{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0001","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[1,2,3],\"index\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3], a_n=0_after_the_list, 0, 1, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([1, 2, 3], a_n=0_after_the_list, 0, 1, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133001,"parameters":{"finite_support_coefficients":[1,2,3],"index":0,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[1,1],[2,1],[3,1]],\"index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0002","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[2,4,6,8],\"index\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 4, 6, 8], a_n=0_after_the_list, 1, 4, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([2, 4, 6, 8], a_n=0_after_the_list, 1, 4, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133002,"parameters":{"finite_support_coefficients":[2,4,6,8],"index":1,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[2,1],[4,1],[6,1],[8,1]],\"index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0003","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[3,6,9,12,15],\"index\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 6, 9, 12, 15], a_n=0_after_the_list, 2, 9, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([3, 6, 9, 12, 15], a_n=0_after_the_list, 2, 9, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133003,"parameters":{"finite_support_coefficients":[3,6,9,12,15],"index":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[3,1],[6,1],[9,1],[12,1],[15,1]],\"index\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0004","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[4,8,12,16,20,24],\"index\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 8, 12, 16, 20, 24], a_n=0_after_the_list, 3, 16, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([4, 8, 12, 16, 20, 24], a_n=0_after_the_list, 3, 16, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133004,"parameters":{"finite_support_coefficients":[4,8,12,16,20,24],"index":3,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[4,1],[8,1],[12,1],[16,1],[20,1],[24,1]],\"index\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0005","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[5,10,15],\"index\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, 10, 15], a_n=0_after_the_list, 4, 0, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([5, 10, 15], a_n=0_after_the_list, 4, 0, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133005,"parameters":{"finite_support_coefficients":[5,10,15],"index":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[5,1],[10,1],[15,1]],\"index\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0006","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[6,12,18,24],\"index\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, 12, 18, 24], a_n=0_after_the_list, 5, 0, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([6, 12, 18, 24], a_n=0_after_the_list, 5, 0, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133006,"parameters":{"finite_support_coefficients":[6,12,18,24],"index":5,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[6,1],[12,1],[18,1],[24,1]],\"index\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0007","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[7,14,21,28,35],\"index\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, 14, 21, 28, 35], a_n=0_after_the_list, 6, 0, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([7, 14, 21, 28, 35], a_n=0_after_the_list, 6, 0, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133007,"parameters":{"finite_support_coefficients":[7,14,21,28,35],"index":6,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[7,1],[14,1],[21,1],[28,1],[35,1]],\"index\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0008","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[8,16,24,32,40,48],\"index\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, 16, 24, 32, 40, 48], a_n=0_after_the_list, 7, 0, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([8, 16, 24, 32, 40, 48], a_n=0_after_the_list, 7, 0, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133008,"parameters":{"finite_support_coefficients":[8,16,24,32,40,48],"index":7,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[8,1],[16,1],[24,1],[32,1],[40,1],[48,1]],\"index\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0009","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[9,18,27],\"index\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, 18, 27], a_n=0_after_the_list, 8, 0, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([9, 18, 27], a_n=0_after_the_list, 8, 0, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133009,"parameters":{"finite_support_coefficients":[9,18,27],"index":8,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[9,1],[18,1],[27,1]],\"index\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0010","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[10,20,30,40],\"index\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, 20, 30, 40], a_n=0_after_the_list, 0, 10, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([10, 20, 30, 40], a_n=0_after_the_list, 0, 10, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133010,"parameters":{"finite_support_coefficients":[10,20,30,40],"index":0,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[10,1],[20,1],[30,1],[40,1]],\"index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0011","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[11,22,33,44,55],\"index\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11, 22, 33, 44, 55], a_n=0_after_the_list, 1, 22, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([11, 22, 33, 44, 55], a_n=0_after_the_list, 1, 22, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133011,"parameters":{"finite_support_coefficients":[11,22,33,44,55],"index":1,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[11,1],[22,1],[33,1],[44,1],[55,1]],\"index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0012","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[12,24,36,48,60,72],\"index\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, 24, 36, 48, 60, 72], a_n=0_after_the_list, 2, 36, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([12, 24, 36, 48, 60, 72], a_n=0_after_the_list, 2, 36, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133012,"parameters":{"finite_support_coefficients":[12,24,36,48,60,72],"index":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[12,1],[24,1],[36,1],[48,1],[60,1],[72,1]],\"index\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0013","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[13,26,39],\"index\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([13, 26, 39], a_n=0_after_the_list, 3, 0, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([13, 26, 39], a_n=0_after_the_list, 3, 0, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133013,"parameters":{"finite_support_coefficients":[13,26,39],"index":3,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[13,1],[26,1],[39,1]],\"index\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0014","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[14,28,42,56],\"index\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([14, 28, 42, 56], a_n=0_after_the_list, 4, 0, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([14, 28, 42, 56], a_n=0_after_the_list, 4, 0, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133014,"parameters":{"finite_support_coefficients":[14,28,42,56],"index":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[14,1],[28,1],[42,1],[56,1]],\"index\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0015","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[15,30,45,60,75],\"index\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([15, 30, 45, 60, 75], a_n=0_after_the_list, 5, 0, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([15, 30, 45, 60, 75], a_n=0_after_the_list, 5, 0, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133015,"parameters":{"finite_support_coefficients":[15,30,45,60,75],"index":5,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[15,1],[30,1],[45,1],[60,1],[75,1]],\"index\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0016","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[16,32,48,64,80,96],\"index\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([16, 32, 48, 64, 80, 96], a_n=0_after_the_list, 6, 0, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([16, 32, 48, 64, 80, 96], a_n=0_after_the_list, 6, 0, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133016,"parameters":{"finite_support_coefficients":[16,32,48,64,80,96],"index":6,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[16,1],[32,1],[48,1],[64,1],[80,1],[96,1]],\"index\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0017","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[17,34,51],\"index\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([17, 34, 51], a_n=0_after_the_list, 7, 0, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([17, 34, 51], a_n=0_after_the_list, 7, 0, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133017,"parameters":{"finite_support_coefficients":[17,34,51],"index":7,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[17,1],[34,1],[51,1]],\"index\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0018","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[18,36,54,72],\"index\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([18, 36, 54, 72], a_n=0_after_the_list, 8, 0, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([18, 36, 54, 72], a_n=0_after_the_list, 8, 0, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133018,"parameters":{"finite_support_coefficients":[18,36,54,72],"index":8,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[18,1],[36,1],[54,1],[72,1]],\"index\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0019","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[19,38,57,76,95],\"index\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([19, 38, 57, 76, 95], a_n=0_after_the_list, 0, 19, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([19, 38, 57, 76, 95], a_n=0_after_the_list, 0, 19, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133019,"parameters":{"finite_support_coefficients":[19,38,57,76,95],"index":0,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[19,1],[38,1],[57,1],[76,1],[95,1]],\"index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0020","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[20,40,60,80,100,120],\"index\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([20, 40, 60, 80, 100, 120], a_n=0_after_the_list, 1, 40, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([20, 40, 60, 80, 100, 120], a_n=0_after_the_list, 1, 40, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133020,"parameters":{"finite_support_coefficients":[20,40,60,80,100,120],"index":1,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[20,1],[40,1],[60,1],[80,1],[100,1],[120,1]],\"index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0021","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[21,42,63],\"index\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([21, 42, 63], a_n=0_after_the_list, 2, 63, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([21, 42, 63], a_n=0_after_the_list, 2, 63, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133021,"parameters":{"finite_support_coefficients":[21,42,63],"index":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[21,1],[42,1],[63,1]],\"index\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0022","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[22,44,66,88],\"index\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([22, 44, 66, 88], a_n=0_after_the_list, 3, 88, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([22, 44, 66, 88], a_n=0_after_the_list, 3, 88, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133022,"parameters":{"finite_support_coefficients":[22,44,66,88],"index":3,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[22,1],[44,1],[66,1],[88,1]],\"index\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0023","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[23,46,69,92,115],\"index\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([23, 46, 69, 92, 115], a_n=0_after_the_list, 4, 115, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([23, 46, 69, 92, 115], a_n=0_after_the_list, 4, 115, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133023,"parameters":{"finite_support_coefficients":[23,46,69,92,115],"index":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[23,1],[46,1],[69,1],[92,1],[115,1]],\"index\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0024","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[24,48,72,96,120,144],\"index\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([24, 48, 72, 96, 120, 144], a_n=0_after_the_list, 5, 144, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([24, 48, 72, 96, 120, 144], a_n=0_after_the_list, 5, 144, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133024,"parameters":{"finite_support_coefficients":[24,48,72,96,120,144],"index":5,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[24,1],[48,1],[72,1],[96,1],[120,1],[144,1]],\"index\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0025","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[25,50,75],\"index\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([25, 50, 75], a_n=0_after_the_list, 6, 0, coefficient_extraction_uses_the_complete_finite_support_law)","explanation_zh":"Wave6第七層核對=([25, 50, 75], a_n=0_after_the_list, 6, 0, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-coefficient-extraction","seed":202611133025,"parameters":{"finite_support_coefficients":[25,50,75],"index":6,"domain":"Analytic_ordinary_generating_function","operation":"ogf-coefficient-extraction","semantic_template":"qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[25,1],[50,1],[75,1]],\"index\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0026","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[1,2,3],\"x\":[1,2],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3], 