{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r06-finite-field-0001","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":2}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 0], 2, additive_order_of_one)","explanation_zh":"Wave4收束核對=(2, [0, 1, 0], 2, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316001,"parameters":{"p":2,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0002","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":3}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0], 3, additive_order_of_one)","explanation_zh":"Wave4收束核對=(3, [0, 1, 2, 0], 3, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316002,"parameters":{"p":3,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0003","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":5}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4, 0], 5, additive_order_of_one)","explanation_zh":"Wave4收束核對=(5, [0, 1, 2, 3, 4, 0], 5, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316003,"parameters":{"p":5,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0004","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":7}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(7, [0, 1, 2, 3, 4, 5, 6, 0], 7, additive_order_of_one)","explanation_zh":"Wave4收束核對=(7, [0, 1, 2, 3, 4, 5, 6, 0], 7, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316004,"parameters":{"p":7,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0005","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":11}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], 11, additive_order_of_one)","explanation_zh":"Wave4收束核對=(11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], 11, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316005,"parameters":{"p":11,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0006","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":13}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], 13, additive_order_of_one)","explanation_zh":"Wave4收束核對=(13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], 13, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316006,"parameters":{"p":13,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0007","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":17}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(17, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], 17, additive_order_of_one)","explanation_zh":"Wave4收束核對=(17, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], 17, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316007,"parameters":{"p":17,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0008","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":19}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(19, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], 19, additive_order_of_one)","explanation_zh":"Wave4收束核對=(19, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], 19, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316008,"parameters":{"p":19,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0009","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":23}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(23, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], 23, additive_order_of_one)","explanation_zh":"Wave4收束核對=(23, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], 23, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316009,"parameters":{"p":23,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0010","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":29}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(29, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 0], 29, additive_order_of_one)","explanation_zh":"Wave4收束核對=(29, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 0], 29, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316010,"parameters":{"p":29,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0011","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":31}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(31, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 0], 31, additive_order_of_one)","explanation_zh":"Wave4收束核對=(31, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 0], 31, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316011,"parameters":{"p":31,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0012","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":37}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(37, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 0], 37, additive_order_of_one)","explanation_zh":"Wave4收束核對=(37, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 0], 37, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316012,"parameters":{"p":37,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0013","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":41}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(41, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 0], 41, additive_order_of_one)","explanation_zh":"Wave4收束核對=(41, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 0], 41, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316013,"parameters":{"p":41,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0014","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":43}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(43, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 0], 43, additive_order_of_one)","explanation_zh":"Wave4收束核對=(43, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 0], 43, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316014,"parameters":{"p":43,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0015","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":47}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(47, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 0], 47, additive_order_of_one)","explanation_zh":"Wave4收束核對=(47, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 0], 47, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316015,"parameters":{"p":47,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0016","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":53}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(53, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 0], 53, additive_order_of_one)","explanation_zh":"Wave4收束核對=(53, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 0], 53, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316016,"parameters":{"p":53,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":53}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0017","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":59}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(59, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 0], 59, additive_order_of_one)","explanation_zh":"Wave4收束核對=(59, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 0], 59, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316017,"parameters":{"p":59,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":59}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0018","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":61}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(61, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 0], 61, additive_order_of_one)","explanation_zh":"Wave4收束核對=(61, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 0], 61, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316018,"parameters":{"p":61,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":61}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0019","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":67}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(67, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 0], 67, additive_order_of_one)","explanation_zh":"Wave4收束核對=(67, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 0], 67, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316019,"parameters":{"p":67,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":67}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0020","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":71}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(71, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 0], 71, additive_order_of_one)","explanation_zh":"Wave4收束核對=(71, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 0], 71, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316020,"parameters":{"p":71,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":71}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0021","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":73}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(73, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 0], 73, additive_order_of_one)","explanation_zh":"Wave4收束核對=(73, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 0], 73, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316021,"parameters":{"p":73,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":73}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0022","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":79}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(79, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 0], 79, additive_order_of_one)","explanation_zh":"Wave4收束核對=(79, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 0], 79, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316022,"parameters":{"p":79,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":79}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0023","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":83}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(83, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 0], 83, additive_order_of_one)","explanation_zh":"Wave4收束核對=(83, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 0], 83, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316023,"parameters":{"p":83,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":83}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0024","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":89}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(89, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 0], 89, additive_order_of_one)","explanation_zh":"Wave4收束核對=(89, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 0], 89, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316024,"parameters":{"p":89,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":89}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0025","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":97}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。","answer":{"kind":"expression","value":"(97, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 0], 97, additive_order_of_one)","explanation_zh":"Wave4收束核對=(97, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 0], 97, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-characteristic","seed":202610316025,"parameters":{"p":97,"domain":"Exact_prime_field_Zp","operation":"fielddeep-characteristic","semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":97}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0026","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":2,\"element\":1}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, [1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(1, [1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316026,"parameters":{"p":2,"element":1,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":1,\"p\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0027","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":3,\"element\":2}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, [2, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(2, [2, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316027,"parameters":{"p":3,"element":2,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":2,\"p\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0028","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":5,\"element\":3}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(3, [3, 4, 2, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(3, [3, 4, 2, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316028,"parameters":{"p":5,"element":3,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":3,\"p\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0029","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":7,\"element\":4}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(4, [4, 2, 1, 4, 2, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(4, [4, 2, 1, 4, 2, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316029,"parameters":{"p":7,"element":4,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":4,\"p\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0030","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":11,\"element\":5}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, [5, 3, 4, 9, 1, 5, 3, 4, 9, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(5, [5, 3, 4, 9, 1, 5, 3, 4, 9, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316030,"parameters":{"p":11,"element":5,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":5,\"p\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0031","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":13,\"element\":6}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(6, [6, 10, 8, 9, 2, 12, 7, 3, 5, 4, 11, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(6, [6, 10, 8, 9, 2, 12, 7, 3, 5, 4, 11, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316031,"parameters":{"p":13,"element":6,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":6,\"p\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0032","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":17,\"element\":7}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(7, [7, 