{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0001","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[-12,-3]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([-12, -3], [-12, 3])","explanation_zh":"現代橋梁核對=([-12, -3], [-12, 3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231001,"parameters":{"value":[-12,-3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-12,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0002","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[-11,-2]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([-11, -2], [-11, 2])","explanation_zh":"現代橋梁核對=([-11, -2], [-11, 2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231002,"parameters":{"value":[-11,-2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-11,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0003","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[-10,-1]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([-10, -1], [-10, 1])","explanation_zh":"現代橋梁核對=([-10, -1], [-10, 1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231003,"parameters":{"value":[-10,-1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-10,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0004","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[-9,0]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([-9, 0], [-9, 0])","explanation_zh":"現代橋梁核對=([-9, 0], [-9, 0])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231004,"parameters":{"value":[-9,0],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-9,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0005","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[-8,1]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([-8, 1], [-8, -1])","explanation_zh":"現代橋梁核對=([-8, 1], [-8, -1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231005,"parameters":{"value":[-8,1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0006","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[-7,2]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([-7, 2], [-7, -2])","explanation_zh":"現代橋梁核對=([-7, 2], [-7, -2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231006,"parameters":{"value":[-7,2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0007","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[-6,3]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([-6, 3], [-6, -3])","explanation_zh":"現代橋梁核對=([-6, 3], [-6, -3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231007,"parameters":{"value":[-6,3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-6,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0008","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[-5,-3]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([-5, -3], [-5, 3])","explanation_zh":"現代橋梁核對=([-5, -3], [-5, 3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231008,"parameters":{"value":[-5,-3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-5,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0009","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[-4,-2]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([-4, -2], [-4, 2])","explanation_zh":"現代橋梁核對=([-4, -2], [-4, 2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231009,"parameters":{"value":[-4,-2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-4,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0010","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[-3,-1]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([-3, -1], [-3, 1])","explanation_zh":"現代橋梁核對=([-3, -1], [-3, 1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231010,"parameters":{"value":[-3,-1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-3,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0011","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[-2,0]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([-2, 0], [-2, 0])","explanation_zh":"現代橋梁核對=([-2, 0], [-2, 0])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231011,"parameters":{"value":[-2,0],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0012","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[-1,1]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([-1, 1], [-1, -1])","explanation_zh":"現代橋梁核對=([-1, 1], [-1, -1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231012,"parameters":{"value":[-1,1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0013","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[0,2]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [0, -2])","explanation_zh":"現代橋梁核對=([0, 2], [0, -2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231013,"parameters":{"value":[0,2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0014","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[1,3]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([1, 3], [1, -3])","explanation_zh":"現代橋梁核對=([1, 3], [1, -3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231014,"parameters":{"value":[1,3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0015","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[2,-3]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([2, -3], [2, 3])","explanation_zh":"現代橋梁核對=([2, -3], [2, 3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231015,"parameters":{"value":[2,-3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[2,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0016","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[3,-2]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([3, -2], [3, 2])","explanation_zh":"現代橋梁核對=([3, -2], [3, 2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231016,"parameters":{"value":[3,-2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[3,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0017","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[4,-1]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([4, -1], [4, 1])","explanation_zh":"現代橋梁核對=([4, -1], [4, 1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231017,"parameters":{"value":[4,-1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[4,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0018","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[5,0]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([5, 0], [5, 0])","explanation_zh":"現代橋梁核對=([5, 0], [5, 0])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231018,"parameters":{"value":[5,0],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[5,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0019","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[6,1]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([6, 1], [6, -1])","explanation_zh":"現代橋梁核對=([6, 