1/2, 2, [[0, 1, 1], [1, 2, 1], [2, 3, 3/4]], 11/4, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([1, 2, 3], 1/2, 2, [[0, 1, 1], [1, 2, 1], [2, 3, 3/4]], 11/4, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133026,"parameters":{"finite_support_coefficients":[1,2,3],"x":[1,2],"cutoff":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[1,1],[2,1],[3,1]],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0027","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[2,4,6,8],\"x\":[1,3],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 4, 6, 8], 1/3, 2, [[0, 2, 2], [1, 4, 4/3], [2, 6, 2/3]], 4, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([2, 4, 6, 8], 1/3, 2, [[0, 2, 2], [1, 4, 4/3], [2, 6, 2/3]], 4, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133027,"parameters":{"finite_support_coefficients":[2,4,6,8],"x":[1,3],"cutoff":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[2,1],[4,1],[6,1],[8,1]],\"x\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0028","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[3,6,9,12,15],\"x\":[1,4],\"cutoff\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 6, 9, 12, 15], 1/4, 4, [[0, 3, 3], [1, 6, 3/2], [2, 9, 9/16], [3, 12, 3/16], [4, 15, 15/256]], 1359/256, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([3, 6, 9, 12, 15], 1/4, 4, [[0, 3, 3], [1, 6, 3/2], [2, 9, 9/16], [3, 12, 3/16], [4, 15, 15/256]], 1359/256, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133028,"parameters":{"finite_support_coefficients":[3,6,9,12,15],"x":[1,4],"cutoff":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":4,\"finite_support_coefficients\":[[3,1],[6,1],[9,1],[12,1],[15,1]],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0029","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[4,8,12,16,20,24],\"x\":[1,2],\"cutoff\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 8, 12, 16, 20, 24], 1/2, 4, [[0, 4, 4], [1, 8, 4], [2, 12, 3], [3, 16, 2], [4, 20, 5/4]], 57/4, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([4, 8, 12, 16, 20, 24], 1/2, 4, [[0, 4, 4], [1, 8, 4], [2, 12, 3], [3, 16, 2], [4, 20, 5/4]], 57/4, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133029,"parameters":{"finite_support_coefficients":[4,8,12,16,20,24],"x":[1,2],"cutoff":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":4,\"finite_support_coefficients\":[[4,1],[8,1],[12,1],[16,1],[20,1],[24,1]],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0030","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[5,10,15],\"x\":[1,3],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, 10, 15], 1/3, 2, [[0, 5, 5], [1, 10, 10/3], [2, 15, 5/3]], 10, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([5, 10, 15], 1/3, 2, [[0, 5, 5], [1, 10, 10/3], [2, 15, 5/3]], 10, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133030,"parameters":{"finite_support_coefficients":[5,10,15],"x":[1,3],"cutoff":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[5,1],[10,1],[15,1]],\"x\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0031","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[6,12,18,24],\"x\":[1,4],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, 12, 18, 24], 1/4, 2, [[0, 6, 6], [1, 12, 3], [2, 18, 9/8]], 81/8, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([6, 12, 18, 24], 1/4, 2, [[0, 6, 6], [1, 12, 3], [2, 18, 9/8]], 81/8, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133031,"parameters":{"finite_support_coefficients":[6,12,18,24],"x":[1,4],"cutoff":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[6,1],[12,1],[18,1],[24,1]],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0032","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[7,14,21,28,35],\"x\":[1,2],\"cutoff\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, 14, 21, 28, 35], 1/2, 4, [[0, 7, 7], [1, 14, 7], [2, 21, 21/4], [3, 28, 7/2], [4, 35, 35/16]], 399/16, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([7, 14, 21, 28, 35], 1/2, 4, [[0, 7, 7], [1, 14, 7], [2, 21, 21/4], [3, 28, 7/2], [4, 35, 35/16]], 399/16, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133032,"parameters":{"finite_support_coefficients":[7,14,21,28,35],"x":[1,2],"cutoff":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":4,\"finite_support_coefficients\":[[7,1],[14,1],[21,1],[28,1],[35,1]],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0033","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[8,16,24,32,40,48],\"x\":[1,3],\"cutoff\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, 16, 24, 32, 40, 48], 1/3, 4, [[0, 8, 8], [1, 16, 16/3], [2, 24, 8/3], [3, 32, 32/27], [4, 40, 40/81]], 1432/81, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([8, 16, 24, 32, 40, 48], 1/3, 4, [[0, 8, 8], [1, 16, 16/3], [2, 24, 8/3], [3, 32, 32/27], [4, 40, 40/81]], 1432/81, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133033,"parameters":{"finite_support_coefficients":[8,16,24,32,40,48],"x":[1,3],"cutoff":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":4,\"finite_support_coefficients\":[[8,1],[16,1],[24,1],[32,1],[40,1],[48,1]],\"x\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0034","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[9,18,27],\"x\":[1,4],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, 18, 27], 1/4, 2, [[0, 9, 9], [1, 18, 9/2], [2, 27, 27/16]], 243/16, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([9, 18, 27], 1/4, 2, [[0, 9, 9], [1, 18, 9/2], [2, 27, 27/16]], 243/16, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133034,"parameters":{"finite_support_coefficients":[9,18,27],"x":[1,4],"cutoff":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[9,1],[18,1],[27,1]],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0035","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[10,20,30,40],\"x\":[1,2],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, 20, 30, 40], 1/2, 2, [[0, 10, 10], [1, 20, 10], [2, 30, 15/2]], 55/2, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([10, 20, 30, 40], 1/2, 2, [[0, 10, 10], [1, 20, 10], [2, 30, 15/2]], 55/2, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133035,"parameters":{"finite_support_coefficients":[10,20,30,40],"x":[1,2],"cutoff":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[10,1],[20,1],[30,1],[40,1]],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0036","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[11,22,33,44,55],\"x\":[1,3],\"cutoff\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11, 22, 33, 44, 55], 1/3, 4, [[0, 11, 11], [1, 22, 22/3], [2, 33, 11/3], [3, 44, 44/27], [4, 55, 55/81]], 1969/81, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([11, 22, 33, 44, 55], 1/3, 4, [[0, 11, 11], [1, 22, 22/3], [2, 33, 11/3], [3, 44, 44/27], [4, 55, 55/81]], 