{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0001","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":-2,\"initial_values\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [1, 2], -2, [0, 1], 是, 是, rule_applies_for_n_ge_2)","explanation_zh":"Wave6第四層核對=(2, [1, 2], -2, [0, 1], 是, 是, rule_applies_for_n_ge_2)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105001,"parameters":{"order":2,"coefficients":[1,2],"constant":-2,"initial_values":[0,1],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2],\"constant\":-2,\"initial_values\":[0,1],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0002","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":3,\"coefficients\":[2,4,6],\"constant\":-1,\"initial_values\":[1,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [2, 4, 6], -1, [1, 2, 3], 是, 是, rule_applies_for_n_ge_3)","explanation_zh":"Wave6第四層核對=(3, [2, 4, 6], -1, [1, 2, 3], 是, 是, rule_applies_for_n_ge_3)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105002,"parameters":{"order":3,"coefficients":[2,4,6],"constant":-1,"initial_values":[1,2,3],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6],\"constant\":-1,\"initial_values\":[1,2,3],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0003","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":4,\"coefficients\":[1,2,3,4],\"constant\":0,\"initial_values\":[2,3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], 0, [2, 3, 4, 5], 是, 是, rule_applies_for_n_ge_4)","explanation_zh":"Wave6第四層核對=(4, [1, 2, 3, 4], 0, [2, 3, 4, 5], 是, 是, rule_applies_for_n_ge_4)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105003,"parameters":{"order":4,"coefficients":[1,2,3,4],"constant":0,"initial_values":[2,3,4,5],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3,4],\"constant\":0,\"initial_values\":[2,3,4,5],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0004","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":2,\"coefficients\":[2,4],\"constant\":1,\"initial_values\":[3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [2, 4], 1, [3, 4], 是, 是, rule_applies_for_n_ge_2)","explanation_zh":"Wave6第四層核對=(2, [2, 4], 1, [3, 4], 是, 是, rule_applies_for_n_ge_2)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105004,"parameters":{"order":2,"coefficients":[2,4],"constant":1,"initial_values":[3,4],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4],\"constant\":1,\"initial_values\":[3,4],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0005","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":3,\"coefficients\":[1,2,3],\"constant\":2,\"initial_values\":[4,5,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [1, 2, 3], 2, [4, 5, 6], 是, 是, rule_applies_for_n_ge_3)","explanation_zh":"Wave6第四層核對=(3, [1, 2, 3], 2, [4, 5, 6], 是, 是, rule_applies_for_n_ge_3)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105005,"parameters":{"order":3,"coefficients":[1,2,3],"constant":2,"initial_values":[4,5,6],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3],\"constant\":2,\"initial_values\":[4,5,6],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0006","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":-2,\"initial_values\":[5,6,7,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [2, 4, 6, 8], -2, [5, 6, 7, 8], 是, 是, rule_applies_for_n_ge_4)","explanation_zh":"Wave6第四層核對=(4, [2, 4, 6, 8], -2, [5, 6, 7, 8], 是, 是, rule_applies_for_n_ge_4)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105006,"parameters":{"order":4,"coefficients":[2,4,6,8],"constant":-2,"initial_values":[5,6,7,8],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6,8],\"constant\":-2,\"initial_values\":[5,6,7,8],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0007","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":-1,\"initial_values\":[6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [1, 2], -1, [6, 7], 是, 是, rule_applies_for_n_ge_2)","explanation_zh":"Wave6第四層核對=(2, [1, 2], -1, [6, 7], 是, 是, rule_applies_for_n_ge_2)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105007,"parameters":{"order":2,"coefficients":[1,2],"constant":-1,"initial_values":[6,7],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2],\"constant\":-1,\"initial_values\":[6,7],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0008","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":3,\"coefficients\":[2,4,6],\"constant\":0,\"initial_values\":[7,8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [2, 4, 6], 0, [7, 8, 9], 是, 是, rule_applies_for_n_ge_3)","explanation_zh":"Wave6第四層核對=(3, [2, 4, 6], 0, [7, 8, 9], 是, 是, rule_applies_for_n_ge_3)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105008,"parameters":{"order":3,"coefficients":[2,4,6],"constant":0,"initial_values":[7,8,9],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6],\"constant\":0,\"initial_values\":[7,8,9],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0009","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":4,\"coefficients\":[1,2,3,4],\"constant\":1,\"initial_values\":[8,9,10,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], 1, [8, 9, 10, 11], 是, 是, rule_applies_for_n_ge_4)","explanation_zh":"Wave6第四層核對=(4, [1, 2, 3, 4], 1, [8, 9, 10, 11], 是, 是, rule_applies_for_n_ge_4)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105009,"parameters":{"order":4,"coefficients":[1,2,3,4],"constant":1,"initial_values":[8,9,10,11],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3,4],\"constant\":1,\"initial_values\":[8,9,10,11],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0010","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":2,\"coefficients\":[2,4],\"constant\":2,\"initial_values\":[9,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [2, 4], 2, [9, 10], 是, 是, rule_applies_for_n_ge_2)","explanation_zh":"Wave6第四層核對=(2, [2, 4], 2, [9, 10], 是, 是, rule_applies_for_n_ge_2)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105010,"parameters":{"order":2,"coefficients":[2,4],"constant":2,"initial_values":[9,10],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4],\"constant\":2,\"initial_values\":[9,10],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0011","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":3,\"coefficients\":[1,2,3],\"constant\":-2,\"initial_values\":[10,11,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [1, 2, 3], -2, [10, 11, 12], 是, 是, rule_applies_for_n_ge_3)","explanation_zh":"Wave6第四層核對=(3, [1, 2, 3], -2, [10, 11, 12], 是, 是, rule_applies_for_n_ge_3)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105011,"parameters":{"order":3,"coefficients":[1,2,3],"constant":-2,"initial_values":[10,11,12],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3],\"constant\":-2,\"initial_values\":[10,11,12],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0012","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":-1,\"initial_values\":[11,12,13,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [2, 4, 6, 8], -1, [11, 12, 13, 14], 是, 是, rule_applies_for_n_ge_4)","explanation_zh":"Wave6第四層核對=(4, [2, 4, 6, 8], -1, [11, 12, 13, 14], 是, 是, rule_applies_for_n_ge_4)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105012,"parameters":{"order":4,"coefficients":[2,4,6,8],"constant":-1,"initial_values":[11,12,13,14],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6,8],\"constant\":-1,\"initial_values\":[11,12,13,14],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0013","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":0,\"initial_values\":[12,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [1, 2], 0, [12, 13], 是, 是, rule_applies_for_n_ge_2)","explanation_zh":"Wave6第四層核對=(2, [1, 2], 0, [12, 13], 是, 是, rule_applies_for_n_ge_2)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105013,"parameters":{"order":2,"coefficients":[1,2],"constant":0,"initial_values":[12,13],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2],\"constant\":0,\"initial_values\":[12,13],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0014","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":3,\"coefficients\":[2,4,6],\"constant\":1,\"initial_values\":[13,14,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [2, 4, 6], 1, [13, 14, 15], 是, 是, rule_applies_for_n_ge_3)","explanation_zh":"Wave6第四層核對=(3, [2, 4, 6], 1, [13, 14, 15], 是, 