1], [6, -1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231019,"parameters":{"value":[6,1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0020","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[7,2]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([7, 2], [7, -2])","explanation_zh":"現代橋梁核對=([7, 2], [7, -2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231020,"parameters":{"value":[7,2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0021","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[8,3]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([8, 3], [8, -3])","explanation_zh":"現代橋梁核對=([8, 3], [8, -3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231021,"parameters":{"value":[8,3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[8,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0022","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[9,-3]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([9, -3], [9, 3])","explanation_zh":"現代橋梁核對=([9, -3], [9, 3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231022,"parameters":{"value":[9,-3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[9,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0023","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[10,-2]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([10, -2], [10, 2])","explanation_zh":"現代橋梁核對=([10, -2], [10, 2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231023,"parameters":{"value":[10,-2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[10,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0024","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[11,-1]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([11, -1], [11, 1])","explanation_zh":"現代橋梁核對=([11, -1], [11, 1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231024,"parameters":{"value":[11,-1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[11,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0025","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugate；資料={\"value\":[12,0]}。請依 [value,conjugate] 欄序回答。","answer":{"kind":"expression","value":"([12, 0], [12, 0])","explanation_zh":"現代橋梁核對=([12, 0], [12, 0])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugate","seed":202610231025,"parameters":{"value":[12,0],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugate","semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[12,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0026","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[-12,-3]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-12, -3], 153)","explanation_zh":"現代橋梁核對=([-12, -3], 153)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231026,"parameters":{"value":[-12,-3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-12,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0027","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[-11,-2]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-11, -2], 125)","explanation_zh":"現代橋梁核對=([-11, -2], 125)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231027,"parameters":{"value":[-11,-2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-11,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0028","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[-10,-1]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-10, -1], 101)","explanation_zh":"現代橋梁核對=([-10, -1], 101)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231028,"parameters":{"value":[-10,-1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-10,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0029","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[-9,0]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-9, 0], 81)","explanation_zh":"現代橋梁核對=([-9, 0], 81)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231029,"parameters":{"value":[-9,0],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-9,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0030","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[-8,1]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-8, 1], 65)","explanation_zh":"現代橋梁核對=([-8, 1], 65)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231030,"parameters":{"value":[-8,1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0031","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[-7,2]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-7, 2], 53)","explanation_zh":"現代橋梁核對=([-7, 2], 53)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231031,"parameters":{"value":[-7,2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0032","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[-6,3]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-6, 3], 45)","explanation_zh":"現代橋梁核對=([-6, 3], 45)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231032,"parameters":{"value":[-6,3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-6,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0033","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[-5,-3]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-5, -3], 34)","explanation_zh":"現代橋梁核對=([-5, -3], 34)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231033,"parameters":{"value":[-5,-3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-5,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0034","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[-4,-2]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-4, -2], 20)","explanation_zh":"現代橋梁核對=([-4, -2], 20)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231034,"parameters":{"value":[-4,-2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-4,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0035","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[-3,-1]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-3, -1], 10)","explanation_zh":"現代橋梁核對=([-3, -1], 10)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231035,"parameters":{"value":[-3,-1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-3,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0036","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[-2,0]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-2, 0], 4)","explanation_zh":"現代橋梁核對=([-2, 0], 4)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231036,"parameters":{"value":[-2,0],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0037","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[-1,1]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-1, 1], 2)","explanation_zh":"現代橋梁核對=([-1, 1], 2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231037,"parameters":{"value":[-1,1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0038","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[0,2]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], 4)","explanation_zh":"現代橋梁核對=([0, 2], 