{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0001","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]],\"masses\":[[1,8],[1,4],[1,8],[3,8]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[-12, -1], [-11, 0], [-10, 1], [-9, 2]], [1/8, 1/4, 1/8, 3/8], 是, 7/8, 否)","explanation_zh":"結構深度核對=([[-12, -1], [-11, 0], [-10, 1], [-9, 2]], [1/8, 1/4, 1/8, 3/8], 是, 7/8, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264001,"parameters":{"support":[[-12,-1],[-11,0],[-10,1],[-9,2]],"masses":[[1,8],[1,4],[1,8],[3,8]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[3,8]],\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0002","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[-11,-1],[-10,0],[-9,1],[-8,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[-11, -1], [-10, 0], [-9, 1], [-8, 3]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[-11, -1], [-10, 0], [-9, 1], [-8, 3]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264002,"parameters":{"support":[[-11,-1],[-10,0],[-9,1],[-8,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-11,-1],[-10,0],[-9,1],[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0003","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[-10, -1], [-9, 0], [-8, 1], [-7, 4]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[-10, -1], [-9, 0], [-8, 1], [-7, 4]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264003,"parameters":{"support":[[-10,-1],[-9,0],[-8,1],[-7,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0004","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[-9, -1], [-8, 0], [-7, 1], [-6, 5]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[-9, -1], [-8, 0], [-7, 1], [-6, 5]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264004,"parameters":{"support":[[-9,-1],[-8,0],[-7,1],[-6,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0005","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[-8, -1], [-7, 0], [-6, 1], [-5, 6]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[-8, -1], [-7, 0], [-6, 1], [-5, 6]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264005,"parameters":{"support":[[-8,-1],[-7,0],[-6,1],[-5,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0006","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[-7,-1],[-6,0],[-5,1],[-4,2]],\"masses\":[[1,8],[1,4],[1,8],[3,8]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[-7, -1], [-6, 0], [-5, 1], [-4, 2]], [1/8, 1/4, 1/8, 3/8], 是, 7/8, 否)","explanation_zh":"結構深度核對=([[-7, -1], [-6, 0], [-5, 1], [-4, 2]], [1/8, 1/4, 1/8, 3/8], 是, 7/8, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264006,"parameters":{"support":[[-7,-1],[-6,0],[-5,1],[-4,2]],"masses":[[1,8],[1,4],[1,8],[3,8]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[3,8]],\"support\":[[-7,-1],[-6,0],[-5,1],[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0007","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[-6, -1], [-5, 0], [-4, 1], [-3, 3]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[-6, -1], [-5, 0], [-4, 1], [-3, 3]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264007,"parameters":{"support":[[-6,-1],[-5,0],[-4,1],[-3,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0008","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[-5, -1], [-4, 0], [-3, 1], [-2, 4]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[-5, -1], [-4, 0], [-3, 1], [-2, 4]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264008,"parameters":{"support":[[-5,-1],[-4,0],[-3,1],[-2,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0009","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[-4, -1], [-3, 0], [-2, 1], [-1, 5]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[-4, -1], [-3, 0], [-2, 1], [-1, 5]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264009,"parameters":{"support":[[-4,-1],[-3,0],[-2,1],[-1,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0010","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[-3, -1], [-2, 0], [-1, 1], [0, 6]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[-3, -1], [-2, 0], [-1, 1], [0, 6]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264010,"parameters":{"support":[[-3,-1],[-2,0],[-1,1],[0,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0011","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[-2,-1],[-1,0],[0,1],[1,2]],\"masses\":[[1,8],[1,4],[1,8],[3,8]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[-2, -1], [-1, 0], [0, 1], [1, 2]], [1/8, 1/4, 1/8, 3/8], 是, 7/8, 否)","explanation_zh":"結構深度核對=([[-2, -1], [-1, 0], [0, 1], [1, 2]], [1/8, 1/4, 1/8, 3/8], 是, 7/8, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264011,"parameters":{"support":[[-2,-1],[-1,0],[0,1],[1,2]],"masses":[[1,8],[1,4],[1,8],[3,8]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[3,8]],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0012","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[-1,-1],[0,0],[1,1],[2,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[-1, -1], [0, 0], [1, 1], [2, 3]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[-1, -1], [0, 0], [1, 1], [2, 3]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264012,"parameters":{"support":[[-1,-1],[0,0],[1,1],[2,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-1,-1],[0,0],[1,1],[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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0013","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[0,-1],[1,0],[2,1],[3,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[0, -1], [1, 0], [2, 1], [3, 4]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[0, -1], [1, 0], [2, 1], [3, 4]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264013,"parameters":{"support":[[0,-1],[1,0],[2,1],[3,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[0,-1],[1,0],[2,1],[3,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0014","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[1,-1],[2,0],[3,1],[4,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[1, -1], [2, 0], [3, 1], [4, 5]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[1, -1], [2, 0], [3, 1], [4, 5]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264014,"parameters":{"support":[[1,-1],[2,0],[3,1],[4,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[1,-1],[2,0],[3,1],[4,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0015","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[2,-1],[3,0],[4,1],[5,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[2, -1], [3, 0], [4, 1], [5, 6]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[2, -1], [3, 0], [4, 