{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0001","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []], 2, 2, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []], 2, 2, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174001,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0002","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174002,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0003","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174003,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0004","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174004,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0005","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174005,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0006","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174006,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0007","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1],[0],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174007,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0008","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174008,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0009","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174009,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0010","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174010,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0011","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174011,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0012","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2],[0],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], [[2, 100], [2, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], [[2, 100], [2, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174012,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0013","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174013,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0014","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174014,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0015","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174015,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0016","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2],[1],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], [[2, 100], [2, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], [[2, 100], [2, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174016,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0017","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174017,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0018","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174018,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0019","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174019,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0020","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174020,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0021","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174021,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0022","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174022,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0023","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], 6, 6, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], 6, 6, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174023,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0024","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174024,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0025","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[2, 100], [2, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)","explanation_zh":"Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[2, 100], [2, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-rectangle-generation","seed":202611174025,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-rectangle-generation","semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0026","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174026,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0027","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174027,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0028","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174028,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0029","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174029,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0030","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174030,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0031","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174031,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0032","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1],[0],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174032,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0033","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174033,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0034","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174034,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0035","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174035,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0036","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174036,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0037","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2],[0],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是], [[0], [[0, 100], [0, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是], [[0], [[0, 100], [0, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174037,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0038","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174038,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0039","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174039,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0040","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174040,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0041","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2],[1],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是], [[1], [[1, 100], [1, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是], [[1], [[1, 