4)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231038,"parameters":{"value":[0,2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0039","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[1,3]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([1, 3], 10)","explanation_zh":"現代橋梁核對=([1, 3], 10)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231039,"parameters":{"value":[1,3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0040","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[2,-3]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([2, -3], 13)","explanation_zh":"現代橋梁核對=([2, -3], 13)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231040,"parameters":{"value":[2,-3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[2,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0041","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[3,-2]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([3, -2], 13)","explanation_zh":"現代橋梁核對=([3, -2], 13)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231041,"parameters":{"value":[3,-2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[3,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0042","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[4,-1]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([4, -1], 17)","explanation_zh":"現代橋梁核對=([4, -1], 17)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231042,"parameters":{"value":[4,-1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[4,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0043","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[5,0]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([5, 0], 25)","explanation_zh":"現代橋梁核對=([5, 0], 25)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231043,"parameters":{"value":[5,0],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[5,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0044","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[6,1]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([6, 1], 37)","explanation_zh":"現代橋梁核對=([6, 1], 37)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231044,"parameters":{"value":[6,1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0045","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[7,2]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([7, 2], 53)","explanation_zh":"現代橋梁核對=([7, 2], 53)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231045,"parameters":{"value":[7,2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0046","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[8,3]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([8, 3], 73)","explanation_zh":"現代橋梁核對=([8, 3], 73)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231046,"parameters":{"value":[8,3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[8,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0047","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[9,-3]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([9, -3], 90)","explanation_zh":"現代橋梁核對=([9, -3], 90)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231047,"parameters":{"value":[9,-3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[9,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0048","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[10,-2]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([10, -2], 104)","explanation_zh":"現代橋梁核對=([10, -2], 104)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231048,"parameters":{"value":[10,-2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[10,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0049","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[11,-1]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([11, -1], 122)","explanation_zh":"現代橋梁核對=([11, -1], 122)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231049,"parameters":{"value":[11,-1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[11,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0050","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-modulus-square；資料={\"value\":[12,0]}。請依 [value,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([12, 0], 144)","explanation_zh":"現代橋梁核對=([12, 0], 144)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-modulus-square","seed":202610231050,"parameters":{"value":[12,0],"domain":"Integer_pair_complex_invariants","operation":"complexmod-modulus-square","semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[12,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0051","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[-12,-3]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-12, -3], [-12, 3], [153, 0], 153)","explanation_zh":"現代橋梁核對=([-12, -3], [-12, 3], [153, 0], 153)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231051,"parameters":{"value":[-12,-3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-12,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0052","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[-11,-2]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-11, -2], [-11, 2], [125, 0], 125)","explanation_zh":"現代橋梁核對=([-11, -2], [-11, 2], [125, 0], 125)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231052,"parameters":{"value":[-11,-2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-11,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0053","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[-10,-1]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-10, -1], [-10, 1], [101, 0], 101)","explanation_zh":"現代橋梁核對=([-10, -1], [-10, 1], [101, 0], 101)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231053,"parameters":{"value":[-10,-1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-10,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0054","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[-9,0]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-9, 0], [-9, 0], [81, 0], 81)","explanation_zh":"現代橋梁核對=([-9, 0], [-9, 0], [81, 0], 81)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231054,"parameters":{"value":[-9,0],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-9,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0055","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[-8,1]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-8, 1], [-8, -1], [65, 0], 65)","explanation_zh":"現代橋梁核對=([-8, 1], [-8, -1], [65, 0], 65)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231055,"parameters":{"value":[-8,1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0056","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[-7,2]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-7, 2], [-7, -2], [53, 0], 53)","explanation_zh":"現代橋梁核對=([-7, 2], [-7, -2], [53, 0], 