1], [5, 6]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264015,"parameters":{"support":[[2,-1],[3,0],[4,1],[5,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[2,-1],[3,0],[4,1],[5,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0016","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[3,-1],[4,0],[5,1],[6,2]],\"masses\":[[1,8],[1,4],[1,8],[3,8]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[3, -1], [4, 0], [5, 1], [6, 2]], [1/8, 1/4, 1/8, 3/8], 是, 7/8, 否)","explanation_zh":"結構深度核對=([[3, -1], [4, 0], [5, 1], [6, 2]], [1/8, 1/4, 1/8, 3/8], 是, 7/8, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264016,"parameters":{"support":[[3,-1],[4,0],[5,1],[6,2]],"masses":[[1,8],[1,4],[1,8],[3,8]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[3,8]],\"support\":[[3,-1],[4,0],[5,1],[6,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0017","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[4,-1],[5,0],[6,1],[7,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[4, -1], [5, 0], [6, 1], [7, 3]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[4, -1], [5, 0], [6, 1], [7, 3]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264017,"parameters":{"support":[[4,-1],[5,0],[6,1],[7,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[4,-1],[5,0],[6,1],[7,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0018","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[5,-1],[6,0],[7,1],[8,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[5, -1], [6, 0], [7, 1], [8, 4]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[5, -1], [6, 0], [7, 1], [8, 4]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264018,"parameters":{"support":[[5,-1],[6,0],[7,1],[8,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[5,-1],[6,0],[7,1],[8,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0019","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[6,-1],[7,0],[8,1],[9,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[6, -1], [7, 0], [8, 1], [9, 5]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[6, -1], [7, 0], [8, 1], [9, 5]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264019,"parameters":{"support":[[6,-1],[7,0],[8,1],[9,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[6,-1],[7,0],[8,1],[9,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0020","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[7,-1],[8,0],[9,1],[10,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[7, -1], [8, 0], [9, 1], [10, 6]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[7, -1], [8, 0], [9, 1], [10, 6]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264020,"parameters":{"support":[[7,-1],[8,0],[9,1],[10,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[7,-1],[8,0],[9,1],[10,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0021","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[8,-1],[9,0],[10,1],[11,2]],\"masses\":[[1,8],[1,4],[1,8],[3,8]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[8, -1], [9, 0], [10, 1], [11, 2]], [1/8, 1/4, 1/8, 3/8], 是, 7/8, 否)","explanation_zh":"結構深度核對=([[8, -1], [9, 0], [10, 1], [11, 2]], [1/8, 1/4, 1/8, 3/8], 是, 7/8, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264021,"parameters":{"support":[[8,-1],[9,0],[10,1],[11,2]],"masses":[[1,8],[1,4],[1,8],[3,8]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[3,8]],\"support\":[[8,-1],[9,0],[10,1],[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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0022","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[9,-1],[10,0],[11,1],[12,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[9, -1], [10, 0], [11, 1], [12, 3]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[9, -1], [10, 0], [11, 1], [12, 3]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264022,"parameters":{"support":[[9,-1],[10,0],[11,1],[12,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[9,-1],[10,0],[11,1],[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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0023","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[10,-1],[11,0],[12,1],[13,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[10, -1], [11, 0], [12, 1], [13, 4]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[10, -1], [11, 0], [12, 1], [13, 4]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264023,"parameters":{"support":[[10,-1],[11,0],[12,1],[13,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[10,-1],[11,0],[12,1],[13,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0024","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[11,-1],[12,0],[13,1],[14,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[11, -1], [12, 0], [13, 1], [14, 5]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[11, -1], [12, 0], [13, 1], [14, 5]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264024,"parameters":{"support":[[11,-1],[12,0],[13,1],[14,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[11,-1],[12,0],[13,1],[14,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0025","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-qualification；資料={\"support\":[[12,-1],[13,0],[14,1],[15,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [support,masses,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[12, -1], [13, 0], [14, 1], [15, 6]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)","explanation_zh":"結構深度核對=([[12, -1], [13, 0], [14, 1], [15, 6]], [1/8, 1/4, 1/8, 1/2], 是, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-qualification","seed":202610264025,"parameters":{"support":[[12,-1],[13,0],[14,1],[15,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-qualification","semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[12,-1],[13,0],[14,1],[15,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0026","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[-12, 1/8], [-11, 1/4], [-10, 1/8], [-9, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])","explanation_zh":"結構深度核對=([[-12, 1/8], [-11, 1/4], [-10, 1/8], [-9, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264026,"parameters":{"support":[[-12,-1],[-11,0],[-10,1],[-9,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0027","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[-11,-1],[-10,0],[-9,1],[-8,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[-11, 1/8], [-10, 1/4], [-9, 1/8], [-8, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])","explanation_zh":"結構深度核對=([[-11, 1/8], [-10, 1/4], [-9, 