100], [1, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174041,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0042","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174042,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0043","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174043,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0044","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174044,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0045","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174045,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0046","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174046,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0047","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174047,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0048","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174048,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0049","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174049,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0050","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)","explanation_zh":"Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-projection-continuity","seed":202611174050,"parameters":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"domain":"Complete_finite_product_topology","operation":"product-projection-continuity","semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0051","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174051,"parameters":{"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0052","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0],[1],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174052,"parameters":{"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0053","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174053,"parameters":{"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0054","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0],[1],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174054,"parameters":{"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0055","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174055,"parameters":{"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0056","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0],[1],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174056,"parameters":{"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0057","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1],[0],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174057,"parameters":{"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0058","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0],[1],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174058,"parameters":{"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0059","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174059,"parameters":{"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0060","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0],[1],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174060,"parameters":{"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0061","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174061,"parameters":{"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0062","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0],[1],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2],[0],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174062,"parameters":{"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0063","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174063,"parameters":{"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0064","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0],[1],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174064,"parameters":{"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0065","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174065,"parameters":{"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0066","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0],[1],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2],[1],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174066,"parameters":{"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0067","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174067,"parameters":{"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0068","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0],[1],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174068,"parameters":{"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0069","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174069,"parameters":{"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0070","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0],[1],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174070,"parameters":{"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0071","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[0]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174071,"parameters":{"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0072","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0],[1],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174072,"parameters":{"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0073","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174073,"parameters":{"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0074","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0],[1],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174074,"parameters":{"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0075","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={\"z_points\":[0,1],\"z_topology\":[[],[0,1]],\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"map_x\":[0,1],\"map_y\":[100,101]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation_zh":"Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-coordinate-criterion","seed":202611174075,"parameters":{"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101],"domain":"Complete_finite_product_topology","operation":"product-coordinate-criterion","semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0076","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"nonrectangle_open\",\"model\":{\"x_points\":[0,1],\"x_topology\":[[],[0],[1],[0,1]],\"y_points\":[100,101],\"y_topology\":[[],[100],[101],[100,101]],\"candidate_open\":[[0,100],[1,101]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonrectangle_open, [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [1, 101]], 是, 否, [[[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[0, 100], [1, 100]], [[0, 100]], [[0, 101], [1, 101]], [[0, 101]], [[1, 100], [1, 101]], [[1, 100]], [[1, 101]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation_zh":"Wave6第十一層核對=(nonrectangle_open, [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [1, 101]], 是, 否, [[[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[0, 100], [1, 100]], [[0, 100]], [[0, 101], [1, 101]], [[0, 101]], [[1, 100], [1, 101]], [[1, 100]], [[1, 