{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r05-catalan-numbers-0001","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [1], 1, C_zero_equals_one)","explanation_zh":"Wave6第五層核對=(0, [1], 1, C_zero_equals_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115001,"parameters":{"n":0,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0002","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [1, 1], 1, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(1, [1, 1], 1, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115002,"parameters":{"n":1,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0003","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [1, 1, 2], 2, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(2, [1, 1, 2], 2, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115003,"parameters":{"n":2,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0004","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [1, 1, 2, 5], 5, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(3, [1, 1, 2, 5], 5, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115004,"parameters":{"n":3,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0005","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 1, 2, 5, 14], 14, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(4, [1, 1, 2, 5, 14], 14, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115005,"parameters":{"n":4,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0006","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [1, 1, 2, 5, 14, 42], 42, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(5, [1, 1, 2, 5, 14, 42], 42, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115006,"parameters":{"n":5,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0007","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 1, 2, 5, 14, 42, 132], 132, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(6, [1, 1, 2, 5, 14, 42, 132], 132, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115007,"parameters":{"n":6,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0008","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, [1, 1, 2, 5, 14, 42, 132, 429], 429, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(7, [1, 1, 2, 5, 14, 42, 132, 429], 429, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115008,"parameters":{"n":7,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0009","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [1, 1, 2, 5, 14, 42, 132, 429, 1430], 1430, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(8, [1, 1, 2, 5, 14, 42, 132, 429, 1430], 1430, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115009,"parameters":{"n":8,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0010","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862], 4862, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(9, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862], 4862, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115010,"parameters":{"n":9,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0011","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796], 16796, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(10, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796], 16796, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115011,"parameters":{"n":10,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0012","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786], 58786, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(11, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786], 58786, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115012,"parameters":{"n":11,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0013","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012], 208012, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(12, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012], 208012, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115013,"parameters":{"n":12,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0014","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900], 742900, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(13, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900], 742900, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115014,"parameters":{"n":13,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0015","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440], 2674440, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(14, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440], 2674440, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115015,"parameters":{"n":14,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0016","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845], 9694845, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(15, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845], 9694845, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115016,"parameters":{"n":15,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0017","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670], 35357670, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(16, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670], 35357670, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115017,"parameters":{"n":16,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0018","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790], 129644790, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(17, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790], 129644790, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115018,"parameters":{"n":17,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0019","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700], 477638700, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(18, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700], 477638700, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115019,"parameters":{"n":18,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0020","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190], 1767263190, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(19, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190], 1767263190, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115020,"parameters":{"n":19,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0021","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420], 6564120420, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(20, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420], 6564120420, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115021,"parameters":{"n":20,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0022","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020], 24466267020, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(21, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020], 24466267020, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115022,"parameters":{"n":21,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0023","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640], 91482563640, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(22, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640], 91482563640, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115023,"parameters":{"n":22,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0024","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650], 343059613650, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(23, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650], 343059613650, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115024,"parameters":{"n":23,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0025","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-generation；資料={\"n\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324], 1289904147324, C_next_sum_i_zero_to_n_minus_one)","explanation_zh":"Wave6第五層核對=(24, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324], 1289904147324, C_next_sum_i_zero_to_n_minus_one)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115025,"parameters":{"n":24,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-generation","semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0026","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, 0, 1, 1, 是)","explanation_zh":"Wave6第五層核對=(0, 1, 1, 0, 1, 1, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115026,"parameters":{"n":0,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0027","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 2, 0, 1, 1, 是)","explanation_zh":"Wave6第五層核對=(1, 2, 2, 0, 1, 1, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115027,"parameters":{"n":1,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0028","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 6, 3, 0, 2, 2, 是)","explanation_zh":"Wave6第五層核對=(2, 6, 3, 0, 2, 2, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115028,"parameters":{"n":2,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0029","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 20, 4, 0, 5, 5, 是)","explanation_zh":"Wave6第五層核對=(3, 20, 4, 0, 5, 5, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115029,"parameters":{"n":3,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0030","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 70, 5, 0, 14, 14, 是)","explanation_zh":"Wave6第五層核對=(4, 70, 5, 0, 14, 14, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115030,"parameters":{"n":4,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0031","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 252, 6, 0, 42, 42, 是)","explanation_zh":"Wave6第五層核對=(5, 252, 6, 0, 42, 42, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115031,"parameters":{"n":5,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0032","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 