{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0001","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[0,1,2,0],\"codomain_count\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 3, [0, 3, 0], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(4, 3, [0, 3, 0], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075001,"parameters":{"mapping":[0,1,2,0],"codomain_count":3,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":3,\"mapping\":[0,1,2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0002","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[1,3,1,3,1],\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 4, [0, 2, 1], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(5, 4, [0, 2, 1], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075002,"parameters":{"mapping":[1,3,1,3,1],"codomain_count":4,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":4,\"mapping\":[1,3,1,3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0003","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[2,0,3,1,4,2],\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 5, [0, 5, 2], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(6, 5, [0, 5, 2], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075003,"parameters":{"mapping":[2,0,3,1,4,2],"codomain_count":5,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":5,\"mapping\":[2,0,3,1,4,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0004","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[3,4,5,0,1,2,3],\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 6, [0, 6, 3], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(7, 6, [0, 6, 3], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075004,"parameters":{"mapping":[3,4,5,0,1,2,3],"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":6,\"mapping\":[3,4,5,0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0005","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[4,6,1,3,5,0,2,4],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 7, [0, 7, 4], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(8, 7, [0, 7, 4], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075005,"parameters":{"mapping":[4,6,1,3,5,0,2,4],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":7,\"mapping\":[4,6,1,3,5,0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0006","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[2,2,2,2,2],\"codomain_count\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 3, [0, 1, 2], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(5, 3, [0, 1, 2], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075006,"parameters":{"mapping":[2,2,2,2,2],"codomain_count":3,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":3,\"mapping\":[2,2,2,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0007","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[2,3,0,1,2,3],\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 4, [0, 4, 2], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(6, 4, [0, 4, 2], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075007,"parameters":{"mapping":[2,3,0,1,2,3],"codomain_count":4,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":4,\"mapping\":[2,3,0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0008","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[2,4,1,3,0,2,4],\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 5, [0, 5, 2], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(7, 5, [0, 5, 2], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075008,"parameters":{"mapping":[2,4,1,3,0,2,4],"codomain_count":5,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":5,\"mapping\":[2,4,1,3,0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0009","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[2,5,2,5,2,5,2,5],\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 6, [0, 2, 2], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(8, 6, [0, 2, 2], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075009,"parameters":{"mapping":[2,5,2,5,2,5,2,5],"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":6,\"mapping\":[2,5,2,5,2,5,2,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0010","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[2,3,4,5,6,0,1,2,3],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 7, [0, 7, 2], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(9, 7, [0, 7, 2], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075010,"parameters":{"mapping":[2,3,4,5,6,0,1,2,3],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":7,\"mapping\":[2,3,4,5,6,0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0011","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[1,0,2,1,0,2],\"codomain_count\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [0, 3, 1], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(6, 3, [0, 3, 1], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075011,"parameters":{"mapping":[1,0,2,1,0,2],"codomain_count":3,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":3,\"mapping\":[1,0,2,1,0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0012","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[3,2,1,0,3,2,1],\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 4, [0, 4, 3], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(7, 4, [0, 4, 3], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075012,"parameters":{"mapping":[3,2,1,0,3,2,1],"codomain_count":4,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":4,\"mapping\":[3,2,1,0,3,2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0013","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[2,3,4,0,1,2,3,4],\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 5, [0, 5, 2], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(8, 5, [0, 5, 2], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075013,"parameters":{"mapping":[2,3,4,0,1,2,3,4],"codomain_count":5,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":5,\"mapping\":[2,3,4,0,1,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0014","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[1,3,5,1,3,5,1,3,5],\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 6, [0, 3, 1], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(9, 6, [0, 3, 1], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075014,"parameters":{"mapping":[1,3,5,1,3,5,1,3,5],"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":6,\"mapping\":[1,3,5,1,3,5,1,3,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0015","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[0,3,6,2,5,1,4,0,3,6],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 7, [0, 7, 0], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(10, 7, [0, 7, 0], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075015,"parameters":{"mapping":[0,3,6,2,5,1,4,0,3,6],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":7,\"mapping\":[0,3,6,2,5,1,4,0,3,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0016","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[0,1,2,0,1,2,0],\"codomain_count\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 3, [0, 3, 0], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(7, 3, [0, 3, 0], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075016,"parameters":{"mapping":[0,1,2,0,1,2,0],"codomain_count":3,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":3,\"mapping\":[0,1,2,0,1,2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0017","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[0,2,0,2,0,2,0,2],\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 4, [0, 2, 0], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(8, 4, [0, 2, 0], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075017,"parameters":{"mapping":[0,2,0,2,0,2,0,2],"codomain_count":4,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":4,\"mapping\":[0,2,0,2,0,2,0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0018","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[2,0,3,1,4,2,0,3,1],\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 5, [0, 5, 2], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(9, 5, [0, 5, 2], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075018,"parameters":{"mapping":[2,0,3,1,4,2,0,3,1],"codomain_count":5,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":5,\"mapping\":[2,0,3,1,4,2,0,3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0019","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[0,1,2,3,4,5,0,1,2,3],\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 6, [0, 6, 0], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(10, 6, [0, 6, 0], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075019,"parameters":{"mapping":[0,1,2,3,4,5,0,1,2,3],"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":6,\"mapping\":[0,1,2,3,4,5,0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0020","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[5,0,2,4,6,1,3,5,0,2,4],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 7, [0, 7, 5], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(11, 7, [0, 7, 5], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075020,"parameters":{"mapping":[5,0,2,4,6,1,3,5,0,2,4],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":7,\"mapping\":[5,0,2,4,6,1,3,5,0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0021","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[2,2,2,2,2,2,2,2],\"codomain_count\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 3, [0, 1, 2], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(8, 3, [0, 1, 2], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075021,"parameters":{"mapping":[2,2,2,2,2,2,2,2],"codomain_count":3,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":3,\"mapping\":[2,2,2,2,2,2,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0022","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[1,2,3,0,1,2,3,0,1],\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 4, [0, 4, 1], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(9, 4, [0, 4, 1], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075022,"parameters":{"mapping":[1,2,3,0,1,2,3,0,1],"codomain_count":4,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":4,\"mapping\":[1,2,3,0,1,2,3,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0023","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[2,4,1,3,0,2,4,1,3,0],\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 5, [0, 5, 2], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(10, 5, [0, 5, 2], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075023,"parameters":{"mapping":[2,4,1,3,0,2,4,1,3,0],"codomain_count":5,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":5,\"mapping\":[2,4,1,3,0,2,4,1,3,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0024","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[5,2,5,2,5,2,5,2,5,2,5],\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 6, [0, 2, 5], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(11, 6, [0, 2, 5], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075024,"parameters":{"mapping":[5,2,5,2,5,2,5,2,5,2,5],"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":6,\"mapping\":[5,2,5,2,5,2,5,2,5,2,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0025","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[3,4,5,6,0,1,2,3,4,5,6,0],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 7, [0, 7, 3], 是, complete_total_map_collision)","explanation_zh":"Wave6根層核對=(12, 7, [0, 7, 3], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-collision-witness","seed":202611075025,"parameters":{"mapping":[3,4,5,6,0,1,2,3,4,5,6,0],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-collision-witness","semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":7,\"mapping\":[3,4,5,6,0,1,2,3,4,5,6,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0026","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[0,1,2,0],\"codomain_count\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 1], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([2, 1, 1], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075026,"parameters":{"mapping":[0,1,2,0],"codomain_count":3,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":3,\"mapping\":[0,1,2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0027","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[1,3,1,3,1],\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 3, 0, 2], 3, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([0, 3, 0, 2], 3, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075027,"parameters":{"mapping":[1,3,1,3,1],"codomain_count":4,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":4,\"mapping\":[1,3,1,3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0028","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[2,0,3,1,4,2],\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 2, 1, 1], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([1, 1, 2, 1, 1], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075028,"parameters":{"mapping":[2,0,3,1,4,2],"codomain_count":5,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":5,\"mapping\":[2,0,3,1,4,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0029","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[3,4,5,0,1,2,3],\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 2, 1, 1], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([1, 1, 1, 2, 1, 1], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075029,"parameters":{"mapping":[3,4,5,0,1,2,3],"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":6,\"mapping\":[3,4,5,0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0030","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[4,6,1,3,5,0,2,4],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 1, 2, 1, 1], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([1, 1, 1, 1, 2, 1, 1], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075030,"parameters":{"mapping":[4,6,1,3,5,0,2,4],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":7,\"mapping\":[4,6,1,3,5,0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0031","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[2,2,2,2,2],\"codomain_count\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 5], 5, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([0, 0, 5], 5, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075031,"parameters":{"mapping":[2,2,2,2,2],"codomain_count":3,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":3,\"mapping\":[2,2,2,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0034","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[2,5,2,5,2,5,2,5],\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 4, 0, 0, 4], 4, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([0, 0, 4, 0, 0, 4], 4, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075034,"parameters":{"mapping":[2,5,2,5,2,5,2,5],"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":6,\"mapping\":[2,5,2,5,2,5,2,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0035","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[2,3,4,5,6,0,1,2,3],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 2, 2, 1, 1, 1], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([1, 1, 2, 2, 1, 1, 1], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075035,"parameters":{"mapping":[2,3,4,5,6,0,1,2,3],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":7,\"mapping\":[2,3,4,5,6,0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0036","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[1,0,2,1,0,2],\"codomain_count\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 2], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([2, 2, 2], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075036,"parameters":{"mapping":[1,0,2,1,0,2],"codomain_count":3,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":3,\"mapping\":[1,0,2,1,0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0037","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[3,2,1,0,3,2,1],\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([1, 2, 2, 2], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075037,"parameters":{"mapping":[3,2,1,0,3,2,1],"codomain_count":4,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":4,\"mapping\":[3,2,1,0,3,2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0038","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[2,3,4,0,1,2,3,4],\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 