15, 3, 4, 11, 9, 12, 16, 10, 2, 14, 13, 6, 8, 5, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(7, [7, 15, 3, 4, 11, 9, 12, 16, 10, 2, 14, 13, 6, 8, 5, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316032,"parameters":{"p":17,"element":7,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":7,\"p\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0033","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":19,\"element\":8}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(8, [8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(8, [8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316033,"parameters":{"p":19,"element":8,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":8,\"p\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0034","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":23,\"element\":9}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(9, [9, 12, 16, 6, 8, 3, 4, 13, 2, 18, 1, 9, 12, 16, 6, 8, 3, 4, 13, 2, 18, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(9, [9, 12, 16, 6, 8, 3, 4, 13, 2, 18, 1, 9, 12, 16, 6, 8, 3, 4, 13, 2, 18, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316034,"parameters":{"p":23,"element":9,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":9,\"p\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0035","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":29,\"element\":10}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(10, [10, 13, 14, 24, 8, 22, 17, 25, 18, 6, 2, 20, 26, 28, 19, 16, 15, 5, 21, 7, 12, 4, 11, 23, 27, 9, 3, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(10, [10, 13, 14, 24, 8, 22, 17, 25, 18, 6, 2, 20, 26, 28, 19, 16, 15, 5, 21, 7, 12, 4, 11, 23, 27, 9, 3, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316035,"parameters":{"p":29,"element":10,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":10,\"p\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0036","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":31,\"element\":11}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(11, [11, 28, 29, 9, 6, 4, 13, 19, 23, 5, 24, 16, 21, 14, 30, 20, 3, 2, 22, 25, 27, 18, 12, 8, 26, 7, 15, 10, 17, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(11, [11, 28, 29, 9, 6, 4, 13, 19, 23, 5, 24, 16, 21, 14, 30, 20, 3, 2, 22, 25, 27, 18, 12, 8, 26, 7, 15, 10, 17, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316036,"parameters":{"p":31,"element":11,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":11,\"p\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0037","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":37,\"element\":12}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(12, [12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(12, [12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316037,"parameters":{"p":37,"element":12,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":12,\"p\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0038","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":41,\"element\":13}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(13, [13, 5, 24, 25, 38, 2, 26, 10, 7, 9, 35, 4, 11, 20, 14, 18, 29, 8, 22, 40, 28, 36, 17, 16, 3, 39, 15, 31, 34, 32, 6, 37, 30, 21, 27, 23, 12, 33, 19, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(13, [13, 5, 24, 25, 38, 2, 26, 10, 7, 9, 35, 4, 11, 20, 14, 18, 29, 8, 22, 40, 28, 36, 17, 16, 3, 39, 15, 31, 34, 32, 6, 37, 30, 21, 27, 23, 12, 33, 19, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316038,"parameters":{"p":41,"element":13,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":13,\"p\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0039","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":43,\"element\":14}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(14, [14, 24, 35, 17, 23, 21, 36, 31, 4, 13, 10, 11, 25, 6, 41, 15, 38, 16, 9, 40, 1, 14, 24, 35, 17, 23, 21, 36, 31, 4, 13, 10, 11, 25, 6, 41, 15, 38, 16, 9, 40, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(14, [14, 24, 35, 17, 23, 21, 36, 31, 4, 13, 10, 11, 25, 6, 41, 15, 38, 16, 9, 40, 1, 14, 24, 35, 17, 23, 21, 36, 31, 4, 13, 10, 11, 25, 6, 41, 15, 38, 16, 9, 40, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316039,"parameters":{"p":43,"element":14,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":14,\"p\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0040","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":47,\"element\":15}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(15, [15, 37, 38, 6, 43, 34, 40, 36, 23, 16, 5, 28, 44, 2, 30, 27, 29, 12, 39, 21, 33, 25, 46, 32, 10, 9, 41, 4, 13, 7, 11, 24, 31, 42, 19, 3, 45, 17, 20, 18, 35, 8, 26, 14, 22, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(15, [15, 37, 38, 6, 43, 34, 40, 36, 23, 16, 5, 28, 44, 2, 30, 27, 29, 12, 39, 21, 33, 25, 46, 32, 10, 9, 41, 4, 13, 7, 11, 24, 31, 42, 19, 3, 45, 17, 20, 18, 35, 8, 26, 14, 22, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316040,"parameters":{"p":47,"element":15,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":15,\"p\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0041","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":53,\"element\":16}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(16, [16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(16, [16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316041,"parameters":{"p":53,"element":16,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":16,\"p\":53}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0042","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":59,\"element\":17}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(17, [17, 53, 16, 36, 22, 20, 45, 57, 25, 12, 27, 46, 15, 19, 28, 4, 9, 35, 5, 26, 29, 21, 3, 51, 41, 48, 49, 7, 1, 17, 53, 16, 36, 22, 20, 45, 57, 25, 12, 27, 46, 15, 19, 28, 4, 9, 35, 5, 26, 29, 21, 3, 51, 41, 48, 49, 7, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(17, [17, 53, 16, 36, 22, 20, 45, 57, 25, 12, 27, 46, 15, 19, 28, 4, 9, 35, 5, 26, 29, 21, 3, 51, 41, 48, 49, 7, 1, 17, 53, 16, 36, 22, 20, 45, 57, 25, 12, 27, 46, 15, 19, 28, 4, 9, 35, 5, 26, 29, 21, 3, 51, 41, 48, 49, 7, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316042,"parameters":{"p":59,"element":17,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":17,\"p\":59}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0043","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":61,\"element\":18}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(18, [18, 19, 37, 56, 32, 27, 59, 25, 23, 48, 10, 58, 7, 4, 11, 15, 26, 41, 6, 47, 53, 39, 31, 9, 40, 49, 28, 16, 44, 60, 43, 42, 24, 5, 29, 34, 2, 36, 38, 13, 51, 3, 54, 57, 50, 46, 35, 20, 55, 14, 8, 22, 30, 52, 21, 12, 33, 45, 17, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(18, [18, 19, 37, 56, 32, 27, 59, 25, 23, 48, 10, 58, 7, 4, 11, 15, 26, 41, 6, 47, 53, 39, 31, 