1969/81, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133036,"parameters":{"finite_support_coefficients":[11,22,33,44,55],"x":[1,3],"cutoff":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":4,\"finite_support_coefficients\":[[11,1],[22,1],[33,1],[44,1],[55,1]],\"x\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0037","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[12,24,36,48,60,72],\"x\":[1,4],\"cutoff\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, 24, 36, 48, 60, 72], 1/4, 4, [[0, 12, 12], [1, 24, 6], [2, 36, 9/4], [3, 48, 3/4], [4, 60, 15/64]], 1359/64, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([12, 24, 36, 48, 60, 72], 1/4, 4, [[0, 12, 12], [1, 24, 6], [2, 36, 9/4], [3, 48, 3/4], [4, 60, 15/64]], 1359/64, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133037,"parameters":{"finite_support_coefficients":[12,24,36,48,60,72],"x":[1,4],"cutoff":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":4,\"finite_support_coefficients\":[[12,1],[24,1],[36,1],[48,1],[60,1],[72,1]],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0038","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[13,26,39],\"x\":[1,2],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([13, 26, 39], 1/2, 2, [[0, 13, 13], [1, 26, 13], [2, 39, 39/4]], 143/4, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([13, 26, 39], 1/2, 2, [[0, 13, 13], [1, 26, 13], [2, 39, 39/4]], 143/4, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133038,"parameters":{"finite_support_coefficients":[13,26,39],"x":[1,2],"cutoff":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[13,1],[26,1],[39,1]],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0039","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[14,28,42,56],\"x\":[1,3],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([14, 28, 42, 56], 1/3, 2, [[0, 14, 14], [1, 28, 28/3], [2, 42, 14/3]], 28, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([14, 28, 42, 56], 1/3, 2, [[0, 14, 14], [1, 28, 28/3], [2, 42, 14/3]], 28, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133039,"parameters":{"finite_support_coefficients":[14,28,42,56],"x":[1,3],"cutoff":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[14,1],[28,1],[42,1],[56,1]],\"x\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0040","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[15,30,45,60,75],\"x\":[1,4],\"cutoff\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([15, 30, 45, 60, 75], 1/4, 4, [[0, 15, 15], [1, 30, 15/2], [2, 45, 45/16], [3, 60, 15/16], [4, 75, 75/256]], 6795/256, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([15, 30, 45, 60, 75], 1/4, 4, [[0, 15, 15], [1, 30, 15/2], [2, 45, 45/16], [3, 60, 15/16], [4, 75, 75/256]], 6795/256, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133040,"parameters":{"finite_support_coefficients":[15,30,45,60,75],"x":[1,4],"cutoff":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":4,\"finite_support_coefficients\":[[15,1],[30,1],[45,1],[60,1],[75,1]],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0041","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[16,32,48,64,80,96],\"x\":[1,2],\"cutoff\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([16, 32, 48, 64, 80, 96], 1/2, 4, [[0, 16, 16], [1, 32, 16], [2, 48, 12], [3, 64, 8], [4, 80, 5]], 57, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([16, 32, 48, 64, 80, 96], 1/2, 4, [[0, 16, 16], [1, 32, 16], [2, 48, 12], [3, 64, 8], [4, 80, 5]], 57, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133041,"parameters":{"finite_support_coefficients":[16,32,48,64,80,96],"x":[1,2],"cutoff":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":4,\"finite_support_coefficients\":[[16,1],[32,1],[48,1],[64,1],[80,1],[96,1]],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0042","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[17,34,51],\"x\":[1,3],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([17, 34, 51], 1/3, 2, [[0, 17, 17], [1, 34, 34/3], [2, 51, 17/3]], 34, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([17, 34, 51], 1/3, 2, [[0, 17, 17], [1, 34, 34/3], [2, 51, 17/3]], 34, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133042,"parameters":{"finite_support_coefficients":[17,34,51],"x":[1,3],"cutoff":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[17,1],[34,1],[51,1]],\"x\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0043","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[18,36,54,72],\"x\":[1,4],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([18, 36, 54, 72], 1/4, 2, [[0, 18, 18], [1, 36, 9], [2, 54, 27/8]], 243/8, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([18, 36, 54, 72], 1/4, 2, [[0, 18, 18], [1, 36, 9], [2, 54, 27/8]], 243/8, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133043,"parameters":{"finite_support_coefficients":[18,36,54,72],"x":[1,4],"cutoff":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[18,1],[36,1],[54,1],[72,1]],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0044","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[19,38,57,76,95],\"x\":[1,2],\"cutoff\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([19, 38, 57, 76, 95], 1/2, 4, [[0, 19, 19], [1, 38, 19], [2, 57, 57/4], [3, 76, 19/2], [4, 95, 95/16]], 1083/16, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([19, 38, 57, 76, 95], 1/2, 4, [[0, 19, 19], [1, 38, 19], [2, 57, 57/4], [3, 76, 19/2], [4, 95, 95/16]], 1083/16, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133044,"parameters":{"finite_support_coefficients":[19,38,57,76,95],"x":[1,2],"cutoff":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":4,\"finite_support_coefficients\":[[19,1],[38,1],[57,1],[76,1],[95,1]],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0045","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[20,40,60,80,100,120],\"x\":[1,3],\"cutoff\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([20, 40, 60, 80, 100, 120], 1/3, 4, [[0, 20, 20], [1, 40, 40/3], [2, 60, 20/3], [3, 80, 80/27], [4, 100, 100/81]], 3580/81, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([20, 40, 60, 80, 100, 120], 1/3, 4, [[0, 20, 20], [1, 40, 40/3], [2, 60, 20/3], [3, 80, 80/27], [4, 100, 100/81]], 3580/81, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133045,"parameters":{"finite_support_coefficients":[20,40,60,80,100,120],"x":[1,3],"cutoff":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":4,\"finite_support_coefficients\":[[20,1],[40,1],[60,1],[80,1],[100,1],[120,1]],\"x\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0046","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[21,42,63],\"x\":[1,4],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([21, 42, 63], 1/4, 2, [[0, 21, 21], [1, 42, 21/2], [2, 63, 63/16]], 567/16, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([21, 42, 63], 1/4, 2, [[0, 21, 21], [1, 42, 21/2], [2, 63, 63/16]], 567/16, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133046,"parameters":{"finite_support_coefficients":[21,42,63],"x":[1,4],"cutoff":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[21,1],[42,1],[63,1]],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0047","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[22,44,66,88],\"x\":[1,2],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([22, 