53)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231056,"parameters":{"value":[-7,2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0057","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[-6,3]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-6, 3], [-6, -3], [45, 0], 45)","explanation_zh":"現代橋梁核對=([-6, 3], [-6, -3], [45, 0], 45)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231057,"parameters":{"value":[-6,3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-6,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0058","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[-5,-3]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-5, -3], [-5, 3], [34, 0], 34)","explanation_zh":"現代橋梁核對=([-5, -3], [-5, 3], [34, 0], 34)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231058,"parameters":{"value":[-5,-3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-5,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0059","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[-4,-2]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-4, -2], [-4, 2], [20, 0], 20)","explanation_zh":"現代橋梁核對=([-4, -2], [-4, 2], [20, 0], 20)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231059,"parameters":{"value":[-4,-2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-4,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0060","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[-3,-1]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-3, -1], [-3, 1], [10, 0], 10)","explanation_zh":"現代橋梁核對=([-3, -1], [-3, 1], [10, 0], 10)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231060,"parameters":{"value":[-3,-1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-3,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0061","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[-2,0]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-2, 0], [-2, 0], [4, 0], 4)","explanation_zh":"現代橋梁核對=([-2, 0], [-2, 0], [4, 0], 4)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231061,"parameters":{"value":[-2,0],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0062","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[-1,1]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([-1, 1], [-1, -1], [2, 0], 2)","explanation_zh":"現代橋梁核對=([-1, 1], [-1, -1], [2, 0], 2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231062,"parameters":{"value":[-1,1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0063","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[0,2]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [0, -2], [4, 0], 4)","explanation_zh":"現代橋梁核對=([0, 2], [0, -2], [4, 0], 4)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231063,"parameters":{"value":[0,2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0064","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[1,3]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([1, 3], [1, -3], [10, 0], 10)","explanation_zh":"現代橋梁核對=([1, 3], [1, -3], [10, 0], 10)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231064,"parameters":{"value":[1,3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0065","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[2,-3]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([2, -3], [2, 3], [13, 0], 13)","explanation_zh":"現代橋梁核對=([2, -3], [2, 3], [13, 0], 13)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231065,"parameters":{"value":[2,-3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[2,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0066","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[3,-2]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([3, -2], [3, 2], [13, 0], 13)","explanation_zh":"現代橋梁核對=([3, -2], [3, 2], [13, 0], 13)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231066,"parameters":{"value":[3,-2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[3,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0067","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[4,-1]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([4, -1], [4, 1], [17, 0], 17)","explanation_zh":"現代橋梁核對=([4, -1], [4, 1], [17, 0], 17)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231067,"parameters":{"value":[4,-1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[4,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0068","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[5,0]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([5, 0], [5, 0], [25, 0], 25)","explanation_zh":"現代橋梁核對=([5, 0], [5, 0], [25, 0], 25)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231068,"parameters":{"value":[5,0],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[5,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0069","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[6,1]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([6, 1], [6, -1], [37, 0], 37)","explanation_zh":"現代橋梁核對=([6, 1], [6, -1], [37, 0], 37)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231069,"parameters":{"value":[6,1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0070","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[7,2]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([7, 2], [7, -2], [53, 0], 53)","explanation_zh":"現代橋梁核對=([7, 2], [7, -2], [53, 0], 53)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231070,"parameters":{"value":[7,2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0071","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[8,3]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([8, 3], [8, -3], [73, 0], 73)","explanation_zh":"現代橋梁核對=([8, 3], [8, -3], [73, 0], 73)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231071,"parameters":{"value":[8,3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[8,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0072","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[9,-3]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([9, -3], [9, 3], [90, 0], 90)","explanation_zh":"現代橋梁核對=([9, -3], [9, 3], [90, 0], 90)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231072,"parameters":{"value":[9,-3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[9,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0073","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[10,-2]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([10, -2], [10, 2], [104, 0], 104)","explanation_zh":"現代橋梁核對=([10, -2], [10, 2], [104, 0], 104)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231073,"parameters":{"value":[10,-2],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[10,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0074","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[11,-1]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([11, -1], [11, 1], [122, 0], 122)","explanation_zh":"現代橋梁核對=([11, -1], [11, 1], [122, 0], 122)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231074,"parameters":{"value":[11,-1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[11,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0075","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-product-conjugate；資料={\"value\":[12,0]}。