1/8], [-8, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264027,"parameters":{"support":[[-11,-1],[-10,0],[-9,1],[-8,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-11,-1],[-10,0],[-9,1],[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0028","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[-10, 1/8], [-9, 1/4], [-8, 1/8], [-7, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])","explanation_zh":"結構深度核對=([[-10, 1/8], [-9, 1/4], [-8, 1/8], [-7, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264028,"parameters":{"support":[[-10,-1],[-9,0],[-8,1],[-7,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0029","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[-9, 1/8], [-8, 1/4], [-7, 1/8], [-6, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])","explanation_zh":"結構深度核對=([[-9, 1/8], [-8, 1/4], [-7, 1/8], [-6, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264029,"parameters":{"support":[[-9,-1],[-8,0],[-7,1],[-6,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0030","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[-8, 1/8], [-7, 1/4], [-6, 1/8], [-5, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])","explanation_zh":"結構深度核對=([[-8, 1/8], [-7, 1/4], [-6, 1/8], [-5, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264030,"parameters":{"support":[[-8,-1],[-7,0],[-6,1],[-5,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0031","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[-7,-1],[-6,0],[-5,1],[-4,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[-7, 1/8], [-6, 1/4], [-5, 1/8], [-4, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])","explanation_zh":"結構深度核對=([[-7, 1/8], [-6, 1/4], [-5, 1/8], [-4, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264031,"parameters":{"support":[[-7,-1],[-6,0],[-5,1],[-4,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-7,-1],[-6,0],[-5,1],[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0032","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[-6, 1/8], [-5, 1/4], [-4, 1/8], [-3, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])","explanation_zh":"結構深度核對=([[-6, 1/8], [-5, 1/4], [-4, 1/8], [-3, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264032,"parameters":{"support":[[-6,-1],[-5,0],[-4,1],[-3,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0033","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[-5, 1/8], [-4, 1/4], [-3, 1/8], [-2, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])","explanation_zh":"結構深度核對=([[-5, 1/8], [-4, 1/4], [-3, 1/8], [-2, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264033,"parameters":{"support":[[-5,-1],[-4,0],[-3,1],[-2,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0034","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[-4, 1/8], [-3, 1/4], [-2, 1/8], [-1, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])","explanation_zh":"結構深度核對=([[-4, 1/8], [-3, 1/4], [-2, 1/8], [-1, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264034,"parameters":{"support":[[-4,-1],[-3,0],[-2,1],[-1,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0035","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[-3, 1/8], [-2, 1/4], [-1, 1/8], [0, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])","explanation_zh":"結構深度核對=([[-3, 1/8], [-2, 1/4], [-1, 1/8], [0, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264035,"parameters":{"support":[[-3,-1],[-2,0],[-1,1],[0,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0036","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[-2,-1],[-1,0],[0,1],[1,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[-2, 1/8], [-1, 1/4], [0, 1/8], [1, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])","explanation_zh":"結構深度核對=([[-2, 1/8], [-1, 1/4], [0, 1/8], [1, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264036,"parameters":{"support":[[-2,-1],[-1,0],[0,1],[1,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0037","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[-1,-1],[0,0],[1,1],[2,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])","explanation_zh":"結構深度核對=([[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264037,"parameters":{"support":[[-1,-1],[0,0],[1,1],[2,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-1,-1],[0,0],[1,1],[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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0038","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[0,-1],[1,0],[2,1],[3,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1/8], [1, 1/4], [2, 1/8], [3, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])","explanation_zh":"結構深度核對=([[0, 1/8], [1, 1/4], [2, 1/8], [3, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264038,"parameters":{"support":[[0,-1],[1,0],[2,1],[3,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[0,-1],[1,0],[2,1],[3,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0039","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[1,-1],[2,0],[3,1],[4,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1/8], [2, 1/4], [3, 1/8], [4, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])","explanation_zh":"結構深度核對=([[1, 1/8], [2, 1/4], [3, 1/8], [4, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264039,"parameters":{"support":[[1,-1],[2,0],[3,1],[4,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[1,-1],[2,0],[3,1],[4,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0040","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[2,-1],[3,0],[4,1],[5,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[2, 1/8], [3, 1/4], [4, 1/8], [5, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])","explanation_zh":"結構深度核對=([[2, 1/8], [3, 1/4], [4, 1/8], [5, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264040,"parameters":{"support":[[2,-1],[3,0],[4,1],[5,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[2,-1],[3,0],[4,1],[5,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0041","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[3,-1],[4,0],[5,1],[6,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[3, 1/8], [4, 1/4], [5, 1/8], [6, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])","explanation_zh":"結構深度核對=([[3, 1/8], [4, 1/4], [5, 1/8], [6, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264041,"parameters":{"support":[[3,-1],[4,0],[5,1],[6,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[3,-1],[4,0],[5,1],[6,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0042","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[4,-1],[5,0],[6,1],[7,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[4, 1/8], [5, 1/4], [6, 