44, 66, 88], 1/2, 2, [[0, 22, 22], [1, 44, 22], [2, 66, 33/2]], 121/2, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([22, 44, 66, 88], 1/2, 2, [[0, 22, 22], [1, 44, 22], [2, 66, 33/2]], 121/2, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133047,"parameters":{"finite_support_coefficients":[22,44,66,88],"x":[1,2],"cutoff":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[22,1],[44,1],[66,1],[88,1]],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0048","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[23,46,69,92,115],\"x\":[1,3],\"cutoff\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([23, 46, 69, 92, 115], 1/3, 4, [[0, 23, 23], [1, 46, 46/3], [2, 69, 23/3], [3, 92, 92/27], [4, 115, 115/81]], 4117/81, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([23, 46, 69, 92, 115], 1/3, 4, [[0, 23, 23], [1, 46, 46/3], [2, 69, 23/3], [3, 92, 92/27], [4, 115, 115/81]], 4117/81, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133048,"parameters":{"finite_support_coefficients":[23,46,69,92,115],"x":[1,3],"cutoff":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":4,\"finite_support_coefficients\":[[23,1],[46,1],[69,1],[92,1],[115,1]],\"x\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0049","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[24,48,72,96,120,144],\"x\":[1,4],\"cutoff\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([24, 48, 72, 96, 120, 144], 1/4, 4, [[0, 24, 24], [1, 48, 12], [2, 72, 9/2], [3, 96, 3/2], [4, 120, 15/32]], 1359/32, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([24, 48, 72, 96, 120, 144], 1/4, 4, [[0, 24, 24], [1, 48, 12], [2, 72, 9/2], [3, 96, 3/2], [4, 120, 15/32]], 1359/32, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133049,"parameters":{"finite_support_coefficients":[24,48,72,96,120,144],"x":[1,4],"cutoff":4,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":4,\"finite_support_coefficients\":[[24,1],[48,1],[72,1],[96,1],[120,1],[144,1]],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0050","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={\"finite_support_coefficients\":[25,50,75],\"x\":[1,2],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([25, 50, 75], 1/2, 2, [[0, 25, 25], [1, 50, 25], [2, 75, 75/4]], 275/4, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)","explanation_zh":"Wave6第七層核對=([25, 50, 75], 1/2, 2, [[0, 25, 25], [1, 50, 25], [2, 75, 75/4]], 275/4, 是, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-finite-partial-sum","seed":202611133050,"parameters":{"finite_support_coefficients":[25,50,75],"x":[1,2],"cutoff":2,"domain":"Analytic_ordinary_generating_function","operation":"ogf-finite-partial-sum","semantic_template":"qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[25,1],[50,1],[75,1]],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0051","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[1,2],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(1/2)^n_for_every_n>=0, 1/2, 1/4, 4/3, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(1/2)^n_for_every_n>=0, 1/2, 1/4, 4/3, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133051,"parameters":{"ratio":[1,2],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[1,2],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0052","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[2,3],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(2/3)^n_for_every_n>=0, 1/2, 1/3, 3/2, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(2/3)^n_for_every_n>=0, 1/2, 1/3, 3/2, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133052,"parameters":{"ratio":[2,3],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[2,3],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0053","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[3,4],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(3/4)^n_for_every_n>=0, 1/2, 3/8, 8/5, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(3/4)^n_for_every_n>=0, 1/2, 3/8, 8/5, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133053,"parameters":{"ratio":[3,4],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[3,4],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0054","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[4,5],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(4/5)^n_for_every_n>=0, 1/2, 2/5, 5/3, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(4/5)^n_for_every_n>=0, 1/2, 2/5, 5/3, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133054,"parameters":{"ratio":[4,5],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[4,5],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0055","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[5,6],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(5/6)^n_for_every_n>=0, 1/2, 5/12, 12/7, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(5/6)^n_for_every_n>=0, 1/2, 5/12, 12/7, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133055,"parameters":{"ratio":[5,6],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[5,6],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0056","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[6,7],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(6/7)^n_for_every_n>=0, 1/2, 3/7, 7/4, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(6/7)^n_for_every_n>=0, 1/2, 3/7, 7/4, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133056,"parameters":{"ratio":[6,7],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[6,7],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0057","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[7,8],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(7/8)^n_for_every_n>=0, 1/2, 7/16, 16/9, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(7/8)^n_for_every_n>=0, 1/2, 7/16, 16/9, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133057,"parameters":{"ratio":[7,8],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[7,8],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0058","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[8,9],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(8/9)^n_for_every_n>=0, 1/2, 4/9, 9/5, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(8/9)^n_for_every_n>=0, 1/2, 4/9, 9/5, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133058,"parameters":{"ratio":[8,9],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[8,9],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0061","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[11,12],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(11/12)^n_for_every_n>=0, 1/2, 11/24, 24/13, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(11/12)^n_for_every_n>=0, 1/2, 11/24, 24/13, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133061,"parameters":{"ratio":[11,12],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[11,12],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0062","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[12,13],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(12/13)^n_for_every_n>=0, 