是, rule_applies_for_n_ge_3)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105014,"parameters":{"order":3,"coefficients":[2,4,6],"constant":1,"initial_values":[13,14,15],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6],\"constant\":1,\"initial_values\":[13,14,15],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0015","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":4,\"coefficients\":[1,2,3,4],\"constant\":2,\"initial_values\":[14,15,16,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], 2, [14, 15, 16, 17], 是, 是, rule_applies_for_n_ge_4)","explanation_zh":"Wave6第四層核對=(4, [1, 2, 3, 4], 2, [14, 15, 16, 17], 是, 是, rule_applies_for_n_ge_4)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105015,"parameters":{"order":4,"coefficients":[1,2,3,4],"constant":2,"initial_values":[14,15,16,17],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3,4],\"constant\":2,\"initial_values\":[14,15,16,17],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0016","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":2,\"coefficients\":[2,4],\"constant\":-2,\"initial_values\":[15,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [2, 4], -2, [15, 16], 是, 是, rule_applies_for_n_ge_2)","explanation_zh":"Wave6第四層核對=(2, [2, 4], -2, [15, 16], 是, 是, rule_applies_for_n_ge_2)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105016,"parameters":{"order":2,"coefficients":[2,4],"constant":-2,"initial_values":[15,16],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4],\"constant\":-2,\"initial_values\":[15,16],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0017","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":3,\"coefficients\":[1,2,3],\"constant\":-1,\"initial_values\":[16,17,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [1, 2, 3], -1, [16, 17, 18], 是, 是, rule_applies_for_n_ge_3)","explanation_zh":"Wave6第四層核對=(3, [1, 2, 3], -1, [16, 17, 18], 是, 是, rule_applies_for_n_ge_3)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105017,"parameters":{"order":3,"coefficients":[1,2,3],"constant":-1,"initial_values":[16,17,18],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3],\"constant\":-1,\"initial_values\":[16,17,18],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0018","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":0,\"initial_values\":[17,18,19,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [2, 4, 6, 8], 0, [17, 18, 19, 20], 是, 是, rule_applies_for_n_ge_4)","explanation_zh":"Wave6第四層核對=(4, [2, 4, 6, 8], 0, [17, 18, 19, 20], 是, 是, rule_applies_for_n_ge_4)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105018,"parameters":{"order":4,"coefficients":[2,4,6,8],"constant":0,"initial_values":[17,18,19,20],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6,8],\"constant\":0,\"initial_values\":[17,18,19,20],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0019","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":1,\"initial_values\":[18,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [1, 2], 1, [18, 19], 是, 是, rule_applies_for_n_ge_2)","explanation_zh":"Wave6第四層核對=(2, [1, 2], 1, [18, 19], 是, 是, rule_applies_for_n_ge_2)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105019,"parameters":{"order":2,"coefficients":[1,2],"constant":1,"initial_values":[18,19],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2],\"constant\":1,\"initial_values\":[18,19],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0020","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":3,\"coefficients\":[2,4,6],\"constant\":2,\"initial_values\":[19,20,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [2, 4, 6], 2, [19, 20, 21], 是, 是, rule_applies_for_n_ge_3)","explanation_zh":"Wave6第四層核對=(3, [2, 4, 6], 2, [19, 20, 21], 是, 是, rule_applies_for_n_ge_3)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105020,"parameters":{"order":3,"coefficients":[2,4,6],"constant":2,"initial_values":[19,20,21],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6],\"constant\":2,\"initial_values\":[19,20,21],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0021","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":4,\"coefficients\":[1,2,3,4],\"constant\":-2,\"initial_values\":[20,21,22,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], -2, [20, 21, 22, 23], 是, 是, rule_applies_for_n_ge_4)","explanation_zh":"Wave6第四層核對=(4, [1, 2, 3, 4], -2, [20, 21, 22, 23], 是, 是, rule_applies_for_n_ge_4)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105021,"parameters":{"order":4,"coefficients":[1,2,3,4],"constant":-2,"initial_values":[20,21,22,23],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3,4],\"constant\":-2,\"initial_values\":[20,21,22,23],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0022","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":2,\"coefficients\":[2,4],\"constant\":-1,\"initial_values\":[21,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [2, 4], -1, [21, 22], 是, 是, rule_applies_for_n_ge_2)","explanation_zh":"Wave6第四層核對=(2, [2, 4], -1, [21, 22], 是, 是, rule_applies_for_n_ge_2)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105022,"parameters":{"order":2,"coefficients":[2,4],"constant":-1,"initial_values":[21,22],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4],\"constant\":-1,\"initial_values\":[21,22],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0023","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":3,\"coefficients\":[1,2,3],\"constant\":0,\"initial_values\":[22,23,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [1, 2, 3], 0, [22, 23, 24], 是, 是, rule_applies_for_n_ge_3)","explanation_zh":"Wave6第四層核對=(3, [1, 2, 3], 0, [22, 23, 24], 是, 是, rule_applies_for_n_ge_3)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105023,"parameters":{"order":3,"coefficients":[1,2,3],"constant":0,"initial_values":[22,23,24],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3],\"constant\":0,\"initial_values\":[22,23,24],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0024","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":1,\"initial_values\":[23,24,25,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [2, 4, 6, 8], 1, [23, 24, 25, 26], 是, 是, rule_applies_for_n_ge_4)","explanation_zh":"Wave6第四層核對=(4, [2, 4, 6, 8], 1, [23, 24, 25, 26], 是, 是, rule_applies_for_n_ge_4)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105024,"parameters":{"order":4,"coefficients":[2,4,6,8],"constant":1,"initial_values":[23,24,25,26],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6,8],\"constant\":1,\"initial_values\":[23,24,25,26],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0025","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":2,\"initial_values\":[24,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [1, 2], 2, [24, 25], 是, 是, rule_applies_for_n_ge_2)","explanation_zh":"Wave6第四層核對=(2, [1, 2], 2, [24, 25], 是, 是, rule_applies_for_n_ge_2)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105025,"parameters":{"order":2,"coefficients":[1,2],"constant":2,"initial_values":[24,25],"domain":"Exact_total_linear_recurrence","operation":"recurrence-specification-qualification","semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2],\"constant\":2,\"initial_values\":[24,25],\"order\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0026","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":-2,\"initial_values\":[0,1],\"target_n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1], 5, [0, 1, 0, -1, -4, -11], -11, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(2, [0, 1], 5, [0, 1, 0, -1, -4, -11], -11, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105026,"parameters":{"order":2,"coefficients":[1,2],"constant":-2,"initial_values":[0,1],"target_n":5,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2],\"constant\":-2,\"initial_values\":[0,1],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0027","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":3,\"coefficients\":[2,4,6],\"constant\":-1,\"initial_values\":[1,2,3],\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [1, 2, 3], 7, [1, 2, 3, 27, 177, 1175, 7811, 51919], 51919, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(3, [1, 2, 3], 7, [1, 2, 3, 27, 177, 1175, 7811, 51919], 51919, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105027,"parameters":{"order":3,"coefficients":[2,4,6],"constant":-1,"initial_values":[1,2,3],"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6],\"constant\":-1,\"initial_values\":[1,2,3],\"order\":3,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0028","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":4,\"coefficients\":[1,2,3,4],\"constant\":0,\"initial_values\":[2,3,4,5],\"target_n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [2, 