [36]], [[240, 256], [272]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051091,"parameters":{"left":[15,16],"right":[17],"other_left":[18,31],"other_right":[19],"scalar":16,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[15,16],\"other_left\":[18,31],\"other_right\":[19],\"right\":[17],\"scalar\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0092","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[16,17],\"right\":[18],\"other_left\":[19,33],\"other_right\":[20],\"scalar\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[16, 17], [18]], [[35, 50], [38]], [[272, 289], [306]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[16, 17], [18]], [[35, 50], [38]], [[272, 289], [306]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051092,"parameters":{"left":[16,17],"right":[18],"other_left":[19,33],"other_right":[20],"scalar":17,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[16,17],\"other_left\":[19,33],\"other_right\":[20],\"right\":[18],\"scalar\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0093","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[17,18],\"right\":[19],\"other_left\":[20,35],\"other_right\":[21],\"scalar\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[17, 18], [19]], [[37, 53], [40]], [[306, 324], [342]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[17, 18], [19]], [[37, 53], [40]], [[306, 324], [342]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051093,"parameters":{"left":[17,18],"right":[19],"other_left":[20,35],"other_right":[21],"scalar":18,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[17,18],\"other_left\":[20,35],\"other_right\":[21],\"right\":[19],\"scalar\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0094","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[18,19],\"right\":[20],\"other_left\":[21,37],\"other_right\":[22],\"scalar\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[18, 19], [20]], [[39, 56], [42]], [[342, 361], [380]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[18, 19], [20]], [[39, 56], [42]], [[342, 361], [380]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051094,"parameters":{"left":[18,19],"right":[20],"other_left":[21,37],"other_right":[22],"scalar":19,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[18,19],\"other_left\":[21,37],\"other_right\":[22],\"right\":[20],\"scalar\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0095","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[19,20],\"right\":[21],\"other_left\":[22,39],\"other_right\":[23],\"scalar\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[19, 20], [21]], [[41, 59], [44]], [[380, 400], [420]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[19, 20], [21]], [[41, 59], [44]], [[380, 400], [420]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051095,"parameters":{"left":[19,20],"right":[21],"other_left":[22,39],"other_right":[23],"scalar":20,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[19,20],\"other_left\":[22,39],\"other_right\":[23],\"right\":[21],\"scalar\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0096","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[20,21],\"right\":[22],\"other_left\":[23,41],\"other_right\":[24],\"scalar\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[20, 21], [22]], [[43, 62], [46]], [[420, 441], [462]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[20, 21], [22]], [[43, 62], [46]], [[420, 441], [462]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051096,"parameters":{"left":[20,21],"right":[22],"other_left":[23,41],"other_right":[24],"scalar":21,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[20,21],\"other_left\":[23,41],\"other_right\":[24],\"right\":[22],\"scalar\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0097","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[21,22],\"right\":[23],\"other_left\":[24,43],\"other_right\":[25],\"scalar\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[21, 22], [23]], [[45, 65], [48]], [[462, 484], [506]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[21, 22], [23]], [[45, 65], [48]], [[462, 484], [506]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051097,"parameters":{"left":[21,22],"right":[23],"other_left":[24,43],"other_right":[25],"scalar":22,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[21,22],\"other_left\":[24,43],\"other_right\":[25],\"right\":[23],\"scalar\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0098","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[22,23],\"right\":[24],\"other_left\":[25,45],\"other_right\":[26],\"scalar\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[22, 23], [24]], [[47, 68], [50]], [[506, 529], [552]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[22, 23], [24]], [[47, 68], [50]], [[506, 529], [552]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051098,"parameters":{"left":[22,23],"right":[24],"other_left":[25,45],"other_right":[26],"scalar":23,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[22,23],\"other_left\":[25,45],\"other_right\":[26],\"right\":[24],\"scalar\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0099","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[23,24],\"right\":[25],\"other_left\":[26,47],\"other_right\":[27],\"scalar\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[23, 24], [25]], [[49, 71], [52]], [[552, 576], [600]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[23, 24], [25]], [[49, 71], [52]], [[552, 576], [600]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051099,"parameters":{"left":[23,24],"right":[25],"other_left":[26,47],"other_right":[27],"scalar":24,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[23,24],\"other_left\":[26,47],\"other_right\":[27],\"right\":[25],\"scalar\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r01-direct-sum-0100","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","language":"zh-Hant","source_mko_ids":["mko-direct-sum"],"title_zh":"子空間直和","stem_zh":"有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[24,25],\"right\":[26],\"other_left\":[27,49],\"other_right\":[28],\"scalar\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[24, 25], [26]], [[51, 74], [54]], [[600, 625], [650]], external_product_carrier_not_internal_subspaces)","explanation_zh":"Wave5第五層核對=([[24, 25], [26]], [[51, 74], [54]], [[600, 625], [650]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["子空間直和"]},"variant":{"family_id":"qf-mod36-directsum-external-coordinate","seed":202611051100,"parameters":{"left":[24,25],"right":[26],"other_left":[27,49],"other_right":[28],"scalar":25,"domain":"Exact_finite_dimensional_real_subspaces","operation":"directsum-external-coordinate","semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[24,25],\"other_left\":[27,49],\"other_right\":[28],\"right\":[26],\"scalar\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}