9, 40, 49, 28, 16, 44, 60, 43, 42, 24, 5, 29, 34, 2, 36, 38, 13, 51, 3, 54, 57, 50, 46, 35, 20, 55, 14, 8, 22, 30, 52, 21, 12, 33, 45, 17, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316043,"parameters":{"p":61,"element":18,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":18,\"p\":61}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0044","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":67,\"element\":19}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(19, [19, 26, 25, 6, 47, 22, 16, 36, 14, 65, 29, 15, 17, 55, 40, 23, 35, 62, 39, 4, 9, 37, 33, 24, 54, 21, 64, 10, 56, 59, 49, 60, 1, 19, 26, 25, 6, 47, 22, 16, 36, 14, 65, 29, 15, 17, 55, 40, 23, 35, 62, 39, 4, 9, 37, 33, 24, 54, 21, 64, 10, 56, 59, 49, 60, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(19, [19, 26, 25, 6, 47, 22, 16, 36, 14, 65, 29, 15, 17, 55, 40, 23, 35, 62, 39, 4, 9, 37, 33, 24, 54, 21, 64, 10, 56, 59, 49, 60, 1, 19, 26, 25, 6, 47, 22, 16, 36, 14, 65, 29, 15, 17, 55, 40, 23, 35, 62, 39, 4, 9, 37, 33, 24, 54, 21, 64, 10, 56, 59, 49, 60, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316044,"parameters":{"p":67,"element":19,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":19,\"p\":67}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0045","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":71,\"element\":20}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(20, [20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(20, [20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316045,"parameters":{"p":71,"element":20,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":20,\"p\":71}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0046","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":73,\"element\":21}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(21, [21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(21, [21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316046,"parameters":{"p":73,"element":21,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":21,\"p\":73}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0047","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":79,\"element\":22}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(22, [22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(22, [22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316047,"parameters":{"p":79,"element":22,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":22,\"p\":79}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0048","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":83,\"element\":23}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(23, [23, 31, 49, 48, 25, 77, 28, 63, 38, 44, 16, 36, 81, 37, 21, 68, 70, 33, 12, 27, 40, 7, 78, 51, 11, 4, 9, 41, 30, 26, 17, 59, 29, 3, 69, 10, 64, 61, 75, 65, 1, 23, 31, 49, 48, 25, 77, 28, 63, 38, 44, 16, 36, 81, 37, 21, 68, 70, 33, 12, 27, 40, 7, 78, 51, 11, 4, 9, 41, 30, 26, 17, 59, 29, 3, 69, 10, 64, 61, 75, 65, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(23, [23, 31, 49, 48, 25, 77, 28, 63, 38, 44, 16, 36, 81, 37, 21, 68, 70, 33, 12, 27, 40, 7, 78, 51, 11, 4, 9, 41, 30, 26, 17, 59, 29, 3, 69, 10, 64, 61, 75, 65, 1, 23, 31, 49, 48, 25, 77, 28, 63, 38, 44, 16, 36, 81, 37, 21, 68, 70, 33, 12, 27, 40, 7, 78, 51, 11, 4, 9, 41, 30, 26, 17, 59, 29, 3, 69, 10, 64, 61, 75, 65, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316048,"parameters":{"p":83,"element":23,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":23,\"p\":83}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0049","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":89,\"element\":24}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(24, [24, 42, 29, 73, 61, 40, 70, 78, 3, 72, 37, 87, 41, 5, 31, 32, 56, 9, 38, 22, 83, 34, 15, 4, 7, 79, 27, 25, 66, 71, 13, 45, 12, 21, 59, 81, 75, 20, 35, 39, 46, 36, 63, 88, 65, 47, 60, 16, 28, 49, 19, 11, 86, 17, 52, 2, 48, 84, 58, 57, 33, 80, 51, 67, 6, 55, 74, 85, 82, 10, 62, 64, 23, 18, 76, 44, 77, 68, 30, 8, 14, 69, 54, 50, 43, 53, 26, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(24, [24, 42, 29, 73, 61, 40, 70, 78, 3, 72, 37, 87, 41, 5, 31, 32, 56, 9, 38, 22, 83, 34, 15, 4, 7, 79, 27, 25, 66, 71, 13, 45, 12, 21, 59, 81, 75, 20, 35, 39, 46, 36, 63, 88, 65, 47, 60, 16, 28, 49, 19, 11, 86, 17, 52, 2, 48, 84, 58, 57, 33, 80, 51, 67, 6, 55, 74, 85, 82, 10, 62, 64, 23, 18, 76, 44, 77, 68, 30, 8, 14, 69, 54, 50, 43, 53, 26, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316049,"parameters":{"p":89,"element":24,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":24,\"p\":89}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0050","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-power-cycle；資料={\"p\":97,\"element\":25}。請依 [element,powers,last,is_one,certificate] 欄序回答。","answer":{"kind":"expression","value":"(25, [25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1, 25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1], 1, 是, nonzero_multiplicative_group_cycle)","explanation_zh":"Wave4收束核對=(25, [25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1, 25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1], 1, 是, nonzero_multiplicative_group_cycle)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316050,"parameters":{"p":97,"element":25,"domain":"Exact_prime_field_Zp","operation":"fielddeep-power-cycle","semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":25,\"p\":97}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0051","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":2,\"a\":1,\"b\":1,\"c\":1}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 1, 0, 1, 是)","explanation_zh":"Wave4收束核對=(1, 1, 1, 1, 0, 1, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316051,"parameters":{"p":2,"a":1,"b":1,"c":1,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":1,\"b\":1,\"c\":1,\"p\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0052","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":3,\"a\":2,\"b\":2,\"c\":0}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 0, 2, 2, 0, 是)","explanation_zh":"Wave4收束核對=(2, 2, 0, 2, 2, 0, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316052,"parameters":{"p":3,"a":2,"b":2,"c":0,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":2,\"b\":2,\"c\":0,\"p\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0053","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":5,\"a\":3,\"b\":3,\"c\":0}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 0, 2, 4, 0, 是)","explanation_zh":"Wave4收束核對=(3, 3, 0, 2, 4, 0, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316053,"parameters":{"p":5,"a":3,"b":3,"c":0,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":3,\"b\":3,\"c\":0,\"p\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0054","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":7,\"a\":4,\"b\":4,\"c\":0}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 