3, 4, 5], 9, [2, 3, 4, 5, 40, 186, 878, 4155, 19666, 93071], 93071, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(4, [2, 3, 4, 5], 9, [2, 3, 4, 5, 40, 186, 878, 4155, 19666, 93071], 93071, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105028,"parameters":{"order":4,"coefficients":[1,2,3,4],"constant":0,"initial_values":[2,3,4,5],"target_n":9,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3,4],\"constant\":0,\"initial_values\":[2,3,4,5],\"order\":4,\"target_n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0029","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":2,\"coefficients\":[2,4],\"constant\":1,\"initial_values\":[3,4],\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [3, 4], 8, [3, 4, 23, 101, 451, 2007, 8931, 39739, 176819], 176819, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(2, [3, 4], 8, [3, 4, 23, 101, 451, 2007, 8931, 39739, 176819], 176819, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105029,"parameters":{"order":2,"coefficients":[2,4],"constant":1,"initial_values":[3,4],"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4],\"constant\":1,\"initial_values\":[3,4],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0030","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":3,\"coefficients\":[1,2,3],\"constant\":2,\"initial_values\":[4,5,6],\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [4, 5, 6], 6, [4, 5, 6, 34, 121, 439, 1595], 1595, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(3, [4, 5, 6], 6, [4, 5, 6, 34, 121, 439, 1595], 1595, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105030,"parameters":{"order":3,"coefficients":[1,2,3],"constant":2,"initial_values":[4,5,6],"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3],\"constant\":2,\"initial_values\":[4,5,6],\"order\":3,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0031","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":-2,\"initial_values\":[5,6,7,8],\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [5, 6, 7, 8], 8, [5, 6, 7, 8, 138, 1190, 10392, 90842, 794122], 794122, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(4, [5, 6, 7, 8], 8, [5, 6, 7, 8, 138, 1190, 10392, 90842, 794122], 794122, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105031,"parameters":{"order":4,"coefficients":[2,4,6,8],"constant":-2,"initial_values":[5,6,7,8],"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6,8],\"constant\":-2,\"initial_values\":[5,6,7,8],\"order\":4,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0032","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":-1,\"initial_values\":[6,7],\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [6, 7], 7, [6, 7, 19, 44, 106, 255, 615, 1484], 1484, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(2, [6, 7], 7, [6, 7, 19, 44, 106, 255, 615, 1484], 1484, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105032,"parameters":{"order":2,"coefficients":[1,2],"constant":-1,"initial_values":[6,7],"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2],\"constant\":-1,\"initial_values\":[6,7],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0033","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":3,\"coefficients\":[2,4,6],\"constant\":0,\"initial_values\":[7,8,9],\"target_n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [7, 8, 9], 9, [7, 8, 9, 100, 652, 4330, 28788, 191352, 1271924, 8454528], 8454528, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(3, [7, 8, 9], 9, [7, 8, 9, 100, 652, 4330, 28788, 191352, 1271924, 8454528], 8454528, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105033,"parameters":{"order":3,"coefficients":[2,4,6],"constant":0,"initial_values":[7,8,9],"target_n":9,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6],\"constant\":0,\"initial_values\":[7,8,9],\"order\":3,\"target_n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0034","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":4,\"coefficients\":[1,2,3,4],\"constant\":1,\"initial_values\":[8,9,10,11],\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [8, 9, 10, 11], 7, [8, 9, 10, 11, 101, 467, 2204, 10431], 10431, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(4, [8, 9, 10, 11], 7, [8, 9, 10, 11, 101, 467, 2204, 10431], 10431, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105034,"parameters":{"order":4,"coefficients":[1,2,3,4],"constant":1,"initial_values":[8,9,10,11],"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3,4],\"constant\":1,\"initial_values\":[8,9,10,11],\"order\":4,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0035","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":2,\"coefficients\":[2,4],\"constant\":2,\"initial_values\":[9,10],\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [9, 10], 6, [9, 10, 60, 262, 1170, 5206, 23166], 23166, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(2, [9, 10], 6, [9, 10, 60, 262, 1170, 5206, 23166], 23166, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105035,"parameters":{"order":2,"coefficients":[2,4],"constant":2,"initial_values":[9,10],"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4],\"constant\":2,\"initial_values\":[9,10],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0036","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":3,\"coefficients\":[1,2,3],\"constant\":-2,\"initial_values\":[10,11,12],\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [10, 11, 12], 8, [10, 11, 12, 66, 231, 835, 3031, 10992, 39871], 39871, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(3, [10, 11, 12], 8, [10, 11, 12, 66, 231, 835, 3031, 10992, 39871], 39871, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105036,"parameters":{"order":3,"coefficients":[1,2,3],"constant":-2,"initial_values":[10,11,12],"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3],\"constant\":-2,\"initial_values\":[10,11,12],\"order\":3,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0037","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":-1,\"initial_values\":[11,12,13,14],\"target_n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [11, 12, 13, 14], 10, [11, 12, 13, 14, 259, 2231, 19483, 170313, 1488843, 13015015, 113773395], 113773395, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(4, [11, 12, 13, 14], 10, [11, 12, 13, 14, 259, 2231, 19483, 170313, 1488843, 13015015, 113773395], 113773395, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105037,"parameters":{"order":4,"coefficients":[2,4,6,8],"constant":-1,"initial_values":[11,12,13,14],"target_n":10,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6,8],\"constant\":-1,\"initial_values\":[11,12,13,14],\"order\":4,\"target_n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0038","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":0,\"initial_values\":[12,13],\"target_n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [12, 13], 5, [12, 13, 38, 89, 216, 521], 521, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(2, [12, 13], 5, [12, 13, 38, 89, 216, 521], 521, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105038,"parameters":{"order":2,"coefficients":[1,2],"constant":0,"initial_values":[12,13],"target_n":5,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2],\"constant\":0,\"initial_values\":[12,13],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0039","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":3,\"coefficients\":[2,4,6],\"constant\":1,\"initial_values\":[13,14,15],\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [13, 14, 15], 7, [13, 14, 15, 173, 1127, 7485, 49765, 330785], 330785, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(3, [13, 14, 15], 7, [13, 14, 15, 173, 1127, 7485, 49765, 330785], 330785, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105039,"parameters":{"order":3,"coefficients":[2,4,6],"constant":1,"initial_values":[13,14,15],"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6],\"constant\":1,\"initial_values\":[13,14,15],\"order\":3,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0040","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":4,\"coefficients\":[1,2,3,4],\"constant\":2,\"initial_values\":[14,15,16,17],\"target_n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [14, 15, 16, 17], 9, [14, 15, 16, 17, 162, 748, 3530, 16707, 79078, 374243], 374243, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(4, [14, 15, 