101]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174076,"parameters":{"mode":"nonrectangle_open","model":{"x_points":[0,1],"x_topology":[[],[0],[1],[0,1]],"y_points":[100,101],"y_topology":[[],[100],[101],[100,101]],"candidate_open":[[0,100],[1,101]]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[0,100],[1,101]],x_points:[0,1],x_topology:[[],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101],[100],[101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0077","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"invalid_factor\",\"model\":{\"x_points\":[0,1],\"x_topology\":[[]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"invalid_side\":\"x\"}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)","explanation_zh":"Wave6第十一層核對=(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174077,"parameters":{"mode":"invalid_factor","model":{"x_points":[0,1],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"invalid_factor\",model:{invalid_side:\"x\",x_points:[0,1],x_topology:[[]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0078","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"finite_box_no_extrapolation\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_box_no_extrapolation, 8, 3, 3, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)","explanation_zh":"Wave6第十一層核對=(finite_box_no_extrapolation, 8, 3, 3, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174078,"parameters":{"mode":"finite_box_no_extrapolation","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"finite_box_no_extrapolation\",model:{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0079","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"ordered_pair_direction\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"pair\":[0,101]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)","explanation_zh":"Wave6第十一層核對=(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174079,"parameters":{"mode":"ordered_pair_direction","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]],"pair":[0,101]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"ordered_pair_direction\",model:{pair:[0,101],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0080","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"nonrectangle_open\",\"model\":{\"x_points\":[40,41],\"x_topology\":[[],[40],[41],[40,41]],\"y_points\":[140,141],\"y_topology\":[[],[140],[141],[140,141]],\"candidate_open\":[[40,140],[41,141]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonrectangle_open, [[40, 140], [40, 141], [41, 140], [41, 141]], [[40, 140], [41, 141]], 是, 否, [[[40, 140], [40, 141], [41, 140], [41, 141]], [[40, 140], [40, 141]], [[40, 140], [41, 140]], [[40, 140]], [[40, 141], [41, 141]], [[40, 141]], [[41, 140], [41, 141]], [[41, 140]], [[41, 141]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation_zh":"Wave6第十一層核對=(nonrectangle_open, [[40, 140], [40, 141], [41, 140], [41, 141]], [[40, 140], [41, 141]], 是, 否, [[[40, 140], [40, 141], [41, 140], [41, 141]], [[40, 140], [40, 141]], [[40, 140], [41, 140]], [[40, 140]], [[40, 141], [41, 141]], [[40, 141]], [[41, 140], [41, 141]], [[41, 140]], [[41, 141]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174080,"parameters":{"mode":"nonrectangle_open","model":{"x_points":[40,41],"x_topology":[[],[40],[41],[40,41]],"y_points":[140,141],"y_topology":[[],[140],[141],[140,141]],"candidate_open":[[40,140],[41,141]]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[40,140],[41,141]],x_points:[40,41],x_topology:[[],[40,41],[40],[41]],y_points:[140,141],y_topology:[[],[140,141],[140],[141]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0081","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"invalid_factor\",\"model\":{\"x_points\":[0,1,2],\"x_topology\":[[]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"invalid_side\":\"x\"}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)","explanation_zh":"Wave6第十一層核對=(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174081,"parameters":{"mode":"invalid_factor","model":{"x_points":[0,1,2],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"invalid_factor\",model:{invalid_side:\"x\",x_points:[0,1,2],x_topology:[[]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0082","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"finite_box_no_extrapolation\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1],[0],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_box_no_extrapolation, 8, 5, 5, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)","explanation_zh":"Wave6第十一層核對=(finite_box_no_extrapolation, 8, 5, 5, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174082,"parameters":{"mode":"finite_box_no_extrapolation","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"finite_box_no_extrapolation\",model:{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0083","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"ordered_pair_direction\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"pair\":[0,101]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)","explanation_zh":"Wave6第十一層核對=(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174083,"parameters":{"mode":"ordered_pair_direction","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"pair":[0,101]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"ordered_pair_direction\",model:{pair:[0,101],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[2]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0084","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"nonrectangle_open\",\"model\":{\"x_points\":[80,81],\"x_topology\":[[],[80],[81],[80,81]],\"y_points\":[180,181],\"y_topology\":[[],[180],[181],[180,181]],\"candidate_open\":[[80,180],[81,181]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonrectangle_open, [[80, 180], [80, 181], [81, 180], [81, 181]], [[80, 180], [81, 181]], 是, 否, [[[80, 180], [80, 181], [81, 180], [81, 181]], [[80, 180], [80, 181]], [[80, 180], [81, 180]], [[80, 180]], [[80, 181], [81, 181]], [[80, 181]], [[81, 180], [81, 181]], [[81, 180]], [[81, 181]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation_zh":"Wave6第十一層核對=(nonrectangle_open, [[80, 180], [80, 181], [81, 180], [81, 181]], [[80, 180], [81, 181]], 是, 否, [[[80, 180], [80, 181], [81, 180], [81, 181]], [[80, 180], [80, 181]], [[80, 180], [81, 180]], [[80, 180]], [[80, 181], [81, 181]], [[80, 181]], [[81, 180], [81, 181]], [[81, 180]], [[81, 