0, 2, 6, 0, 是)","explanation_zh":"Wave4收束核對=(4, 4, 0, 2, 6, 0, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316054,"parameters":{"p":7,"a":4,"b":4,"c":0,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":4,\"b\":4,\"c\":0,\"p\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0055","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":11,\"a\":5,\"b\":5,\"c\":9}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, 9, 9, 3, 9, 是)","explanation_zh":"Wave4收束核對=(5, 5, 9, 9, 3, 9, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316055,"parameters":{"p":11,"a":5,"b":5,"c":9,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":5,\"b\":5,\"c\":9,\"p\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0056","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":13,\"a\":6,\"b\":6,\"c\":11}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(6, 6, 11, 11, 3, 11, 是)","explanation_zh":"Wave4收束核對=(6, 6, 11, 11, 3, 11, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316056,"parameters":{"p":13,"a":6,"b":6,"c":11,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":6,\"b\":6,\"c\":11,\"p\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0057","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":17,\"a\":7,\"b\":7,\"c\":13}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(7, 7, 13, 5, 13, 13, 是)","explanation_zh":"Wave4收束核對=(7, 7, 13, 5, 13, 13, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316057,"parameters":{"p":17,"a":7,"b":7,"c":13,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":7,\"b\":7,\"c\":13,\"p\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0058","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":19,\"a\":8,\"b\":8,\"c\":15}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(8, 8, 15, 12, 8, 15, 是)","explanation_zh":"Wave4收束核對=(8, 8, 15, 12, 8, 15, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316058,"parameters":{"p":19,"a":8,"b":8,"c":15,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":8,\"b\":8,\"c\":15,\"p\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0059","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":23,\"a\":9,\"b\":9,\"c\":17}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(9, 9, 17, 18, 6, 17, 是)","explanation_zh":"Wave4收束核對=(9, 9, 17, 18, 6, 17, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316059,"parameters":{"p":23,"a":9,"b":9,"c":17,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":9,\"b\":9,\"c\":17,\"p\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0060","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":29,\"a\":10,\"b\":10,\"c\":19}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(10, 10, 19, 3, 27, 19, 是)","explanation_zh":"Wave4收束核對=(10, 10, 19, 3, 27, 19, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316060,"parameters":{"p":29,"a":10,"b":10,"c":19,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":10,\"b\":10,\"c\":19,\"p\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0061","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":31,\"a\":11,\"b\":11,\"c\":21}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(11, 11, 21, 17, 15, 21, 是)","explanation_zh":"Wave4收束核對=(11, 11, 21, 17, 15, 21, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316061,"parameters":{"p":31,"a":11,"b":11,"c":21,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":11,\"b\":11,\"c\":21,\"p\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0062","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":37,\"a\":12,\"b\":12,\"c\":23}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(12, 12, 23, 34, 4, 23, 是)","explanation_zh":"Wave4收束核對=(12, 12, 23, 34, 4, 23, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316062,"parameters":{"p":37,"a":12,"b":12,"c":23,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":12,\"b\":12,\"c\":23,\"p\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0063","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":41,\"a\":13,\"b\":13,\"c\":25}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(13, 13, 25, 19, 23, 25, 是)","explanation_zh":"Wave4收束核對=(13, 13, 25, 19, 23, 25, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316063,"parameters":{"p":41,"a":13,"b":13,"c":25,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":13,\"b\":13,\"c\":25,\"p\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0064","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":43,\"a\":14,\"b\":14,\"c\":27}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(14, 14, 27, 40, 4, 27, 是)","explanation_zh":"Wave4收束核對=(14, 14, 27, 40, 4, 27, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316064,"parameters":{"p":43,"a":14,"b":14,"c":27,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":14,\"b\":14,\"c\":27,\"p\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0065","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":47,\"a\":15,\"b\":15,\"c\":29}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(15, 15, 29, 22, 26, 29, 是)","explanation_zh":"Wave4收束核對=(15, 15, 29, 22, 26, 29, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316065,"parameters":{"p":47,"a":15,"b":15,"c":29,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":15,\"b\":15,\"c\":29,\"p\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0066","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":53,\"a\":16,\"b\":16,\"c\":31}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(16, 16, 31, 10, 44, 31, 是)","explanation_zh":"Wave4收束核對=(16, 16, 31, 10, 44, 31, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316066,"parameters":{"p":53,"a":16,"b":16,"c":31,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":16,\"b\":16,\"c\":31,\"p\":53}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0067","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":59,\"a\":17,\"b\":17,\"c\":33}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(17, 17, 33, 7, 53, 33, 是)","explanation_zh":"Wave4收束核對=(17, 17, 33, 7, 53, 33, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316067,"parameters":{"p":59,"a":17,"b":17,"c":33,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":17,\"b\":17,\"c\":33,\"p\":59}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0068","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":61,\"a\":18,\"b\":18,\"c\":35}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(18, 18, 35, 17, 45, 35, 是)","explanation_zh":"Wave4收束核對=(18, 18, 35, 17, 45, 35, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316068,"parameters":{"p":61,"a":18,"b":18,"c":35,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":18,\"b\":18,\"c\":35,\"p\":61}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0069","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":67,\"a\":19,\"b\":19,\"c\":37}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(19, 19, 37, 60, 8, 37, 是)","explanation_zh":"Wave4收束核對=(19, 19, 37, 60, 8, 37, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316069,"parameters":{"p":67,"a":19,"b":19,"c":37,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":19,\"b\":19,\"c\":37,\"p\":67}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0070","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":71,\"a\":20,\"b\":20,\"c\":39}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(20, 20, 39, 32, 40, 39, 是)","explanation_zh":"Wave4收束核對=(20, 20, 39, 32, 40, 39, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316070,"parameters":{"p":71,"a":20,"b":20,"c":39,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":20,\"b\":20,\"c\":39,\"p\":71}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0071","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":73,\"a\":21,\"b\":21,\"c\":41}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(21, 21, 41, 7, 67, 41, 是)","explanation_zh":"Wave4收束核對=(21, 21, 41, 7, 67, 41, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316071,"parameters":{"p":73,"a":21,"b":21,"c":41,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":21,\"b\":21,\"c\":41,\"p\":73}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0072","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":79,\"a\":22,\"b\":22,\"c\":43}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(22, 22, 43, 18, 62, 43, 是)","explanation_zh":"Wave4收束核對=(22, 22, 43, 18, 62, 43, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316072,"parameters":{"p":79,"a":22,"b":22,"c":43,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":22,\"b\":22,\"c\":43,\"p\":79}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0073","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":83,\"a\":23,\"b\":23,\"c\":45}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(23, 23, 45, 65, 19, 45, 是)","explanation_zh":"Wave4收束核對=(23, 23, 45, 65, 19, 45, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316073,"parameters":{"p":83,"a":23,"b":23,"c":45,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":23,\"b\":23,\"c\":45,\"p\":83}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0074","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":89,\"a\":24,\"b\":24,\"c\":47}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(24, 24, 47, 26, 64, 47, 是)","explanation_zh":"Wave4收束核對=(24, 24, 47, 26, 64, 47, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316074,"parameters":{"p":89,"a":24,"b":24,"c":47,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":24,\"b\":24,\"c\":47,\"p\":89}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0075","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":97,\"a\":25,\"b\":25,\"c\":49}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。","answer":{"kind":"expression","value":"(25, 25, 49, 66, 32, 49, 是)","explanation_zh":"Wave4收束核對=(25, 25, 49, 66, 32, 49, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316075,"parameters":{"p":97,"a":25,"b":25,"c":49,"domain":"Exact_prime_field_Zp","operation":"fielddeep-affine-solve","semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":25,\"b\":25,\"c\":49,\"p\":97}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0076","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":2,\"element\":0}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 0, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(0, 2, 0, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316076,"parameters":{"p":2,"element":0,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":0,\"p\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0077","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":3,\"element\":1}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 1, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(1, 3, 1, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316077,"parameters":{"p":3,"element":1,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":1,\"p\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0078","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":5,\"element\":2}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 5, 2, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(2, 5, 2, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316078,"parameters":{"p":5,"element":2,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":2,\"p\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0079","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":7,\"element\":3}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(3, 7, 3, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(3, 7, 3, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316079,"parameters":{"p":7,"element":3,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":3,\"p\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0080","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":11,\"element\":4}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(4, 11, 4, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(4, 11, 4, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316080,"parameters":{"p":11,"element":4,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":4,\"p\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0081","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":13,\"element\":5}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 13, 5, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(5, 13, 5, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316081,"parameters":{"p":13,"element":5,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":5,\"p\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0082","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":17,\"element\":6}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(6, 17, 6, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(6, 17, 6, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316082,"parameters":{"p":17,"element":6,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":6,\"p\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0083","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":19,\"element\":7}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(7, 19, 7, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(7, 19, 7, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316083,"parameters":{"p":19,"element":7,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":7,\"p\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0084","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":23,\"element\":8}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(8, 23, 8, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(8, 23, 8, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316084,"parameters":{"p":23,"element":8,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":8,\"p\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0085","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":29,\"element\":9}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(9, 