924, 7, 0, 132, 132, 是)","explanation_zh":"Wave6第五層核對=(6, 924, 7, 0, 132, 132, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115032,"parameters":{"n":6,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0033","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 3432, 8, 0, 429, 429, 是)","explanation_zh":"Wave6第五層核對=(7, 3432, 8, 0, 429, 429, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115033,"parameters":{"n":7,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0034","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 12870, 9, 0, 1430, 1430, 是)","explanation_zh":"Wave6第五層核對=(8, 12870, 9, 0, 1430, 1430, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115034,"parameters":{"n":8,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0035","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 48620, 10, 0, 4862, 4862, 是)","explanation_zh":"Wave6第五層核對=(9, 48620, 10, 0, 4862, 4862, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115035,"parameters":{"n":9,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0036","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 184756, 11, 0, 16796, 16796, 是)","explanation_zh":"Wave6第五層核對=(10, 184756, 11, 0, 16796, 16796, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115036,"parameters":{"n":10,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0037","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 705432, 12, 0, 58786, 58786, 是)","explanation_zh":"Wave6第五層核對=(11, 705432, 12, 0, 58786, 58786, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115037,"parameters":{"n":11,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0038","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2704156, 13, 0, 208012, 208012, 是)","explanation_zh":"Wave6第五層核對=(12, 2704156, 13, 0, 208012, 208012, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115038,"parameters":{"n":12,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0039","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 10400600, 14, 0, 742900, 742900, 是)","explanation_zh":"Wave6第五層核對=(13, 10400600, 14, 0, 742900, 742900, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115039,"parameters":{"n":13,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0040","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 40116600, 15, 0, 2674440, 2674440, 是)","explanation_zh":"Wave6第五層核對=(14, 40116600, 15, 0, 2674440, 2674440, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115040,"parameters":{"n":14,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0041","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 155117520, 16, 0, 9694845, 9694845, 是)","explanation_zh":"Wave6第五層核對=(15, 155117520, 16, 0, 9694845, 9694845, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115041,"parameters":{"n":15,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0042","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 601080390, 17, 0, 35357670, 35357670, 是)","explanation_zh":"Wave6第五層核對=(16, 601080390, 17, 0, 35357670, 35357670, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115042,"parameters":{"n":16,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0043","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 2333606220, 18, 0, 129644790, 129644790, 是)","explanation_zh":"Wave6第五層核對=(17, 2333606220, 18, 0, 129644790, 129644790, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115043,"parameters":{"n":17,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0044","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 9075135300, 19, 0, 477638700, 477638700, 是)","explanation_zh":"Wave6第五層核對=(18, 9075135300, 19, 0, 477638700, 477638700, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115044,"parameters":{"n":18,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0045","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 35345263800, 20, 0, 1767263190, 1767263190, 是)","explanation_zh":"Wave6第五層核對=(19, 35345263800, 20, 0, 1767263190, 1767263190, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115045,"parameters":{"n":19,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0046","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 137846528820, 21, 0, 6564120420, 6564120420, 是)","explanation_zh":"Wave6第五層核對=(20, 137846528820, 21, 0, 6564120420, 6564120420, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115046,"parameters":{"n":20,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0047","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 538257874440, 22, 0, 24466267020, 24466267020, 是)","explanation_zh":"Wave6第五層核對=(21, 538257874440, 22, 0, 24466267020, 24466267020, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115047,"parameters":{"n":21,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0048","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 2104098963720, 23, 0, 91482563640, 91482563640, 是)","explanation_zh":"Wave6第五層核對=(22, 2104098963720, 23, 0, 91482563640, 91482563640, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115048,"parameters":{"n":22,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0049","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 8233430727600, 24, 0, 343059613650, 343059613650, 是)","explanation_zh":"Wave6第五層核對=(23, 8233430727600, 24, 0, 343059613650, 343059613650, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115049,"parameters":{"n":23,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0050","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 32247603683100, 25, 0, 1289904147324, 1289904147324, 是)","explanation_zh":"Wave6第五層核對=(24, 32247603683100, 25, 0, 1289904147324, 1289904147324, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-closed-form","seed":202611115050,"parameters":{"n":24,"domain":"Exact_Catalan_sequence","operation":"catalan-closed-form","semantic_template":"qf-mod42-catalan-closed-form/{\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0051","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, 1, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(0, 1, 1, 1, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115051,"parameters":{"n":0,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0052","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 1, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(1, 1, 1, 1, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115052,"parameters":{"n":1,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0053","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 2, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(2, 2, 2, 2, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115053,"parameters":{"n":2,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0054","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 5, 5, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(3, 5, 5, 5, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115054,"parameters":{"n":3,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0055","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 14, 14, 14, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(4, 14, 14, 14, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115055,"parameters":{"n":4,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0056","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 42, 42, 42, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(5, 42, 42, 42, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115056,"parameters":{"n":5,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0057","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 132, 132, 132, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(6, 132, 132, 132, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115057,"parameters":{"n":6,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0058","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 429, 429, 429, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(7, 429, 429, 429, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115058,"parameters":{"n":7,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0059","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 1430, 1430, 1430, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(8, 1430, 1430, 1430, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115059,"parameters":{"n":8,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0060","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 4862, 4862, 4862, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(9, 4862, 4862, 4862, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115060,"parameters":{"n":9,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0061","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 16796, 16796, 16796, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(10, 16796, 16796, 16796, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115061,"parameters":{"n":10,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0062","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 58786, 58786, 58786, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(11, 58786, 58786, 58786, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115062,"parameters":{"n":11,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0063","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 208012, 208012, 208012, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(12, 208012, 208012, 208012, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115063,"parameters":{"n":12,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0064","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 742900, 742900, 742900, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(13, 742900, 742900, 742900, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115064,"parameters":{"n":13,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0065","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 