16, 17], 9, [14, 15, 16, 17, 162, 748, 3530, 16707, 79078, 374243], 374243, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105040,"parameters":{"order":4,"coefficients":[1,2,3,4],"constant":2,"initial_values":[14,15,16,17],"target_n":9,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3,4],\"constant\":2,\"initial_values\":[14,15,16,17],\"order\":4,\"target_n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0041","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":2,\"coefficients\":[2,4],\"constant\":-2,\"initial_values\":[15,16],\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [15, 16], 8, [15, 16, 92, 398, 1774, 7890, 35106, 156202, 695018], 695018, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(2, [15, 16], 8, [15, 16, 92, 398, 1774, 7890, 35106, 156202, 695018], 695018, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105041,"parameters":{"order":2,"coefficients":[2,4],"constant":-2,"initial_values":[15,16],"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4],\"constant\":-2,\"initial_values\":[15,16],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0042","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":3,\"coefficients\":[1,2,3],\"constant\":-1,\"initial_values\":[16,17,18],\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [16, 17, 18], 6, [16, 17, 18, 103, 361, 1306, 4742], 4742, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(3, [16, 17, 18], 6, [16, 17, 18, 103, 361, 1306, 4742], 4742, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105042,"parameters":{"order":3,"coefficients":[1,2,3],"constant":-1,"initial_values":[16,17,18],"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3],\"constant\":-1,\"initial_values\":[16,17,18],\"order\":3,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0043","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":0,\"initial_values\":[17,18,19,20],\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [17, 18, 19, 20], 8, [17, 18, 19, 20, 380, 3272, 28574, 249784, 2183564], 2183564, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(4, [17, 18, 19, 20], 8, [17, 18, 19, 20, 380, 3272, 28574, 249784, 2183564], 2183564, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105043,"parameters":{"order":4,"coefficients":[2,4,6,8],"constant":0,"initial_values":[17,18,19,20],"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6,8],\"constant\":0,\"initial_values\":[17,18,19,20],\"order\":4,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0044","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":1,\"initial_values\":[18,19],\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [18, 19], 7, [18, 19, 57, 134, 326, 787, 1901, 4590], 4590, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(2, [18, 19], 7, [18, 19, 57, 134, 326, 787, 1901, 4590], 4590, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105044,"parameters":{"order":2,"coefficients":[1,2],"constant":1,"initial_values":[18,19],"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2],\"constant\":1,\"initial_values\":[18,19],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0045","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":3,\"coefficients\":[2,4,6],\"constant\":2,\"initial_values\":[19,20,21],\"target_n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [19, 20, 21], 9, [19, 20, 21, 246, 1602, 10640, 70742, 470218, 3125558, 20775706], 20775706, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(3, [19, 20, 21], 9, [19, 20, 21, 246, 1602, 10640, 70742, 470218, 3125558, 20775706], 20775706, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105045,"parameters":{"order":3,"coefficients":[2,4,6],"constant":2,"initial_values":[19,20,21],"target_n":9,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6],\"constant\":2,\"initial_values\":[19,20,21],\"order\":3,\"target_n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0046","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":4,\"coefficients\":[1,2,3,4],\"constant\":-2,\"initial_values\":[20,21,22,23],\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [20, 21, 22, 23], 7, [20, 21, 22, 23, 218, 1004, 4736, 22413], 22413, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(4, [20, 21, 22, 23], 7, [20, 21, 22, 23, 218, 1004, 4736, 22413], 22413, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105046,"parameters":{"order":4,"coefficients":[1,2,3,4],"constant":-2,"initial_values":[20,21,22,23],"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3,4],\"constant\":-2,\"initial_values\":[20,21,22,23],\"order\":4,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0047","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":2,\"coefficients\":[2,4],\"constant\":-1,\"initial_values\":[21,22],\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [21, 22], 6, [21, 22, 129, 559, 2493, 11089, 49341], 49341, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(2, [21, 22], 6, [21, 22, 129, 559, 2493, 11089, 49341], 49341, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105047,"parameters":{"order":2,"coefficients":[2,4],"constant":-1,"initial_values":[21,22],"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4],\"constant\":-1,\"initial_values\":[21,22],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0048","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":3,\"coefficients\":[1,2,3],\"constant\":0,\"initial_values\":[22,23,24],\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [22, 23, 24], 8, [22, 23, 24, 140, 491, 1777, 6453, 23404, 84895], 84895, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(3, [22, 23, 24], 8, [22, 23, 24, 140, 491, 1777, 6453, 23404, 84895], 84895, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105048,"parameters":{"order":3,"coefficients":[1,2,3],"constant":0,"initial_values":[22,23,24],"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3],\"constant\":0,\"initial_values\":[22,23,24],\"order\":3,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0049","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":1,\"initial_values\":[23,24,25,26],\"target_n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [23, 24, 25, 26], 10, [23, 24, 25, 26, 501, 4313, 37665, 329255, 2878285, 25161097, 219950837], 219950837, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(4, [23, 24, 25, 26], 10, [23, 24, 25, 26, 501, 4313, 37665, 329255, 2878285, 25161097, 219950837], 219950837, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105049,"parameters":{"order":4,"coefficients":[2,4,6,8],"constant":1,"initial_values":[23,24,25,26],"target_n":10,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6,8],\"constant\":1,\"initial_values\":[23,24,25,26],\"order\":4,\"target_n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0050","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-forward-generation；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":2,\"initial_values\":[24,25],\"target_n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [24, 25], 5, [24, 25, 76, 179, 436, 1053], 1053, exact_forward_application_only)","explanation_zh":"Wave6第四層核對=(2, [24, 25], 5, [24, 25, 76, 179, 436, 1053], 1053, exact_forward_application_only)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-forward-generation","seed":202611105050,"parameters":{"order":2,"coefficients":[1,2],"constant":2,"initial_values":[24,25],"target_n":5,"domain":"Exact_total_linear_recurrence","operation":"recurrence-forward-generation","semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2],\"constant\":2,\"initial_values\":[24,25],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0051","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":-2,\"initial_values\":[0,1],\"target_n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 5, -11, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(2, 2, 5, -11, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105051,"parameters":{"order":2,"coefficients":[1,2],"constant":-2,"initial_values":[0,1],"target_n":5,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2],\"constant\":-2,\"initial_values\":[0,1],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0052","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":3,\"coefficients\":[2,4,6],\"constant\":-1,\"initial_values\":[1,2,3],\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 7, 51919, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(3, 3, 7, 