181]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174084,"parameters":{"mode":"nonrectangle_open","model":{"x_points":[80,81],"x_topology":[[],[80],[81],[80,81]],"y_points":[180,181],"y_topology":[[],[180],[181],[180,181]],"candidate_open":[[80,180],[81,181]]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[80,180],[81,181]],x_points:[80,81],x_topology:[[],[80,81],[80],[81]],y_points:[180,181],y_topology:[[],[180,181],[180],[181]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0085","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"invalid_factor\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"invalid_side\":\"x\"}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)","explanation_zh":"Wave6第十一層核對=(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174085,"parameters":{"mode":"invalid_factor","model":{"x_points":[0,1,2,3],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"invalid_factor\",model:{invalid_side:\"x\",x_points:[0,1,2,3],x_topology:[[]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0086","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"finite_box_no_extrapolation\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_box_no_extrapolation, 8, 4, 4, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)","explanation_zh":"Wave6第十一層核對=(finite_box_no_extrapolation, 8, 4, 4, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174086,"parameters":{"mode":"finite_box_no_extrapolation","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"finite_box_no_extrapolation\",model:{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0087","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"ordered_pair_direction\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2],[0],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"pair\":[0,101]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)","explanation_zh":"Wave6第十一層核對=(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174087,"parameters":{"mode":"ordered_pair_direction","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"pair":[0,101]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"ordered_pair_direction\",model:{pair:[0,101],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0],[2]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0088","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"nonrectangle_open\",\"model\":{\"x_points\":[120,121],\"x_topology\":[[],[120],[121],[120,121]],\"y_points\":[220,221],\"y_topology\":[[],[220],[221],[220,221]],\"candidate_open\":[[120,220],[121,221]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonrectangle_open, [[120, 220], [120, 221], [121, 220], [121, 221]], [[120, 220], [121, 221]], 是, 否, [[[120, 220], [120, 221], [121, 220], [121, 221]], [[120, 220], [120, 221]], [[120, 220], [121, 220]], [[120, 220]], [[120, 221], [121, 221]], [[120, 221]], [[121, 220], [121, 221]], [[121, 220]], [[121, 221]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation_zh":"Wave6第十一層核對=(nonrectangle_open, [[120, 220], [120, 221], [121, 220], [121, 221]], [[120, 220], [121, 221]], 是, 否, [[[120, 220], [120, 221], [121, 220], [121, 221]], [[120, 220], [120, 221]], [[120, 220], [121, 220]], [[120, 220]], [[120, 221], [121, 221]], [[120, 221]], [[121, 220], [121, 221]], [[121, 220]], [[121, 221]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174088,"parameters":{"mode":"nonrectangle_open","model":{"x_points":[120,121],"x_topology":[[],[120],[121],[120,121]],"y_points":[220,221],"y_topology":[[],[220],[221],[220,221]],"candidate_open":[[120,220],[121,221]]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[120,220],[121,221]],x_points:[120,121],x_topology:[[],[120,121],[120],[121]],y_points:[220,221],y_topology:[[],[220,221],[220],[221]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0089","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"invalid_factor\",\"model\":{\"x_points\":[0,1,2,3,4],\"x_topology\":[[]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"invalid_side\":\"x\"}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)","explanation_zh":"Wave6第十一層核對=(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174089,"parameters":{"mode":"invalid_factor","model":{"x_points":[0,1,2,3,4],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"invalid_factor\",model:{invalid_side:\"x\",x_points:[0,1,2,3,4],x_topology:[[]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0090","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"finite_box_no_extrapolation\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_box_no_extrapolation, 8, 4, 4, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)","explanation_zh":"Wave6第十一層核對=(finite_box_no_extrapolation, 8, 4, 4, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174090,"parameters":{"mode":"finite_box_no_extrapolation","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"finite_box_no_extrapolation\",model:{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0091","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"ordered_pair_direction\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2],[1],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"pair\":[0,101]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)","explanation_zh":"Wave6第十一層核對=(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174091,"parameters":{"mode":"ordered_pair_direction","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"pair":[0,101]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"ordered_pair_direction\",model:{pair:[0,101],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1],[2]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0092","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"nonrectangle_open\",\"model\":{\"x_points\":[160,161],\"x_topology\":[[],[160],[161],[160,161]],\"y_points\":[260,261],\"y_topology\":[[],[260],[261],[260,261]],\"candidate_open\":[[160,260],[161,261]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonrectangle_open, [[160, 260], [160, 261], [161, 260], [161, 261]], [[160, 260], [161, 261]], 是, 否, [[[160, 260], [160, 261], [161, 260], [161, 261]], [[160, 260], [160, 261]], [[160, 260], [161, 260]], [[160, 260]], [[160, 261], [161, 261]], [[160, 261]], [[161, 260], [161, 261]], [[161, 260]], [[161, 261]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation_zh":"Wave6第十一層核對=(nonrectangle_open, [[160, 260], [160, 261], [161, 260], [161, 261]], [[160, 260], [161, 261]], 是, 否, [[[160, 260], [160, 261], [161, 260], [161, 261]], [[160, 260], [160, 261]], [[160, 260], [161, 260]], [[160, 260]], [[160, 261], [161, 261]], [[160, 261]], [[161, 260], [161, 261]], [[161, 260]], [[161, 