請依 [value,conjugate,product,modulus_square] 欄序回答。","answer":{"kind":"expression","value":"([12, 0], [12, 0], [144, 0], 144)","explanation_zh":"現代橋梁核對=([12, 0], [12, 0], [144, 0], 144)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231075,"parameters":{"value":[12,0],"domain":"Integer_pair_complex_invariants","operation":"complexmod-product-conjugate","semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[12,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0076","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-12,-3],\"right\":[-2,-9]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([-12, -3], [-2, -9], [-3, 114], [-3, -114], [-3, -114], 是)","explanation_zh":"現代橋梁核對=([-12, -3], [-2, -9], [-3, 114], [-3, -114], [-3, -114], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231076,"parameters":{"left":[-12,-3],"right":[-2,-9],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-12,-3],\"right\":[-2,-9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0077","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-11,-2],\"right\":[-1,-7]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([-11, -2], [-1, -7], [-3, 79], [-3, -79], [-3, -79], 是)","explanation_zh":"現代橋梁核對=([-11, -2], [-1, -7], [-3, 79], [-3, -79], [-3, -79], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231077,"parameters":{"left":[-11,-2],"right":[-1,-7],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-11,-2],\"right\":[-1,-7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0078","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-10,-1],\"right\":[0,-5]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([-10, -1], [0, -5], [-5, 50], [-5, -50], [-5, -50], 是)","explanation_zh":"現代橋梁核對=([-10, -1], [0, -5], [-5, 50], [-5, -50], [-5, -50], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231078,"parameters":{"left":[-10,-1],"right":[0,-5],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-10,-1],\"right\":[0,-5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0079","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-9,0],\"right\":[1,-3]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([-9, 0], [1, -3], [-9, 27], [-9, -27], [-9, -27], 是)","explanation_zh":"現代橋梁核對=([-9, 0], [1, -3], [-9, 27], [-9, -27], [-9, -27], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231079,"parameters":{"left":[-9,0],"right":[1,-3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-9,0],\"right\":[1,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0080","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-8,1],\"right\":[2,-1]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([-8, 1], [2, -1], [-15, 10], [-15, -10], [-15, -10], 是)","explanation_zh":"現代橋梁核對=([-8, 1], [2, -1], [-15, 10], [-15, -10], [-15, -10], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231080,"parameters":{"left":[-8,1],"right":[2,-1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-8,1],\"right\":[2,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0081","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-7,2],\"right\":[-2,1]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([-7, 2], [-2, 1], [12, -11], [12, 11], [12, 11], 是)","explanation_zh":"現代橋梁核對=([-7, 2], [-2, 1], [12, -11], [12, 11], [12, 11], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231081,"parameters":{"left":[-7,2],"right":[-2,1],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-7,2],\"right\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0082","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-6,3],\"right\":[-1,3]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([-6, 3], [-1, 3], [-3, -21], [-3, 21], [-3, 21], 是)","explanation_zh":"現代橋梁核對=([-6, 3], [-1, 3], [-3, -21], [-3, 21], [-3, 21], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231082,"parameters":{"left":[-6,3],"right":[-1,3],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-6,3],\"right\":[-1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0083","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-5,-3],\"right\":[0,5]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([-5, -3], [0, 5], [15, -25], [15, 25], [15, 25], 是)","explanation_zh":"現代橋梁核對=([-5, -3], [0, 5], [15, -25], [15, 25], [15, 25], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231083,"parameters":{"left":[-5,-3],"right":[0,5],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-5,-3],\"right\":[0,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0084","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-4,-2],\"right\":[1,7]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([-4, -2], [1, 7], [10, -30], [10, 30], [10, 30], 是)","explanation_zh":"現代橋梁核對=([-4, -2], [1, 7], [10, -30], [10, 30], [10, 30], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231084,"parameters":{"left":[-4,-2],"right":[1,7],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-4,-2],\"right\":[1,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0085","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-3,-1],\"right\":[2,9]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([-3, -1], [2, 9], [3, -29], [3, 29], [3, 29], 是)","explanation_zh":"現代橋梁核對=([-3, -1], [2, 9], [3, -29], [3, 29], [3, 29], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231085,"parameters":{"left":[-3,-1],"right":[2,9],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-3,-1],\"right\":[2,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0086","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-2,0],\"right\":[-2,11]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([-2, 0], [-2, 11], [4, -22], [4, 22], [4, 22], 是)","explanation_zh":"現代橋梁核對=([-2, 0], [-2, 11], [4, -22], [4, 22], [4, 22], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231086,"parameters":{"left":[-2,0],"right":[-2,11],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-2,0],\"right\":[-2,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0087","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-1,1],\"right\":[-1,13]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([-1, 1], [-1, 13], [-12, -14], [-12, 14], [-12, 14], 是)","explanation_zh":"現代橋梁核對=([-1, 1], [-1, 13], [-12, -14], [-12, 14], [-12, 14], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231087,"parameters":{"left":[-1,1],"right":[-1,13],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-1,1],\"right\":[-1,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0088","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[0,2],\"right\":[0,15]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [0, 