51919, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105052,"parameters":{"order":3,"coefficients":[2,4,6],"constant":-1,"initial_values":[1,2,3],"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6],\"constant\":-1,\"initial_values\":[1,2,3],\"order\":3,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0053","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":4,\"coefficients\":[1,2,3,4],\"constant\":0,\"initial_values\":[2,3,4,5],\"target_n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 9, 93071, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(4, 4, 9, 93071, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105053,"parameters":{"order":4,"coefficients":[1,2,3,4],"constant":0,"initial_values":[2,3,4,5],"target_n":9,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3,4],\"constant\":0,\"initial_values\":[2,3,4,5],\"order\":4,\"target_n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0054","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":2,\"coefficients\":[2,4],\"constant\":1,\"initial_values\":[3,4],\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 8, 176819, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(2, 2, 8, 176819, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105054,"parameters":{"order":2,"coefficients":[2,4],"constant":1,"initial_values":[3,4],"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4],\"constant\":1,\"initial_values\":[3,4],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0055","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":3,\"coefficients\":[1,2,3],\"constant\":2,\"initial_values\":[4,5,6],\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 6, 1595, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(3, 3, 6, 1595, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105055,"parameters":{"order":3,"coefficients":[1,2,3],"constant":2,"initial_values":[4,5,6],"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3],\"constant\":2,\"initial_values\":[4,5,6],\"order\":3,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0056","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":-2,\"initial_values\":[5,6,7,8],\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 8, 794122, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(4, 4, 8, 794122, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105056,"parameters":{"order":4,"coefficients":[2,4,6,8],"constant":-2,"initial_values":[5,6,7,8],"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6,8],\"constant\":-2,\"initial_values\":[5,6,7,8],\"order\":4,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0057","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":-1,\"initial_values\":[6,7],\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 7, 1484, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(2, 2, 7, 1484, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105057,"parameters":{"order":2,"coefficients":[1,2],"constant":-1,"initial_values":[6,7],"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2],\"constant\":-1,\"initial_values\":[6,7],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0058","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":3,\"coefficients\":[2,4,6],\"constant\":0,\"initial_values\":[7,8,9],\"target_n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 9, 8454528, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(3, 3, 9, 8454528, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105058,"parameters":{"order":3,"coefficients":[2,4,6],"constant":0,"initial_values":[7,8,9],"target_n":9,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6],\"constant\":0,\"initial_values\":[7,8,9],\"order\":3,\"target_n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0059","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":4,\"coefficients\":[1,2,3,4],\"constant\":1,\"initial_values\":[8,9,10,11],\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 7, 10431, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(4, 4, 7, 10431, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105059,"parameters":{"order":4,"coefficients":[1,2,3,4],"constant":1,"initial_values":[8,9,10,11],"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3,4],\"constant\":1,\"initial_values\":[8,9,10,11],\"order\":4,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0060","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":2,\"coefficients\":[2,4],\"constant\":2,\"initial_values\":[9,10],\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 6, 23166, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(2, 2, 6, 23166, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105060,"parameters":{"order":2,"coefficients":[2,4],"constant":2,"initial_values":[9,10],"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4],\"constant\":2,\"initial_values\":[9,10],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0061","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":3,\"coefficients\":[1,2,3],\"constant\":-2,\"initial_values\":[10,11,12],\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 8, 39871, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(3, 3, 8, 39871, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105061,"parameters":{"order":3,"coefficients":[1,2,3],"constant":-2,"initial_values":[10,11,12],"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3],\"constant\":-2,\"initial_values\":[10,11,12],\"order\":3,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0062","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":-1,\"initial_values\":[11,12,13,14],\"target_n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 10, 113773395, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(4, 4, 10, 113773395, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105062,"parameters":{"order":4,"coefficients":[2,4,6,8],"constant":-1,"initial_values":[11,12,13,14],"target_n":10,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6,8],\"constant\":-1,\"initial_values\":[11,12,13,14],\"order\":4,\"target_n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0063","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":0,\"initial_values\":[12,13],\"target_n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 5, 521, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(2, 2, 5, 521, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105063,"parameters":{"order":2,"coefficients":[1,2],"constant":0,"initial_values":[12,13],"target_n":5,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2],\"constant\":0,\"initial_values\":[12,13],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0064","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":3,\"coefficients\":[2,4,6],\"constant\":1,\"initial_values\":[13,14,15],\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 7, 330785, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(3, 3, 7, 330785, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105064,"parameters":{"order":3,"coefficients":[2,4,6],"constant":1,"initial_values":[13,14,15],"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6],\"constant\":1,\"initial_values\":[13,14,15],\"order\":3,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0065","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":4,\"coefficients\":[1,2,3,4],\"constant\":2,\"initial_values\":[14,15,16,17],\"target_n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 9, 374243, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(4, 4, 9, 374243, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105065,"parameters":{"order":4,"coefficients":[1,2,3,4],"constant":2,"initial_values":[14,15,16,17],"target_n":9,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3,4],\"constant\":2,\"initial_values\":[14,15,16,17],\"order\":4,\"target_n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0066","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":2,\"coefficients\":[2,4],\"constant\":-2,\"initial_values\":[15,16],\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 8, 695018, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(2, 2, 8, 695018, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105066,"parameters":{"order":2,"coefficients":[2,4],"constant":-2,"initial_values":[15,16],"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4],\"constant\":-2,\"initial_values\":[15,16],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0067","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":3,\"coefficients\":[1,2,3],\"constant\":-1,\"initial_values\":[16,17,18],\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 