1/2, 6/13, 13/7, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(12/13)^n_for_every_n>=0, 1/2, 6/13, 13/7, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133062,"parameters":{"ratio":[12,13],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[12,13],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0063","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[13,14],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(13/14)^n_for_every_n>=0, 1/2, 13/28, 28/15, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(13/14)^n_for_every_n>=0, 1/2, 13/28, 28/15, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133063,"parameters":{"ratio":[13,14],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[13,14],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0064","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[14,15],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(14/15)^n_for_every_n>=0, 1/2, 7/15, 15/8, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(14/15)^n_for_every_n>=0, 1/2, 7/15, 15/8, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133064,"parameters":{"ratio":[14,15],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[14,15],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0065","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[15,16],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(15/16)^n_for_every_n>=0, 1/2, 15/32, 32/17, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(15/16)^n_for_every_n>=0, 1/2, 15/32, 32/17, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133065,"parameters":{"ratio":[15,16],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[15,16],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0066","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[16,17],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(16/17)^n_for_every_n>=0, 1/2, 8/17, 17/9, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(16/17)^n_for_every_n>=0, 1/2, 8/17, 17/9, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133066,"parameters":{"ratio":[16,17],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[16,17],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0067","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[17,18],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(17/18)^n_for_every_n>=0, 1/2, 17/36, 36/19, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(17/18)^n_for_every_n>=0, 1/2, 17/36, 36/19, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133067,"parameters":{"ratio":[17,18],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[17,18],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0068","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[18,19],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(18/19)^n_for_every_n>=0, 1/2, 9/19, 19/10, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(18/19)^n_for_every_n>=0, 1/2, 9/19, 19/10, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133068,"parameters":{"ratio":[18,19],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[18,19],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0069","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[19,20],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(19/20)^n_for_every_n>=0, 1/2, 19/40, 40/21, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(19/20)^n_for_every_n>=0, 1/2, 19/40, 40/21, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133069,"parameters":{"ratio":[19,20],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[19,20],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0070","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[20,21],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(20/21)^n_for_every_n>=0, 1/2, 10/21, 21/11, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(20/21)^n_for_every_n>=0, 1/2, 10/21, 21/11, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133070,"parameters":{"ratio":[20,21],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[20,21],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0071","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[21,22],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(21/22)^n_for_every_n>=0, 1/2, 21/44, 44/23, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(21/22)^n_for_every_n>=0, 1/2, 21/44, 44/23, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133071,"parameters":{"ratio":[21,22],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[21,22],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0072","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[22,23],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(22/23)^n_for_every_n>=0, 1/2, 11/23, 23/12, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(22/23)^n_for_every_n>=0, 1/2, 11/23, 23/12, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133072,"parameters":{"ratio":[22,23],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[22,23],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0073","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[23,24],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(23/24)^n_for_every_n>=0, 1/2, 23/48, 48/25, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(23/24)^n_for_every_n>=0, 1/2, 23/48, 48/25, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133073,"parameters":{"ratio":[23,24],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[23,24],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0074","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[24,25],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(24/25)^n_for_every_n>=0, 1/2, 12/25, 25/13, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(24/25)^n_for_every_n>=0, 1/2, 12/25, 25/13, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133074,"parameters":{"ratio":[24,25],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[24,25],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0075","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[25,26],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(25/26)^n_for_every_n>=0, 1/2, 25/52, 52/27, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)","explanation_zh":"Wave6第七層核對=(a_n=(25/26)^n_for_every_n>=0, 1/2, 25/52, 52/27, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-geometric-closed-form","seed":202611133075,"parameters":{"ratio":[25,26],"x":[1,2],"domain":"Analytic_ordinary_generating_function","operation":"ogf-geometric-closed-form","semantic_template":"qf-mod44-ogf-geometric-closed-form/{\"ratio\":[25,26],\"x\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0076","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":1,\"a1\":2,\"x\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [1, 2, 3, 5, 8, 13, 21, 34, 55, 89], 1/4, 5/4, 11/16, 20/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(1, 2, [1, 2, 3, 5, 8, 13, 21, 34, 55, 89], 1/4, 5/4, 11/16, 