1/8], [7, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])","explanation_zh":"結構深度核對=([[4, 1/8], [5, 1/4], [6, 1/8], [7, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264042,"parameters":{"support":[[4,-1],[5,0],[6,1],[7,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[4,-1],[5,0],[6,1],[7,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0043","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[5,-1],[6,0],[7,1],[8,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[5, 1/8], [6, 1/4], [7, 1/8], [8, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])","explanation_zh":"結構深度核對=([[5, 1/8], [6, 1/4], [7, 1/8], [8, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264043,"parameters":{"support":[[5,-1],[6,0],[7,1],[8,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[5,-1],[6,0],[7,1],[8,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0044","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[6,-1],[7,0],[8,1],[9,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[6, 1/8], [7, 1/4], [8, 1/8], [9, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])","explanation_zh":"結構深度核對=([[6, 1/8], [7, 1/4], [8, 1/8], [9, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264044,"parameters":{"support":[[6,-1],[7,0],[8,1],[9,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[6,-1],[7,0],[8,1],[9,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0045","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[7,-1],[8,0],[9,1],[10,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[7, 1/8], [8, 1/4], [9, 1/8], [10, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])","explanation_zh":"結構深度核對=([[7, 1/8], [8, 1/4], [9, 1/8], [10, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264045,"parameters":{"support":[[7,-1],[8,0],[9,1],[10,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[7,-1],[8,0],[9,1],[10,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0046","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[8,-1],[9,0],[10,1],[11,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[8, 1/8], [9, 1/4], [10, 1/8], [11, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])","explanation_zh":"結構深度核對=([[8, 1/8], [9, 1/4], [10, 1/8], [11, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264046,"parameters":{"support":[[8,-1],[9,0],[10,1],[11,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[8,-1],[9,0],[10,1],[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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0047","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[9,-1],[10,0],[11,1],[12,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[9, 1/8], [10, 1/4], [11, 1/8], [12, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])","explanation_zh":"結構深度核對=([[9, 1/8], [10, 1/4], [11, 1/8], [12, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264047,"parameters":{"support":[[9,-1],[10,0],[11,1],[12,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[9,-1],[10,0],[11,1],[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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0048","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[10,-1],[11,0],[12,1],[13,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[10, 1/8], [11, 1/4], [12, 1/8], [13, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])","explanation_zh":"結構深度核對=([[10, 1/8], [11, 1/4], [12, 1/8], [13, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264048,"parameters":{"support":[[10,-1],[11,0],[12,1],[13,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[10,-1],[11,0],[12,1],[13,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0049","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[11,-1],[12,0],[13,1],[14,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[11, 1/8], [12, 1/4], [13, 1/8], [14, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])","explanation_zh":"結構深度核對=([[11, 1/8], [12, 1/4], [13, 1/8], [14, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264049,"parameters":{"support":[[11,-1],[12,0],[13,1],[14,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[11,-1],[12,0],[13,1],[14,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0050","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[12,-1],[13,0],[14,1],[15,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。","answer":{"kind":"expression","value":"([[12, 1/8], [13, 1/4], [14, 1/8], [15, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])","explanation_zh":"結構深度核對=([[12, 1/8], [13, 1/4], [14, 1/8], [15, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-marginals","seed":202610264050,"parameters":{"support":[[12,-1],[13,0],[14,1],[15,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"domain":"Exact_finite_joint_rational_law","operation":"joint-marginals","semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[12,-1],[13,0],[14,1],[15,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0051","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[-12,-10],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([-12, -10], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([-12, -10], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264051,"parameters":{"support":[[-12,-1],[-11,0],[-10,1],[-9,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[-12,-10],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]],\"x_event\":[-12,-10],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0052","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[-11,-1],[-10,0],[-9,1],[-8,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[-11,-9],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([-11, -9], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([-11, -9], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264052,"parameters":{"support":[[-11,-1],[-10,0],[-9,1],[-8,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[-11,-9],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-11,-1],[-10,0],[-9,1],[-8,3]],\"x_event\":[-11,-9],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0053","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[-10,-8],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([-10, -8], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([-10, -8], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264053,"parameters":{"support":[[-10,-1],[-9,0],[-8,1],[-7,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[-10,-8],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]],\"x_event\":[-10,-8],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0054","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[-9,-7],