6, 4742, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(3, 3, 6, 4742, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105067,"parameters":{"order":3,"coefficients":[1,2,3],"constant":-1,"initial_values":[16,17,18],"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3],\"constant\":-1,\"initial_values\":[16,17,18],\"order\":3,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0068","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":0,\"initial_values\":[17,18,19,20],\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 8, 2183564, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(4, 4, 8, 2183564, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105068,"parameters":{"order":4,"coefficients":[2,4,6,8],"constant":0,"initial_values":[17,18,19,20],"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6,8],\"constant\":0,\"initial_values\":[17,18,19,20],\"order\":4,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0069","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":1,\"initial_values\":[18,19],\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 7, 4590, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(2, 2, 7, 4590, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105069,"parameters":{"order":2,"coefficients":[1,2],"constant":1,"initial_values":[18,19],"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2],\"constant\":1,\"initial_values\":[18,19],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0070","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":3,\"coefficients\":[2,4,6],\"constant\":2,\"initial_values\":[19,20,21],\"target_n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 9, 20775706, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(3, 3, 9, 20775706, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105070,"parameters":{"order":3,"coefficients":[2,4,6],"constant":2,"initial_values":[19,20,21],"target_n":9,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6],\"constant\":2,\"initial_values\":[19,20,21],\"order\":3,\"target_n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0071","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":4,\"coefficients\":[1,2,3,4],\"constant\":-2,\"initial_values\":[20,21,22,23],\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 7, 22413, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(4, 4, 7, 22413, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105071,"parameters":{"order":4,"coefficients":[1,2,3,4],"constant":-2,"initial_values":[20,21,22,23],"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3,4],\"constant\":-2,\"initial_values\":[20,21,22,23],\"order\":4,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0072","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":2,\"coefficients\":[2,4],\"constant\":-1,\"initial_values\":[21,22],\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 6, 49341, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(2, 2, 6, 49341, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105072,"parameters":{"order":2,"coefficients":[2,4],"constant":-1,"initial_values":[21,22],"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4],\"constant\":-1,\"initial_values\":[21,22],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0073","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":3,\"coefficients\":[1,2,3],\"constant\":0,\"initial_values\":[22,23,24],\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 8, 84895, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(3, 3, 8, 84895, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105073,"parameters":{"order":3,"coefficients":[1,2,3],"constant":0,"initial_values":[22,23,24],"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3],\"constant\":0,\"initial_values\":[22,23,24],\"order\":3,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0074","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":1,\"initial_values\":[23,24,25,26],\"target_n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 10, 219950837, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(4, 4, 10, 219950837, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105074,"parameters":{"order":4,"coefficients":[2,4,6,8],"constant":1,"initial_values":[23,24,25,26],"target_n":10,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6,8],\"constant\":1,\"initial_values\":[23,24,25,26],\"order\":4,\"target_n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0075","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-initial-value-uniqueness；資料={\"order\":2,\"coefficients\":[1,2],\"constant\":2,\"initial_values\":[24,25],\"target_n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 5, 1053, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation_zh":"Wave6第四層核對=(2, 2, 5, 1053, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105075,"parameters":{"order":2,"coefficients":[1,2],"constant":2,"initial_values":[24,25],"target_n":5,"domain":"Exact_total_linear_recurrence","operation":"recurrence-initial-value-uniqueness","semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2],\"constant\":2,\"initial_values\":[24,25],\"order\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0076","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[1,2],\"constant\":0,\"first_initial\":0,\"second_left\":1,\"second_right\":2,\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 2, [0, 1, 2, 5, 12, 29, 70], [0, 2, 4, 10, 24, 58, 140], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(0, 1, 2, [0, 1, 2, 5, 12, 29, 70], [0, 2, 4, 10, 24, 58, 140], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105076,"parameters":{"coefficients":[1,2],"constant":0,"first_initial":0,"second_left":1,"second_right":2,"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,2],\"constant\":0,\"first_initial\":0,\"second_left\":1,\"second_right\":2,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0077","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[2,3],\"constant\":1,\"first_initial\":1,\"second_left\":2,\"second_right\":3,\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 3, [1, 2, 9, 32, 115, 410, 1461, 5204], [1, 3, 12, 43, 154, 549, 1956, 6967], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(1, 2, 3, [1, 2, 9, 32, 115, 410, 1461, 5204], [1, 3, 12, 43, 154, 549, 1956, 6967], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105077,"parameters":{"coefficients":[2,3],"constant":1,"first_initial":1,"second_left":2,"second_right":3,"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,3],\"constant\":1,\"first_initial\":1,\"second_left\":2,\"second_right\":3,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0078","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[3,4],\"constant\":2,\"first_initial\":2,\"second_left\":3,\"second_right\":4,\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 4, [2, 3, 20, 91, 426, 1979, 9196, 42723, 198482], [2, 4, 24, 110, 514, 2388, 11096, 51550, 239490], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(2, 3, 4, [2, 3, 20, 91, 426, 1979, 9196, 42723, 198482], [2, 4, 24, 110, 514, 2388, 11096, 51550, 239490], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105078,"parameters":{"coefficients":[3,4],"constant":2,"first_initial":2,"second_left":3,"second_right":4,"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,4],\"constant\":2,\"first_initial\":2,\"second_left\":3,\"second_right\":4,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0079","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[1,5],\"constant\":3,\"first_initial\":3,\"second_left\":4,\"second_right\":5,\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 5, [3, 4, 26, 137, 714, 3710, 19267], [3, 5, 31, 163, 849, 4411, 22907], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(3, 4, 5, [3, 4, 26, 137, 714, 3710, 19267], [3, 5, 31, 163, 849, 4411, 22907], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105079,"parameters":{"coefficients":[1,5],"constant":3,"first_initial":3,"second_left":4,"second_right":5,"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,5],\"constant\":3,\"first_initial\":3,\"second_left\":4,\"second_right\":5,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0080","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[2,6],\"constant\":0,\"first_initial\":4,\"second_left\":5,\"second_right\":6,\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 5, 6, [4, 5, 38, 238, 1504, 9500, 60008, 379048], [4, 6, 44, 276, 1744, 11016, 69584, 