2674440, 2674440, 2674440, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(14, 2674440, 2674440, 2674440, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115065,"parameters":{"n":14,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0066","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 9694845, 9694845, 9694845, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(15, 9694845, 9694845, 9694845, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115066,"parameters":{"n":15,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0067","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 35357670, 35357670, 35357670, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(16, 35357670, 35357670, 35357670, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115067,"parameters":{"n":16,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0068","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 129644790, 129644790, 129644790, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(17, 129644790, 129644790, 129644790, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115068,"parameters":{"n":17,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0069","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 477638700, 477638700, 477638700, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(18, 477638700, 477638700, 477638700, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115069,"parameters":{"n":18,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0070","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 1767263190, 1767263190, 1767263190, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(19, 1767263190, 1767263190, 1767263190, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115070,"parameters":{"n":19,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0071","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 6564120420, 6564120420, 6564120420, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(20, 6564120420, 6564120420, 6564120420, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115071,"parameters":{"n":20,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0072","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 24466267020, 24466267020, 24466267020, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(21, 24466267020, 24466267020, 24466267020, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115072,"parameters":{"n":21,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0073","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 91482563640, 91482563640, 91482563640, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(22, 91482563640, 91482563640, 91482563640, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115073,"parameters":{"n":22,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0074","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 343059613650, 343059613650, 343059613650, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(23, 343059613650, 343059613650, 343059613650, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115074,"parameters":{"n":23,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0075","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-recurrence-closed-cross-check；資料={\"n\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 1289904147324, 1289904147324, 1289904147324, 是, C_zero_index_convention_preserved)","explanation_zh":"Wave6第五層核對=(24, 1289904147324, 1289904147324, 1289904147324, 是, C_zero_index_convention_preserved)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115075,"parameters":{"n":24,"domain":"Exact_Catalan_sequence","operation":"catalan-recurrence-closed-cross-check","semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0076","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-1, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115076,"parameters":{"n":-1,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0077","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-2, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115077,"parameters":{"n":-2,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0078","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-3, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115078,"parameters":{"n":-3,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0079","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-4, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115079,"parameters":{"n":-4,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0080","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-5, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115080,"parameters":{"n":-5,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0081","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-6, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115081,"parameters":{"n":-6,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0082","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-7, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115082,"parameters":{"n":-7,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0083","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-8, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115083,"parameters":{"n":-8,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0084","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-9, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115084,"parameters":{"n":-9,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r05-catalan-numbers-0086","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-11, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115086,"parameters":{"n":-11,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0087","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-12, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115087,"parameters":{"n":-12,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0088","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-13, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-13, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115088,"parameters":{"n":-13,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0089","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-14, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-14, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115089,"parameters":{"n":-14,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0090","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-15, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-15, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115090,"parameters":{"n":-15,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0091","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-16, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-16, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115091,"parameters":{"n":-16,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0092","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-17, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-17, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115092,"parameters":{"n":-17,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0093","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-18, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-18, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115093,"parameters":{"n":-18,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0094","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-19, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-19, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115094,"parameters":{"n":-19,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0095","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-20, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-20, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115095,"parameters":{"n":-20,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0096","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-21, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-21, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115096,"parameters":{"n":-21,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0097","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-22, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-22, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115097,"parameters":{"n":-22,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0098","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-23, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-23, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115098,"parameters":{"n":-23,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0099","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-24, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-24, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115099,"parameters":{"n":-24,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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-20261111-r05-catalan-numbers-0100","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","language":"zh-Hant","source_mko_ids":["mko-catalan-numbers"],"title_zh":"Catalan數","stem_zh":"C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-25, 否, null, Catalan_domain_is_natural_numbers_including_zero)","explanation_zh":"Wave6第五層核對=(-25, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Catalan數"]},"variant":{"family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115100,"parameters":{"n":-25,"domain":"Exact_Catalan_sequence","operation":"catalan-negative-index-boundary","semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA 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"}}