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([-9, -7], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([-9, -7], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264054,"parameters":{"support":[[-9,-1],[-8,0],[-7,1],[-6,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[-9,-7],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]],\"x_event\":[-9,-7],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0055","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[-8,-6],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([-8, -6], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([-8, -6], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264055,"parameters":{"support":[[-8,-1],[-7,0],[-6,1],[-5,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[-8,-6],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]],\"x_event\":[-8,-6],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0056","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[-7,-1],[-6,0],[-5,1],[-4,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[-7,-5],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([-7, -5], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([-7, -5], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264056,"parameters":{"support":[[-7,-1],[-6,0],[-5,1],[-4,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[-7,-5],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-7,-1],[-6,0],[-5,1],[-4,2]],\"x_event\":[-7,-5],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0057","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[-6,-4],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([-6, -4], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([-6, -4], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264057,"parameters":{"support":[[-6,-1],[-5,0],[-4,1],[-3,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[-6,-4],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]],\"x_event\":[-6,-4],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0058","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[-5,-3],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([-5, -3], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([-5, -3], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264058,"parameters":{"support":[[-5,-1],[-4,0],[-3,1],[-2,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[-5,-3],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]],\"x_event\":[-5,-3],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0059","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[-4,-2],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([-4, -2], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([-4, -2], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264059,"parameters":{"support":[[-4,-1],[-3,0],[-2,1],[-1,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[-4,-2],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]],\"x_event\":[-4,-2],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0060","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[-3,-1],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([-3, -1], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([-3, -1], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264060,"parameters":{"support":[[-3,-1],[-2,0],[-1,1],[0,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[-3,-1],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]],\"x_event\":[-3,-1],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0061","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[-2,-1],[-1,0],[0,1],[1,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[-2,0],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([-2, 0], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([-2, 0], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264061,"parameters":{"support":[[-2,-1],[-1,0],[0,1],[1,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[-2,0],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]],\"x_event\":[-2,0],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0062","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[-1,-1],[0,0],[1,1],[2,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[-1,1],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([-1, 1], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([-1, 1], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264062,"parameters":{"support":[[-1,-1],[0,0],[1,1],[2,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[-1,1],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-1,-1],[0,0],[1,1],[2,3]],\"x_event\":[-1,1],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0063","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[0,-1],[1,0],[2,1],[3,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[0,2],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([0, 2], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264063,"parameters":{"support":[[0,-1],[1,0],[2,1],[3,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[0,2],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[0,-1],[1,0],[2,1],[3,4]],\"x_event\":[0,2],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0064","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[1,-1],[2,0],[3,1],[4,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[1,3],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([1, 3], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([1, 3], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264064,"parameters":{"support":[[1,-1],[2,0],[3,1],[4,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[1,3],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[1,-1],[2,0],[3,1],[4,5]],\"x_event\":[1,3],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0065","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[2,-1],[3,0],[4,1],[5,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[2,4],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([2, 4], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([2, 4], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264065,"parameters":{"support":[[2,-1],[3,0],[4,1],[5,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[2,4],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[2,-1],[3,0],[4,1],[5,6]],\"x_event\":[2,4],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0066","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[3,-1],[4,0],[5,1],[6,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[3,5],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([3, 5], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([3, 5], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264066,"parameters":{"support":[[3,-1],[4,0],[5,1],[6,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[3,5],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[3,-1],[4,0],[5,1],[6,2]],\"x_event\":[3,5],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0067","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[4,-1],[5,0],[6,1],[7,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[4,6],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([4, 6], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([4, 6], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264067,"parameters":{"support":[[4,-1],[5,0],[6,1],[7,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[4,6],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[4,-1],[5,0],[6,1],[7,3]],\"x_event\":[4,6],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0068","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[5,-1],[6,0],[7,1],[8,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[5,7],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([5, 7], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([5, 7], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264068,"parameters":{"support":[[5,-1],[6,0],[7,1],[8,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[5,7],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[5,-1],[6,0],[7,1],[8,4]],\"x_event\":[5,7],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0069","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[6,-1],[7,0],[8,1],[9,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[6,8],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([6, 8], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([6, 8], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264069,"parameters":{"support":[[6,-1],[7,0],[8,1],[9,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[6,8],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[6,-1],[7,0],[8,1],[9,5]],\"x_event\":[6,8],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0070","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[7,-1],[8,0],[9,1],[10,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[7,9],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([7, 9], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([7, 9], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264070,"parameters":{"support":[[7,-1],[8,0],[9,1],[10,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[7,9],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[7,-1],[8,0],[9,1],[10,6]],\"x_event\":[7,9],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0071","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[8,-1],[9,0],[10,1],[11,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[8,10],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([8, 10], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([8, 10], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264071,"parameters":{"support":[[8,-1],[9,0],[10,1],[11,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[8,10],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[8,-1],[9,0],[10,1],[11,2]],\"x_event\":[8,10],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0072","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[9,-1],[10,0],[11,1],[12,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[9,11],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([9, 11], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([9, 11], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264072,"parameters":{"support":[[9,-1],[10,0],[11,1],[12,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[9,11],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[9,-1],[10,0],[11,1],[12,3]],\"x_event\":[9,11],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0073","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[10,-1],[11,0],[12,1],[13,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[10,12],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([10, 12], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([10, 12], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264073,"parameters":{"support":[[10,-1],[11,0],[12,1],[13,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[10,12],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[10,-1],[11,0],[12,1],[13,4]],\"x_event\":[10,12],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0074","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[11,-1],[12,0],[13,1],[14,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[11,13],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([11, 13], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([11, 13], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264074,"parameters":{"support":[[11,-1],[12,0],[13,1],[14,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[11,13],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[11,-1],[12,0],[13,1],[14,5]],\"x_event\":[11,13],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0075","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-rectangle-event；資料={\"support\":[[12,-1],[13,0],[14,1],[15,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"x_event\":[12,14],\"y_event\":[-1,1]}。請依 [x_event,y_event,probability] 欄序回答。","answer":{"kind":"expression","value":"([12, 14], [-1, 1], 1/4)","explanation_zh":"結構深度核對=([12, 14], [-1, 1], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-rectangle-event","seed":202610264075,"parameters":{"support":[[12,-1],[13,0],[14,1],[15,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"x_event":[12,14],"y_event":[-1,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-rectangle-event","semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[12,-1],[13,0],[14,1],[15,6]],\"x_event\":[12,14],\"y_event\":[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0076","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-12,-1]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([-12, -1], 1/8)","explanation_zh":"結構深度核對=([-12, -1], 1/8)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264076,"parameters":{"support":[[-12,-1],[-11,0],[-10,1],[-9,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[-12,-1],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-12,-1],\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0077","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[-11,-1],[-10,0],[-9,1],[-8,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-10,0]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0], 1/4)","explanation_zh":"結構深度核對=([-10, 0], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264077,"parameters":{"support":[[-11,-1],[-10,0],[-9,1],[-8,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[-10,0],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-10,0],\"support\":[[-11,-1],[-10,0],[-9,1],[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0078","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-8,1]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([-8, 1], 1/8)","explanation_zh":"結構深度核對=([-8, 1], 1/8)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264078,"parameters":{"support":[[-10,-1],[-9,0],[-8,1],[-7,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[-8,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-8,1],\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0079","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[3,99]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([3, 99], 0)","explanation_zh":"結構深度核對=([3, 99], 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264079,"parameters":{"support":[[-9,-1],[-8,0],[-7,1],[-6,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[3,99],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[3,99],\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0080","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-8,-1]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([-8, -1], 1/8)","explanation_zh":"結構深度核對=([-8, -1], 1/8)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264080,"parameters":{"support":[[-8,-1],[-7,0],[-6,1],[-5,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[-8,-1],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-8,-1],\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0081","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[-7,-1],[-6,0],[-5,1],[-4,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-6,0]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([-6, 0], 1/4)","explanation_zh":"結構深度核對=([-6, 0], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264081,"parameters":{"support":[[-7,-1],[-6,0],[-5,1],[-4,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[-6,0],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-6,0],\"support\":[[-7,-1],[-6,0],[-5,1],[-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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0082","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-4,1]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1], 1/8)","explanation_zh":"結構深度核對=([-4, 1], 1/8)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264082,"parameters":{"support":[[-6,-1],[-5,0],[-4,1],[-3,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[-4,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-4,1],\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0083","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[7,99]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([7, 99], 0)","explanation_zh":"結構深度核對=([7, 99], 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264083,"parameters":{"support":[[-5,-1],[-4,0],[-3,1],[-2,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[7,99],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[7,99],\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0084","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-4,-1]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([-4, -1], 1/8)","explanation_zh":"結構深度核對=([-4, -1], 1/8)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264084,"parameters":{"support":[[-4,-1],[-3,0],[-2,1],[-1,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[-4,-1],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-4,-1],\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0085","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-2,0]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([-2, 0], 1/4)","explanation_zh":"結構深度核對=([-2, 0], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264085,"parameters":{"support":[[-3,-1],[-2,0],[-1,1],[0,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[-2,0],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-2,0],\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0086","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[-2,-1],[-1,0],[0,1],[1,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[0,1]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 1/8)","explanation_zh":"結構深度核對=([0, 1], 1/8)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264086,"parameters":{"support":[[-2,-1],[-1,0],[0,1],[1,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[0,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[0,1],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0087","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[-1,-1],[0,0],[1,1],[2,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[11,99]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([11, 99], 0)","explanation_zh":"結構深度核對=([11, 99], 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264087,"parameters":{"support":[[-1,-1],[0,0],[1,1],[2,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[11,99],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[11,99],\"support\":[[-1,-1],[0,0],[1,1],[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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0088","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[0,-1],[1,0],[2,1],[3,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[0,-1]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([0, -1], 1/8)","explanation_zh":"結構深度核對=([0, -1], 1/8)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264088,"parameters":{"support":[[0,-1],[1,0],[2,1],[3,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[0,-1],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[0,-1],\"support\":[[0,-1],[1,0],[2,1],[3,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0089","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[1,-1],[2,0],[3,1],[4,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[2,0]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([2, 0], 1/4)","explanation_zh":"結構深度核對=([2, 0], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264089,"parameters":{"support":[[1,-1],[2,0],[3,1],[4,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[2,0],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[2,0],\"support\":[[1,-1],[2,0],[3,1],[4,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0090","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[2,-1],[3,0],[4,1],[5,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[4,1]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([4, 