439536], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(4, 5, 6, [4, 5, 38, 238, 1504, 9500, 60008, 379048], [4, 6, 44, 276, 1744, 11016, 69584, 439536], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105080,"parameters":{"coefficients":[2,6],"constant":0,"first_initial":4,"second_left":5,"second_right":6,"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,6],\"constant\":0,\"first_initial\":4,\"second_left\":5,\"second_right\":6,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0081","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[3,2],\"constant\":1,\"first_initial\":5,\"second_left\":6,\"second_right\":7,\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 7, [5, 6, 28, 75, 235, 696, 2098, 6285, 18865], [5, 7, 30, 82, 255, 757, 2280, 6832, 20505], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(5, 6, 7, [5, 6, 28, 75, 235, 696, 2098, 6285, 18865], [5, 7, 30, 82, 255, 757, 2280, 6832, 20505], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105081,"parameters":{"coefficients":[3,2],"constant":1,"first_initial":5,"second_left":6,"second_right":7,"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,2],\"constant\":1,\"first_initial\":5,\"second_left\":6,\"second_right\":7,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0082","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[1,3],\"constant\":2,\"first_initial\":6,\"second_left\":7,\"second_right\":8,\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 7, 8, [6, 7, 29, 96, 319, 1055, 3486], [6, 8, 32, 106, 352, 1164, 3846], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(6, 7, 8, [6, 7, 29, 96, 319, 1055, 3486], [6, 8, 32, 106, 352, 1164, 3846], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105082,"parameters":{"coefficients":[1,3],"constant":2,"first_initial":6,"second_left":7,"second_right":8,"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,3],\"constant\":2,\"first_initial\":6,\"second_left\":7,\"second_right\":8,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0083","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[2,4],\"constant\":3,\"first_initial\":7,\"second_left\":8,\"second_right\":9,\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 8, 9, [7, 8, 49, 215, 961, 4277, 19033, 84689], [7, 9, 53, 233, 1041, 4633, 20617, 91737], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(7, 8, 9, [7, 8, 49, 215, 961, 4277, 19033, 84689], [7, 9, 53, 233, 1041, 4633, 20617, 91737], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105083,"parameters":{"coefficients":[2,4],"constant":3,"first_initial":7,"second_left":8,"second_right":9,"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,4],\"constant\":3,\"first_initial\":7,\"second_left\":8,\"second_right\":9,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0084","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[3,5],\"constant\":0,\"first_initial\":8,\"second_left\":9,\"second_right\":10,\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 9, 10, [8, 9, 69, 372, 2067, 11451, 63456, 351633, 1948533], [8, 10, 74, 400, 2222, 12310, 68216, 378010, 2094698], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(8, 9, 10, [8, 9, 69, 372, 2067, 11451, 63456, 351633, 1948533], [8, 10, 74, 400, 2222, 12310, 68216, 378010, 2094698], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105084,"parameters":{"coefficients":[3,5],"constant":0,"first_initial":8,"second_left":9,"second_right":10,"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,5],\"constant\":0,\"first_initial\":8,\"second_left\":9,\"second_right\":10,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0085","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[1,6],\"constant\":1,\"first_initial\":9,\"second_left\":10,\"second_right\":11,\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 10, 11, [9, 10, 70, 431, 2657, 16374, 100902], [9, 11, 76, 468, 2885, 17779, 109560], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(9, 10, 11, [9, 10, 70, 431, 2657, 16374, 100902], [9, 11, 76, 468, 2885, 17779, 109560], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105085,"parameters":{"coefficients":[1,6],"constant":1,"first_initial":9,"second_left":10,"second_right":11,"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,6],\"constant\":1,\"first_initial\":9,\"second_left\":10,\"second_right\":11,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0086","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[2,2],\"constant\":2,\"first_initial\":10,\"second_left\":11,\"second_right\":12,\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 11, 12, [10, 11, 44, 112, 314, 854, 2338, 6386], [10, 12, 46, 118, 330, 898, 2458, 6714], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(10, 11, 12, [10, 11, 44, 112, 314, 854, 2338, 6386], [10, 12, 46, 118, 330, 898, 2458, 6714], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105086,"parameters":{"coefficients":[2,2],"constant":2,"first_initial":10,"second_left":11,"second_right":12,"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,2],\"constant\":2,\"first_initial\":10,\"second_left\":11,\"second_right\":12,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0087","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[3,3],\"constant\":3,\"first_initial\":11,\"second_left\":12,\"second_right\":13,\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 12, 13, [11, 12, 72, 255, 984, 3720, 14115, 53508, 202872], [11, 13, 75, 267, 1029, 3891, 14763, 55965, 212187], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(11, 12, 13, [11, 12, 72, 255, 984, 3720, 14115, 53508, 202872], [11, 13, 75, 267, 1029, 3891, 14763, 55965, 212187], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105087,"parameters":{"coefficients":[3,3],"constant":3,"first_initial":11,"second_left":12,"second_right":13,"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,3],\"constant\":3,\"first_initial\":11,\"second_left\":12,\"second_right\":13,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0088","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[1,4],\"constant\":0,\"first_initial\":12,\"second_left\":13,\"second_right\":14,\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 13, 14, [12, 13, 64, 269, 1140, 4829, 20456], [12, 14, 68, 286, 1212, 5134, 21748], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(12, 13, 14, [12, 13, 64, 269, 1140, 4829, 20456], [12, 14, 68, 286, 1212, 5134, 21748], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105088,"parameters":{"coefficients":[1,4],"constant":0,"first_initial":12,"second_left":13,"second_right":14,"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,4],\"constant\":0,\"first_initial\":12,\"second_left\":13,\"second_right\":14,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0089","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[2,5],\"constant\":1,\"first_initial\":13,\"second_left\":14,\"second_right\":15,\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 14, 15, [13, 14, 97, 514, 2765, 14854, 79801, 428714], [13, 15, 102, 541, 2910, 15633, 83986, 451197], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(13, 14, 15, [13, 14, 97, 514, 2765, 14854, 79801, 428714], [13, 15, 102, 541, 2910, 15633, 83986, 451197], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105089,"parameters":{"coefficients":[2,5],"constant":1,"first_initial":13,"second_left":14,"second_right":15,"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,5],\"constant\":1,\"first_initial\":13,\"second_left\":14,\"second_right\":15,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0090","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[3,6],\"constant\":2,\"first_initial\":14,\"second_left\":15,\"second_right\":16,\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 15, 16, [14, 15, 134, 851, 5510, 35615, 230222, 1488179, 9619742], [14, 16, 140, 890, 5762, 37244, 240752, 1556246, 10059734], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(14, 15, 16, [14, 15, 134, 851, 5510, 35615, 230222, 1488179, 9619742], [14, 16, 140, 890, 5762, 37244, 240752, 1556246, 10059734], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105090,"parameters":{"coefficients":[3,6],"constant":2,"first_initial":14,"second_left":15,"second_right":16,"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,6],\"constant\":2,\"first_initial\":14,\"second_left\":15,\"second_right\":16,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0091","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[1,2],\"constant\":3,\"first_initial\":15,\"second_left\":16,\"second_right\":17,\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 