29, 9, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(9, 29, 9, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316085,"parameters":{"p":29,"element":9,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":9,\"p\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0086","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":31,\"element\":10}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(10, 31, 10, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(10, 31, 10, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316086,"parameters":{"p":31,"element":10,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":10,\"p\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0087","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":37,\"element\":11}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(11, 37, 11, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(11, 37, 11, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316087,"parameters":{"p":37,"element":11,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":11,\"p\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0088","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":41,\"element\":12}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(12, 41, 12, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(12, 41, 12, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316088,"parameters":{"p":41,"element":12,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":12,\"p\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0089","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":43,\"element\":13}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(13, 43, 13, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(13, 43, 13, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316089,"parameters":{"p":43,"element":13,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":13,\"p\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0090","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":47,\"element\":14}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(14, 47, 14, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(14, 47, 14, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316090,"parameters":{"p":47,"element":14,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":14,\"p\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0091","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":53,\"element\":15}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(15, 53, 15, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(15, 53, 15, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316091,"parameters":{"p":53,"element":15,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":15,\"p\":53}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0092","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":59,\"element\":16}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(16, 59, 16, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(16, 59, 16, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316092,"parameters":{"p":59,"element":16,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":16,\"p\":59}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0093","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":61,\"element\":17}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(17, 61, 17, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(17, 61, 17, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316093,"parameters":{"p":61,"element":17,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":17,\"p\":61}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0094","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":67,\"element\":18}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(18, 67, 18, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(18, 67, 18, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316094,"parameters":{"p":67,"element":18,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":18,\"p\":67}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0095","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":71,\"element\":19}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(19, 71, 19, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(19, 71, 19, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316095,"parameters":{"p":71,"element":19,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":19,\"p\":71}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0096","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":73,\"element\":20}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(20, 73, 20, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(20, 73, 20, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316096,"parameters":{"p":73,"element":20,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":20,\"p\":73}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0097","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":79,\"element\":21}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(21, 79, 21, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(21, 79, 21, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316097,"parameters":{"p":79,"element":21,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":21,\"p\":79}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0098","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":83,\"element\":22}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(22, 83, 22, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(22, 83, 22, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316098,"parameters":{"p":83,"element":22,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":22,\"p\":83}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0099","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":89,\"element\":23}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(23, 89, 23, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(23, 89, 23, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316099,"parameters":{"p":89,"element":23,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":23,\"p\":89}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r06-finite-field-0100","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域深度","stem_zh":"完整prime field Z_p運算。操作=fielddeep-frobenius；資料={\"p\":97,\"element\":24}。請依 [element,p,power,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(24, 97, 24, 是, Frobenius_on_prime_field)","explanation_zh":"Wave4收束核對=(24, 97, 24, 是, Frobenius_on_prime_field)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域深度"]},"variant":{"family_id":"qf-mod31-fielddeep-frobenius","seed":202610316100,"parameters":{"p":97,"element":24,"domain":"Exact_prime_field_Zp","operation":"fielddeep-frobenius","semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":24,\"p\":97}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}