20/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133076,"parameters":{"a0":1,"a1":2,"x":[1,4],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[1,1],\"a1\":[2,1],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0077","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":2,\"a1\":3,\"x\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [2, 3, 5, 8, 13, 21, 34, 55, 89, 144], 1/5, 11/5, 19/25, 55/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(2, 3, [2, 3, 5, 8, 13, 21, 34, 55, 89, 144], 1/5, 11/5, 19/25, 55/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133077,"parameters":{"a0":2,"a1":3,"x":[1,5],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[2,1],\"a1\":[3,1],\"x\":[1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0078","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":3,\"a1\":4,\"x\":[1,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 4, [3, 4, 7, 11, 18, 29, 47, 76, 123, 199], 1/6, 19/6, 29/36, 114/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(3, 4, [3, 4, 7, 11, 18, 29, 47, 76, 123, 199], 1/6, 19/6, 29/36, 114/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133078,"parameters":{"a0":3,"a1":4,"x":[1,6],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[3,1],\"a1\":[4,1],\"x\":[1,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0079","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":4,\"a1\":5,\"x\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 5, [4, 5, 9, 14, 23, 37, 60, 97, 157, 254], 1/4, 17/4, 11/16, 68/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(4, 5, [4, 5, 9, 14, 23, 37, 60, 97, 157, 254], 1/4, 17/4, 11/16, 68/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133079,"parameters":{"a0":4,"a1":5,"x":[1,4],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[4,1],\"a1\":[5,1],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0080","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":5,\"a1\":6,\"x\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 6, [5, 6, 11, 17, 28, 45, 73, 118, 191, 309], 1/5, 26/5, 19/25, 130/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(5, 6, [5, 6, 11, 17, 28, 45, 73, 118, 191, 309], 1/5, 26/5, 19/25, 130/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133080,"parameters":{"a0":5,"a1":6,"x":[1,5],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[5,1],\"a1\":[6,1],\"x\":[1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0081","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":6,\"a1\":7,\"x\":[1,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 7, [6, 7, 13, 20, 33, 53, 86, 139, 225, 364], 1/6, 37/6, 29/36, 222/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(6, 7, [6, 7, 13, 20, 33, 53, 86, 139, 225, 364], 1/6, 37/6, 29/36, 222/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133081,"parameters":{"a0":6,"a1":7,"x":[1,6],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[6,1],\"a1\":[7,1],\"x\":[1,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0082","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":7,\"a1\":8,\"x\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 8, [7, 8, 15, 23, 38, 61, 99, 160, 259, 419], 1/4, 29/4, 11/16, 116/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(7, 8, [7, 8, 15, 23, 38, 61, 99, 160, 259, 419], 1/4, 29/4, 11/16, 116/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133082,"parameters":{"a0":7,"a1":8,"x":[1,4],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[7,1],\"a1\":[8,1],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0083","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":8,\"a1\":9,\"x\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 9, [8, 9, 17, 26, 43, 69, 112, 181, 293, 474], 1/5, 41/5, 19/25, 205/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(8, 9, [8, 9, 17, 26, 43, 69, 112, 181, 293, 474], 1/5, 41/5, 19/25, 205/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133083,"parameters":{"a0":8,"a1":9,"x":[1,5],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[8,1],\"a1\":[9,1],\"x\":[1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0084","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":9,\"a1\":10,\"x\":[1,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 10, [9, 10, 19, 29, 48, 77, 125, 202, 327, 529], 1/6, 55/6, 29/36, 330/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(9, 10, [9, 10, 19, 29, 48, 77, 125, 202, 327, 529], 1/6, 55/6, 29/36, 330/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133084,"parameters":{"a0":9,"a1":10,"x":[1,6],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[9,1],\"a1\":[10,1],\"x\":[1,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0085","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":10,\"a1\":11,\"x\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 11, [10, 11, 21, 32, 53, 85, 138, 223, 361, 584], 1/4, 41/4, 11/16, 164/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(10, 11, [10, 11, 21, 32, 53, 85, 138, 223, 361, 584], 1/4, 41/4, 11/16, 164/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133085,"parameters":{"a0":10,"a1":11,"x":[1,4],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[10,1],\"a1\":[11,1],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0086","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":11,\"a1\":12,\"x\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 12, [11, 12, 23, 35, 58, 93, 151, 244, 395, 639], 1/5, 56/5, 19/25, 280/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(11, 12, [11, 12, 23, 35, 58, 93, 151, 244, 395, 639], 1/5, 56/5, 19/25, 280/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133086,"parameters":{"a0":11,"a1":12,"x":[1,5],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[11,1],\"a1\":[12,1],\"x\":[1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0087","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":12,\"a1\":13,\"x\":[1,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 13, [12, 13, 25, 38, 63, 101, 164, 265, 429, 694], 1/6, 73/6, 29/36, 438/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(12, 13, [12, 13, 25, 38, 63, 101, 164, 265, 429, 694], 1/6, 73/6, 29/36, 438/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133087,"parameters":{"a0":12,"a1":13,"x":[1,6],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[12,1],\"a1\":[13,1],\"x\":[1,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0088","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":13,\"a1\":14,\"x\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 14, [13, 14, 27, 41, 68, 109, 177, 286, 463, 749], 1/4, 53/4, 11/16, 212/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(13, 14, [13, 14, 27, 41, 68, 109, 177, 286, 463, 749], 1/4, 53/4, 11/16, 212/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133088,"parameters":{"a0":13,"a1":14,"x":[1,4],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[13,1],\"a1\":[14,1],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0089","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":14,\"a1\":15,\"x\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 15, [14, 15, 29, 44, 73, 117, 190, 307, 497, 804], 1/5, 71/5, 19/25, 355/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(14, 15, [14, 