16, 17, [15, 16, 50, 119, 291, 704, 1702], [15, 17, 52, 124, 303, 733, 1772], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(15, 16, 17, [15, 16, 50, 119, 291, 704, 1702], [15, 17, 52, 124, 303, 733, 1772], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105091,"parameters":{"coefficients":[1,2],"constant":3,"first_initial":15,"second_left":16,"second_right":17,"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,2],\"constant\":3,\"first_initial\":15,\"second_left\":16,\"second_right\":17,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0092","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[2,3],\"constant\":0,\"first_initial\":16,\"second_left\":17,\"second_right\":18,\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 17, 18, [16, 17, 83, 283, 1015, 3611, 12863, 45811], [16, 18, 86, 294, 1054, 3750, 13358, 47574], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(16, 17, 18, [16, 17, 83, 283, 1015, 3611, 12863, 45811], [16, 18, 86, 294, 1054, 3750, 13358, 47574], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105092,"parameters":{"coefficients":[2,3],"constant":0,"first_initial":16,"second_left":17,"second_right":18,"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,3],\"constant\":0,\"first_initial\":16,\"second_left\":17,\"second_right\":18,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0093","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[3,4],\"constant\":1,\"first_initial\":17,\"second_left\":18,\"second_right\":19,\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 18, 19, [17, 18, 124, 551, 2577, 11962, 55580, 258207, 1199569], [17, 19, 128, 570, 2665, 12371, 57480, 267034, 1240577], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(17, 18, 19, [17, 18, 124, 551, 2577, 11962, 55580, 258207, 1199569], [17, 19, 128, 570, 2665, 12371, 57480, 267034, 1240577], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105093,"parameters":{"coefficients":[3,4],"constant":1,"first_initial":17,"second_left":18,"second_right":19,"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,4],\"constant\":1,\"first_initial\":17,\"second_left\":18,\"second_right\":19,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0094","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[1,5],\"constant\":2,\"first_initial\":18,\"second_left\":19,\"second_right\":20,\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 19, 20, [18, 19, 115, 596, 3097, 16083, 83514], [18, 20, 120, 622, 3232, 16784, 87154], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(18, 19, 20, [18, 19, 115, 596, 3097, 16083, 83514], [18, 20, 120, 622, 3232, 16784, 87154], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105094,"parameters":{"coefficients":[1,5],"constant":2,"first_initial":18,"second_left":19,"second_right":20,"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,5],\"constant\":2,\"first_initial\":18,\"second_left\":19,\"second_right\":20,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0095","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[2,6],\"constant\":3,\"first_initial\":19,\"second_left\":20,\"second_right\":21,\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 20, 21, [19, 20, 161, 1009, 6379, 40295, 254531, 1607779], [19, 21, 167, 1047, 6619, 41811, 264107, 1668267], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(19, 20, 21, [19, 20, 161, 1009, 6379, 40295, 254531, 1607779], [19, 21, 167, 1047, 6619, 41811, 264107, 1668267], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105095,"parameters":{"coefficients":[2,6],"constant":3,"first_initial":19,"second_left":20,"second_right":21,"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,6],\"constant\":3,\"first_initial\":19,\"second_left\":20,\"second_right\":21,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0096","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[3,2],\"constant\":0,\"first_initial\":20,\"second_left\":21,\"second_right\":22,\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 21, 22, [20, 21, 102, 267, 840, 2481, 7482, 22407, 67260], [20, 22, 104, 274, 860, 2542, 7664, 22954, 68900], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(20, 21, 22, [20, 21, 102, 267, 840, 2481, 7482, 22407, 67260], [20, 22, 104, 274, 860, 2542, 7664, 22954, 68900], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105096,"parameters":{"coefficients":[3,2],"constant":0,"first_initial":20,"second_left":21,"second_right":22,"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,2],\"constant\":0,\"first_initial\":20,\"second_left\":21,\"second_right\":22,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0097","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[1,3],\"constant\":1,\"first_initial\":21,\"second_left\":22,\"second_right\":23,\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 22, 23, [21, 22, 88, 287, 950, 3138, 10365], [21, 23, 91, 297, 983, 3247, 10725], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(21, 22, 23, [21, 22, 88, 287, 950, 3138, 10365], [21, 23, 91, 297, 983, 3247, 10725], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105097,"parameters":{"coefficients":[1,3],"constant":1,"first_initial":21,"second_left":22,"second_right":23,"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,3],\"constant\":1,\"first_initial\":21,\"second_left\":22,\"second_right\":23,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0098","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[2,4],\"constant\":2,\"first_initial\":22,\"second_left\":23,\"second_right\":24,\"target_n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 23, 24, [22, 23, 138, 600, 2678, 11914, 53014, 235886], [22, 24, 142, 618, 2758, 12270, 54598, 242934], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(22, 23, 24, [22, 23, 138, 600, 2678, 11914, 53014, 235886], [22, 24, 142, 618, 2758, 12270, 54598, 242934], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105098,"parameters":{"coefficients":[2,4],"constant":2,"first_initial":22,"second_left":23,"second_right":24,"target_n":7,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,4],\"constant\":2,\"first_initial\":22,\"second_left\":23,\"second_right\":24,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0099","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[3,5],\"constant\":3,\"first_initial\":23,\"second_left\":24,\"second_right\":25,\"target_n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 24, 25, [23, 24, 192, 1035, 5754, 31878, 176655, 978912, 5424528], [23, 25, 197, 1063, 5909, 32737, 181415, 1005289, 5570693], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(23, 24, 25, [23, 24, 192, 1035, 5754, 31878, 176655, 978912, 5424528], [23, 25, 197, 1063, 5909, 32737, 181415, 1005289, 5570693], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105099,"parameters":{"coefficients":[3,5],"constant":3,"first_initial":23,"second_left":24,"second_right":25,"target_n":8,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,5],\"constant\":3,\"first_initial\":23,\"second_left\":24,\"second_right\":25,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r05-recurrence-relation-0100","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-recurrence-relation"],"title_zh":"遞迴關係","stem_zh":"initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={\"coefficients\":[1,6],\"constant\":0,\"first_initial\":24,\"second_left\":25,\"second_right\":26,\"target_n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 25, 26, [24, 25, 174, 1069, 6588, 40597, 250170], [24, 26, 180, 1106, 6816, 42002, 258828], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation_zh":"Wave6第四層核對=(24, 25, 26, [24, 25, 174, 1069, 6588, 40597, 250170], [24, 26, 180, 1106, 6816, 42002, 258828], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["遞迴關係"]},"variant":{"family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105100,"parameters":{"coefficients":[1,6],"constant":0,"first_initial":24,"second_left":25,"second_right":26,"target_n":6,"domain":"Exact_total_linear_recurrence","operation":"recurrence-missing-initial-boundary","semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,6],\"constant\":0,\"first_initial\":24,\"second_left\":25,\"second_right\":26,\"target_n\":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 orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}