15], [-30, 0], [-30, 0], [-30, 0], 是)","explanation_zh":"現代橋梁核對=([0, 2], [0, 15], [-30, 0], [-30, 0], [-30, 0], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231088,"parameters":{"left":[0,2],"right":[0,15],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[0,2],\"right\":[0,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0089","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[1,3],\"right\":[1,17]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([1, 3], [1, 17], [-50, 20], [-50, -20], [-50, -20], 是)","explanation_zh":"現代橋梁核對=([1, 3], [1, 17], [-50, 20], [-50, -20], [-50, -20], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231089,"parameters":{"left":[1,3],"right":[1,17],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[1,3],\"right\":[1,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0090","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[2,-3],\"right\":[2,19]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([2, -3], [2, 19], [61, 32], [61, -32], [61, -32], 是)","explanation_zh":"現代橋梁核對=([2, -3], [2, 19], [61, 32], [61, -32], [61, -32], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231090,"parameters":{"left":[2,-3],"right":[2,19],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[2,-3],\"right\":[2,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0091","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[3,-2],\"right\":[-2,21]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([3, -2], [-2, 21], [36, 67], [36, -67], [36, -67], 是)","explanation_zh":"現代橋梁核對=([3, -2], [-2, 21], [36, 67], [36, -67], [36, -67], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231091,"parameters":{"left":[3,-2],"right":[-2,21],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[3,-2],\"right\":[-2,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0092","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[4,-1],\"right\":[-1,23]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([4, -1], [-1, 23], [19, 93], [19, -93], [19, -93], 是)","explanation_zh":"現代橋梁核對=([4, -1], [-1, 23], [19, 93], [19, -93], [19, -93], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231092,"parameters":{"left":[4,-1],"right":[-1,23],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[4,-1],\"right\":[-1,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0093","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[5,0],\"right\":[0,25]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([5, 0], [0, 25], [0, 125], [0, -125], [0, -125], 是)","explanation_zh":"現代橋梁核對=([5, 0], [0, 25], [0, 125], [0, -125], [0, -125], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231093,"parameters":{"left":[5,0],"right":[0,25],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[5,0],\"right\":[0,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0094","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[6,1],\"right\":[1,27]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([6, 1], [1, 27], [-21, 163], [-21, -163], [-21, -163], 是)","explanation_zh":"現代橋梁核對=([6, 1], [1, 27], [-21, 163], [-21, -163], [-21, -163], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231094,"parameters":{"left":[6,1],"right":[1,27],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[6,1],\"right\":[1,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0095","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[7,2],\"right\":[2,29]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([7, 2], [2, 29], [-44, 207], [-44, -207], [-44, -207], 是)","explanation_zh":"現代橋梁核對=([7, 2], [2, 29], [-44, 207], [-44, -207], [-44, -207], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231095,"parameters":{"left":[7,2],"right":[2,29],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[7,2],\"right\":[2,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0096","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[8,3],\"right\":[-2,31]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([8, 3], [-2, 31], [-109, 242], [-109, -242], [-109, -242], 是)","explanation_zh":"現代橋梁核對=([8, 3], [-2, 31], [-109, 242], [-109, -242], [-109, -242], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231096,"parameters":{"left":[8,3],"right":[-2,31],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[8,3],\"right\":[-2,31]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0097","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[9,-3],\"right\":[-1,33]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([9, -3], [-1, 33], [90, 300], [90, -300], [90, -300], 是)","explanation_zh":"現代橋梁核對=([9, -3], [-1, 33], [90, 300], [90, -300], [90, -300], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231097,"parameters":{"left":[9,-3],"right":[-1,33],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[9,-3],\"right\":[-1,33]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0098","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[10,-2],\"right\":[0,35]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([10, -2], [0, 35], [70, 350], [70, -350], [70, -350], 是)","explanation_zh":"現代橋梁核對=([10, -2], [0, 35], [70, 350], [70, -350], [70, -350], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231098,"parameters":{"left":[10,-2],"right":[0,35],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[10,-2],\"right\":[0,35]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0099","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[11,-1],\"right\":[1,37]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([11, -1], [1, 37], [48, 406], [48, -406], [48, -406], 是)","explanation_zh":"現代橋梁核對=([11, -1], [1, 37], [48, 406], [48, -406], [48, -406], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231099,"parameters":{"left":[11,-1],"right":[1,37],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[11,-1],\"right\":[1,37]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0100","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","language":"zh-Hant","source_mko_ids":["mko-complex-modulus-conjugate"],"title_zh":"複數模與共軛","stem_zh":"複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[12,0],\"right\":[2,39]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。","answer":{"kind":"expression","value":"([12, 0], [2, 39], [24, 468], [24, -468], [24, -468], 是)","explanation_zh":"現代橋梁核對=([12, 0], [2, 39], [24, 468], [24, -468], [24, -468], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["複數模與共軛"]},"variant":{"family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231100,"parameters":{"left":[12,0],"right":[2,39],"domain":"Integer_pair_complex_invariants","operation":"complexmod-conjugation-product","semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[12,0],\"right\":[2,39]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}