1], 1/8)","explanation_zh":"結構深度核對=([4, 1], 1/8)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264090,"parameters":{"support":[[2,-1],[3,0],[4,1],[5,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[4,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[4,1],\"support\":[[2,-1],[3,0],[4,1],[5,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0091","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[3,-1],[4,0],[5,1],[6,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[15,99]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([15, 99], 0)","explanation_zh":"結構深度核對=([15, 99], 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264091,"parameters":{"support":[[3,-1],[4,0],[5,1],[6,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[15,99],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[15,99],\"support\":[[3,-1],[4,0],[5,1],[6,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0092","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[4,-1],[5,0],[6,1],[7,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[4,-1]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([4, -1], 1/8)","explanation_zh":"結構深度核對=([4, -1], 1/8)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264092,"parameters":{"support":[[4,-1],[5,0],[6,1],[7,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[4,-1],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[4,-1],\"support\":[[4,-1],[5,0],[6,1],[7,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0093","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[5,-1],[6,0],[7,1],[8,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[6,0]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([6, 0], 1/4)","explanation_zh":"結構深度核對=([6, 0], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264093,"parameters":{"support":[[5,-1],[6,0],[7,1],[8,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[6,0],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[6,0],\"support\":[[5,-1],[6,0],[7,1],[8,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0094","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[6,-1],[7,0],[8,1],[9,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[8,1]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([8, 1], 1/8)","explanation_zh":"結構深度核對=([8, 1], 1/8)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264094,"parameters":{"support":[[6,-1],[7,0],[8,1],[9,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[8,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[8,1],\"support\":[[6,-1],[7,0],[8,1],[9,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0095","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[7,-1],[8,0],[9,1],[10,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[19,99]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([19, 99], 0)","explanation_zh":"結構深度核對=([19, 99], 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264095,"parameters":{"support":[[7,-1],[8,0],[9,1],[10,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[19,99],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[19,99],\"support\":[[7,-1],[8,0],[9,1],[10,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0096","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[8,-1],[9,0],[10,1],[11,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[8,-1]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([8, -1], 1/8)","explanation_zh":"結構深度核對=([8, -1], 1/8)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264096,"parameters":{"support":[[8,-1],[9,0],[10,1],[11,2]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[8,-1],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[8,-1],\"support\":[[8,-1],[9,0],[10,1],[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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0097","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[9,-1],[10,0],[11,1],[12,3]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[10,0]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([10, 0], 1/4)","explanation_zh":"結構深度核對=([10, 0], 1/4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264097,"parameters":{"support":[[9,-1],[10,0],[11,1],[12,3]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[10,0],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[10,0],\"support\":[[9,-1],[10,0],[11,1],[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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0098","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[10,-1],[11,0],[12,1],[13,4]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[12,1]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([12, 1], 1/8)","explanation_zh":"結構深度核對=([12, 1], 1/8)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264098,"parameters":{"support":[[10,-1],[11,0],[12,1],[13,4]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[12,1],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[12,1],\"support\":[[10,-1],[11,0],[12,1],[13,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0099","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[11,-1],[12,0],[13,1],[14,5]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[23,99]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([23, 99], 0)","explanation_zh":"結構深度核對=([23, 99], 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264099,"parameters":{"support":[[11,-1],[12,0],[13,1],[14,5]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[23,99],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[23,99],\"support\":[[11,-1],[12,0],[13,1],[14,5]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r04-finite-joint-distribution-0100","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-joint-distribution"],"title_zh":"有限聯合分布","stem_zh":"共同有限機率空間上的完整joint support與精確質量。操作=joint-mass-lookup；資料={\"support\":[[12,-1],[13,0],[14,1],[15,6]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[12,-1]}。請依 [pair,mass] 欄序回答。","answer":{"kind":"expression","value":"([12, -1], 1/8)","explanation_zh":"結構深度核對=([12, -1], 1/8)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限聯合分布"]},"variant":{"family_id":"qf-mod26-joint-mass-lookup","seed":202610264100,"parameters":{"support":[[12,-1],[13,0],[14,1],[15,6]],"masses":[[1,8],[1,4],[1,8],[1,2]],"pair":[12,-1],"domain":"Exact_finite_joint_rational_law","operation":"joint-mass-lookup","semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[12,-1],\"support\":[[12,-1],[13,0],[14,1],[15,6]]}"}},"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、獨立性、一般圖同構、不限維分析、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"}}