261]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174092,"parameters":{"mode":"nonrectangle_open","model":{"x_points":[160,161],"x_topology":[[],[160],[161],[160,161]],"y_points":[260,261],"y_topology":[[],[260],[261],[260,261]],"candidate_open":[[160,260],[161,261]]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[160,260],[161,261]],x_points:[160,161],x_topology:[[],[160,161],[160],[161]],y_points:[260,261],y_topology:[[],[260,261],[260],[261]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0093","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"invalid_factor\",\"model\":{\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"invalid_side\":\"x\"}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)","explanation_zh":"Wave6第十一層核對=(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174093,"parameters":{"mode":"invalid_factor","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"invalid_factor\",model:{invalid_side:\"x\",x_points:[0,1,2,3,4,5],x_topology:[[]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0094","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"finite_box_no_extrapolation\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_box_no_extrapolation, 8, 4, 4, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)","explanation_zh":"Wave6第十一層核對=(finite_box_no_extrapolation, 8, 4, 4, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174094,"parameters":{"mode":"finite_box_no_extrapolation","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"finite_box_no_extrapolation\",model:{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0095","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"ordered_pair_direction\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"pair\":[0,101]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)","explanation_zh":"Wave6第十一層核對=(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174095,"parameters":{"mode":"ordered_pair_direction","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]],"pair":[0,101]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"ordered_pair_direction\",model:{pair:[0,101],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0096","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"nonrectangle_open\",\"model\":{\"x_points\":[200,201],\"x_topology\":[[],[200],[201],[200,201]],\"y_points\":[300,301],\"y_topology\":[[],[300],[301],[300,301]],\"candidate_open\":[[200,300],[201,301]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonrectangle_open, [[200, 300], [200, 301], [201, 300], [201, 301]], [[200, 300], [201, 301]], 是, 否, [[[200, 300], [200, 301], [201, 300], [201, 301]], [[200, 300], [200, 301]], [[200, 300], [201, 300]], [[200, 300]], [[200, 301], [201, 301]], [[200, 301]], [[201, 300], [201, 301]], [[201, 300]], [[201, 301]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation_zh":"Wave6第十一層核對=(nonrectangle_open, [[200, 300], [200, 301], [201, 300], [201, 301]], [[200, 300], [201, 301]], 是, 否, [[[200, 300], [200, 301], [201, 300], [201, 301]], [[200, 300], [200, 301]], [[200, 300], [201, 300]], [[200, 300]], [[200, 301], [201, 301]], [[200, 301]], [[201, 300], [201, 301]], [[201, 300]], [[201, 301]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174096,"parameters":{"mode":"nonrectangle_open","model":{"x_points":[200,201],"x_topology":[[],[200],[201],[200,201]],"y_points":[300,301],"y_topology":[[],[300],[301],[300,301]],"candidate_open":[[200,300],[201,301]]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[200,300],[201,301]],x_points:[200,201],x_topology:[[],[200,201],[200],[201]],y_points:[300,301],y_topology:[[],[300,301],[300],[301]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0097","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"invalid_factor\",\"model\":{\"x_points\":[0,1,2,3,4,5,6],\"x_topology\":[[]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"invalid_side\":\"x\"}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)","explanation_zh":"Wave6第十一層核對=(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174097,"parameters":{"mode":"invalid_factor","model":{"x_points":[0,1,2,3,4,5,6],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"invalid_factor\",model:{invalid_side:\"x\",x_points:[0,1,2,3,4,5,6],x_topology:[[]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0098","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"finite_box_no_extrapolation\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_box_no_extrapolation, 8, 6, 6, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)","explanation_zh":"Wave6第十一層核對=(finite_box_no_extrapolation, 8, 6, 6, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174098,"parameters":{"mode":"finite_box_no_extrapolation","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"finite_box_no_extrapolation\",model:{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0099","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"ordered_pair_direction\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"pair\":[0,101]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)","explanation_zh":"Wave6第十一層核對=(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174099,"parameters":{"mode":"ordered_pair_direction","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"pair":[0,101]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"ordered_pair_direction\",model:{pair:[0,101],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r04-finite-product-topology-0100","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-product-topology"],"title_zh":"有限乘積拓撲","stem_zh":"完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={\"mode\":\"nonrectangle_open\",\"model\":{\"x_points\":[240,241],\"x_topology\":[[],[240],[241],[240,241]],\"y_points\":[340,341],\"y_topology\":[[],[340],[341],[340,341]],\"candidate_open\":[[240,340],[241,341]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonrectangle_open, [[240, 340], [240, 341], [241, 340], [241, 341]], [[240, 340], [241, 341]], 是, 否, [[[240, 340], [240, 341], [241, 340], [241, 341]], [[240, 340], [240, 341]], [[240, 340], [241, 340]], [[240, 340]], [[240, 341], [241, 341]], [[240, 341]], [[241, 340], [241, 341]], [[241, 340]], [[241, 341]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation_zh":"Wave6第十一層核對=(nonrectangle_open, [[240, 340], [240, 341], [241, 340], [241, 341]], [[240, 340], [241, 341]], 是, 否, [[[240, 340], [240, 341], [241, 340], [241, 341]], [[240, 340], [240, 341]], [[240, 340], [241, 340]], [[240, 340]], [[240, 341], [241, 341]], [[240, 341]], [[241, 340], [241, 341]], [[241, 340]], [[241, 341]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限乘積拓撲"]},"variant":{"family_id":"qf-mod48-product-topology-boundary","seed":202611174100,"parameters":{"mode":"nonrectangle_open","model":{"x_points":[240,241],"x_topology":[[],[240],[241],[240,241]],"y_points":[340,341],"y_topology":[[],[340],[341],[340,341]],"candidate_open":[[240,340],[241,341]]},"domain":"Complete_finite_product_topology","operation":"product-topology-boundary","semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[240,340],[241,341]],x_points:[240,241],x_topology:[[],[240,241],[240],[241]],y_points:[340,341],y_topology:[[],[340,341],[340],[341]]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}