15, 29, 44, 73, 117, 190, 307, 497, 804], 1/5, 71/5, 19/25, 355/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133089,"parameters":{"a0":14,"a1":15,"x":[1,5],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[14,1],\"a1\":[15,1],\"x\":[1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0090","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":15,\"a1\":16,\"x\":[1,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 16, [15, 16, 31, 47, 78, 125, 203, 328, 531, 859], 1/6, 91/6, 29/36, 546/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(15, 16, [15, 16, 31, 47, 78, 125, 203, 328, 531, 859], 1/6, 91/6, 29/36, 546/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133090,"parameters":{"a0":15,"a1":16,"x":[1,6],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[15,1],\"a1\":[16,1],\"x\":[1,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0091","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":16,\"a1\":17,\"x\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 17, [16, 17, 33, 50, 83, 133, 216, 349, 565, 914], 1/4, 65/4, 11/16, 260/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(16, 17, [16, 17, 33, 50, 83, 133, 216, 349, 565, 914], 1/4, 65/4, 11/16, 260/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133091,"parameters":{"a0":16,"a1":17,"x":[1,4],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[16,1],\"a1\":[17,1],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0092","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":17,\"a1\":18,\"x\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 18, [17, 18, 35, 53, 88, 141, 229, 370, 599, 969], 1/5, 86/5, 19/25, 430/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(17, 18, [17, 18, 35, 53, 88, 141, 229, 370, 599, 969], 1/5, 86/5, 19/25, 430/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133092,"parameters":{"a0":17,"a1":18,"x":[1,5],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[17,1],\"a1\":[18,1],\"x\":[1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0093","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":18,\"a1\":19,\"x\":[1,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 19, [18, 19, 37, 56, 93, 149, 242, 391, 633, 1024], 1/6, 109/6, 29/36, 654/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(18, 19, [18, 19, 37, 56, 93, 149, 242, 391, 633, 1024], 1/6, 109/6, 29/36, 654/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133093,"parameters":{"a0":18,"a1":19,"x":[1,6],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[18,1],\"a1\":[19,1],\"x\":[1,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0094","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":19,\"a1\":20,\"x\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 20, [19, 20, 39, 59, 98, 157, 255, 412, 667, 1079], 1/4, 77/4, 11/16, 28, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(19, 20, [19, 20, 39, 59, 98, 157, 255, 412, 667, 1079], 1/4, 77/4, 11/16, 28, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133094,"parameters":{"a0":19,"a1":20,"x":[1,4],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[19,1],\"a1\":[20,1],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0095","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":20,\"a1\":21,\"x\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 21, [20, 21, 41, 62, 103, 165, 268, 433, 701, 1134], 1/5, 101/5, 19/25, 505/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(20, 21, [20, 21, 41, 62, 103, 165, 268, 433, 701, 1134], 1/5, 101/5, 19/25, 505/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133095,"parameters":{"a0":20,"a1":21,"x":[1,5],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[20,1],\"a1\":[21,1],\"x\":[1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0096","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":21,\"a1\":22,\"x\":[1,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 22, [21, 22, 43, 65, 108, 173, 281, 454, 735, 1189], 1/6, 127/6, 29/36, 762/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(21, 22, [21, 22, 43, 65, 108, 173, 281, 454, 735, 1189], 1/6, 127/6, 29/36, 762/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133096,"parameters":{"a0":21,"a1":22,"x":[1,6],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[21,1],\"a1\":[22,1],\"x\":[1,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0097","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":22,\"a1\":23,\"x\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 23, [22, 23, 45, 68, 113, 181, 294, 475, 769, 1244], 1/4, 89/4, 11/16, 356/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(22, 23, [22, 23, 45, 68, 113, 181, 294, 475, 769, 1244], 1/4, 89/4, 11/16, 356/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133097,"parameters":{"a0":22,"a1":23,"x":[1,4],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[22,1],\"a1\":[23,1],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0098","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":23,\"a1\":24,\"x\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 24, [23, 24, 47, 71, 118, 189, 307, 496, 803, 1299], 1/5, 116/5, 19/25, 580/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(23, 24, [23, 24, 47, 71, 118, 189, 307, 496, 803, 1299], 1/5, 116/5, 19/25, 580/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133098,"parameters":{"a0":23,"a1":24,"x":[1,5],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[23,1],\"a1\":[24,1],\"x\":[1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0099","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":24,\"a1\":25,\"x\":[1,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 25, [24, 25, 49, 74, 123, 197, 320, 517, 837, 1354], 1/6, 145/6, 29/36, 30, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(24, 25, [24, 25, 49, 74, 123, 197, 320, 517, 837, 1354], 1/6, 145/6, 29/36, 30, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133099,"parameters":{"a0":24,"a1":25,"x":[1,6],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[24,1],\"a1\":[25,1],\"x\":[1,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0100","batch_id":"batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations","language":"zh-Hant","source_mko_ids":["mko-analytic-ordinary-generating-function"],"title_zh":"解析普通生成函數","stem_zh":"A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={\"a0\":25,\"a1\":26,\"x\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, 26, [25, 26, 51, 77, 128, 205, 333, 538, 871, 1409], 1/4, 101/4, 11/16, 404/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)","explanation_zh":"Wave6第七層核對=(25, 26, [25, 26, 51, 77, 128, 205, 333, 538, 871, 1409], 1/4, 101/4, 11/16, 404/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["解析普通生成函數"]},"variant":{"family_id":"qf-mod44-ogf-recurrence-boundary-terms","seed":202611133100,"parameters":{"a0":25,"a1":26,"x":[1,4],"domain":"Analytic_ordinary_generating_function","operation":"ogf-recurrence-boundary-terms","semantic_template":"qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[25,1],\"a1\":[26,1],\"x\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}