2, 2, 2], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([1, 1, 2, 2, 2], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075038,"parameters":{"mapping":[2,3,4,0,1,2,3,4],"codomain_count":5,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":5,\"mapping\":[2,3,4,0,1,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0039","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[1,3,5,1,3,5,1,3,5],\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 3, 0, 3, 0, 3], 3, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([0, 3, 0, 3, 0, 3], 3, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075039,"parameters":{"mapping":[1,3,5,1,3,5,1,3,5],"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":6,\"mapping\":[1,3,5,1,3,5,1,3,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0040","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[0,3,6,2,5,1,4,0,3,6],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 1, 2, 1, 1, 2], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([2, 1, 1, 2, 1, 1, 2], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075040,"parameters":{"mapping":[0,3,6,2,5,1,4,0,3,6],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":7,\"mapping\":[0,3,6,2,5,1,4,0,3,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0041","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[0,1,2,0,1,2,0],\"codomain_count\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 2, 2], 3, 3, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([3, 2, 2], 3, 3, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075041,"parameters":{"mapping":[0,1,2,0,1,2,0],"codomain_count":3,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":3,\"mapping\":[0,1,2,0,1,2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0042","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[0,2,0,2,0,2,0,2],\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 0, 4, 0], 4, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([4, 0, 4, 0], 4, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075042,"parameters":{"mapping":[0,2,0,2,0,2,0,2],"codomain_count":4,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":4,\"mapping\":[0,2,0,2,0,2,0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0043","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[2,0,3,1,4,2,0,3,1],\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 2, 2, 1], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([2, 2, 2, 2, 1], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075043,"parameters":{"mapping":[2,0,3,1,4,2,0,3,1],"codomain_count":5,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":5,\"mapping\":[2,0,3,1,4,2,0,3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0044","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[0,1,2,3,4,5,0,1,2,3],\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 2, 2, 1, 1], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([2, 2, 2, 2, 1, 1], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075044,"parameters":{"mapping":[0,1,2,3,4,5,0,1,2,3],"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":6,\"mapping\":[0,1,2,3,4,5,0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0045","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[5,0,2,4,6,1,3,5,0,2,4],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 2, 1, 2, 2, 1], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([2, 1, 2, 1, 2, 2, 1], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075045,"parameters":{"mapping":[5,0,2,4,6,1,3,5,0,2,4],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":7,\"mapping\":[5,0,2,4,6,1,3,5,0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0046","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[2,2,2,2,2,2,2,2],\"codomain_count\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 8], 8, 3, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([0, 0, 8], 8, 3, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075046,"parameters":{"mapping":[2,2,2,2,2,2,2,2],"codomain_count":3,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":3,\"mapping\":[2,2,2,2,2,2,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0047","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[1,2,3,0,1,2,3,0,1],\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 2, 2], 3, 3, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([2, 3, 2, 2], 3, 3, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075047,"parameters":{"mapping":[1,2,3,0,1,2,3,0,1],"codomain_count":4,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":4,\"mapping\":[1,2,3,0,1,2,3,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0048","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[2,4,1,3,0,2,4,1,3,0],\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 2, 2, 2], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([2, 2, 2, 2, 2], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075048,"parameters":{"mapping":[2,4,1,3,0,2,4,1,3,0],"codomain_count":5,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":5,\"mapping\":[2,4,1,3,0,2,4,1,3,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0049","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[5,2,5,2,5,2,5,2,5,2,5],\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 5, 0, 0, 6], 6, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([0, 0, 5, 0, 0, 6], 6, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075049,"parameters":{"mapping":[5,2,5,2,5,2,5,2,5,2,5],"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":6,\"mapping\":[5,2,5,2,5,2,5,2,5,2,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0050","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[3,4,5,6,0,1,2,3,4,5,6,0],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 1, 2, 2, 2, 2], 2, 2, 是, generalized_pigeonhole_load_bound)","explanation_zh":"Wave6根層核對=([2, 1, 1, 2, 2, 2, 2], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-generalized-load","seed":202611075050,"parameters":{"mapping":[3,4,5,6,0,1,2,3,4,5,6,0],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-generalized-load","semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":7,\"mapping\":[3,4,5,6,0,1,2,3,4,5,6,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0051","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":5,\"codomain_count\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 3, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(5, 3, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075051,"parameters":{"domain_count":5,"codomain_count":3,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":3,\"domain_count\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0052","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":6,\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 4, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(6, 4, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075052,"parameters":{"domain_count":6,"codomain_count":4,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":4,\"domain_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0053","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":7,\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 5, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(7, 5, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075053,"parameters":{"domain_count":7,"codomain_count":5,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":5,\"domain_count\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0054","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":8,\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 6, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(8, 6, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075054,"parameters":{"domain_count":8,"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":6,\"domain_count\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0055","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":9,\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 7, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(9, 7, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075055,"parameters":{"domain_count":9,"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":7,\"domain_count\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0056","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":10,\"codomain_count\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 8, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(10, 8, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075056,"parameters":{"domain_count":10,"codomain_count":8,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":8,\"domain_count\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0057","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":11,\"codomain_count\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 9, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(11, 9, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075057,"parameters":{"domain_count":11,"codomain_count":9,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":9,\"domain_count\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0058","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":12,\"codomain_count\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 10, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(12, 10, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075058,"parameters":{"domain_count":12,"codomain_count":10,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":10,\"domain_count\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0059","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":13,\"codomain_count\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 11, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(13, 11, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075059,"parameters":{"domain_count":13,"codomain_count":11,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":11,\"domain_count\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0060","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":14,\"codomain_count\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 12, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(14, 12, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075060,"parameters":{"domain_count":14,"codomain_count":12,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":12,\"domain_count\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0061","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":15,\"codomain_count\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 13, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(15, 13, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075061,"parameters":{"domain_count":15,"codomain_count":13,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":13,\"domain_count\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0062","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":16,\"codomain_count\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 14, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(16, 14, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075062,"parameters":{"domain_count":16,"codomain_count":14,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":14,\"domain_count\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0063","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":17,\"codomain_count\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 15, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(17, 15, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075063,"parameters":{"domain_count":17,"codomain_count":15,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":15,\"domain_count\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0064","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":18,\"codomain_count\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 16, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(18, 16, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075064,"parameters":{"domain_count":18,"codomain_count":16,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":16,\"domain_count\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0065","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":19,\"codomain_count\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 17, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(19, 17, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075065,"parameters":{"domain_count":19,"codomain_count":17,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":17,\"domain_count\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0066","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":20,\"codomain_count\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 18, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(20, 18, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075066,"parameters":{"domain_count":20,"codomain_count":18,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":18,\"domain_count\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0067","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":21,\"codomain_count\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 19, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(21, 19, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075067,"parameters":{"domain_count":21,"codomain_count":19,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":19,\"domain_count\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0068","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":22,\"codomain_count\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 20, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(22, 20, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075068,"parameters":{"domain_count":22,"codomain_count":20,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":20,\"domain_count\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0069","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":23,\"codomain_count\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 21, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(23, 21, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075069,"parameters":{"domain_count":23,"codomain_count":21,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":21,\"domain_count\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0070","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":24,\"codomain_count\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 22, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(24, 22, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075070,"parameters":{"domain_count":24,"codomain_count":22,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":22,\"domain_count\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0071","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":25,\"codomain_count\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, 23, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(25, 23, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075071,"parameters":{"domain_count":25,"codomain_count":23,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":23,\"domain_count\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0072","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":26,\"codomain_count\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(26, 24, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(26, 24, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075072,"parameters":{"domain_count":26,"codomain_count":24,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":24,\"domain_count\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0073","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":27,\"codomain_count\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(27, 25, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(27, 25, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075073,"parameters":{"domain_count":27,"codomain_count":25,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":25,\"domain_count\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0074","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":28,\"codomain_count\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(28, 26, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(28, 26, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075074,"parameters":{"domain_count":28,"codomain_count":26,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":26,\"domain_count\":28}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0075","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-premise-audit；資料={\"domain_count\":29,\"codomain_count\":27}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 27, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation_zh":"Wave6根層核對=(29, 27, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-premise-audit","seed":202611075075,"parameters":{"domain_count":29,"codomain_count":27,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-premise-audit","semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":27,\"domain_count\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0076","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2],\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(3, 4, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075076,"parameters":{"mapping":[0,0,2],"codomain_count":4,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":4,\"mapping\":[0,0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0077","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3],\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 5, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(4, 5, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075077,"parameters":{"mapping":[0,0,2,3],"codomain_count":5,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":5,\"mapping\":[0,0,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0078","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4],\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(5, 6, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075078,"parameters":{"mapping":[0,0,2,3,4],"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":6,\"mapping\":[0,0,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0079","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4,5],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 7, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(6, 7, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075079,"parameters":{"mapping":[0,0,2,3,4,5],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":7,\"mapping\":[0,0,2,3,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0080","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4,5,6],\"codomain_count\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 8, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(7, 8, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075080,"parameters":{"mapping":[0,0,2,3,4,5,6],"codomain_count":8,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":8,\"mapping\":[0,0,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0081","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2],\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(3, 5, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075081,"parameters":{"mapping":[0,0,2],"codomain_count":5,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":5,\"mapping\":[0,0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0082","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3],\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 6, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(4, 6, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075082,"parameters":{"mapping":[0,0,2,3],"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":6,\"mapping\":[0,0,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0083","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 7, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(5, 7, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075083,"parameters":{"mapping":[0,0,2,3,4],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":7,\"mapping\":[0,0,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0084","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4,5],\"codomain_count\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 8, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(6, 8, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075084,"parameters":{"mapping":[0,0,2,3,4,5],"codomain_count":8,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":8,\"mapping\":[0,0,2,3,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0085","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4,5,6],\"codomain_count\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 9, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(7, 9, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075085,"parameters":{"mapping":[0,0,2,3,4,5,6],"codomain_count":9,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":9,\"mapping\":[0,0,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0086","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2],\"codomain_count\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 6, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(3, 6, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075086,"parameters":{"mapping":[0,0,2],"codomain_count":6,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":6,\"mapping\":[0,0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0087","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 7, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(4, 7, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075087,"parameters":{"mapping":[0,0,2,3],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":7,\"mapping\":[0,0,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0088","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4],\"codomain_count\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 8, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(5, 8, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075088,"parameters":{"mapping":[0,0,2,3,4],"codomain_count":8,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":8,\"mapping\":[0,0,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0089","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4,5],\"codomain_count\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 9, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(6, 9, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075089,"parameters":{"mapping":[0,0,2,3,4,5],"codomain_count":9,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":9,\"mapping\":[0,0,2,3,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0090","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4,5,6],\"codomain_count\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 10, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(7, 10, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075090,"parameters":{"mapping":[0,0,2,3,4,5,6],"codomain_count":10,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":10,\"mapping\":[0,0,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0091","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2],\"codomain_count\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 7, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(3, 7, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075091,"parameters":{"mapping":[0,0,2],"codomain_count":7,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":7,\"mapping\":[0,0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0092","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3],\"codomain_count\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 8, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(4, 8, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075092,"parameters":{"mapping":[0,0,2,3],"codomain_count":8,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":8,\"mapping\":[0,0,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0093","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4],\"codomain_count\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 9, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(5, 9, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075093,"parameters":{"mapping":[0,0,2,3,4],"codomain_count":9,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":9,\"mapping\":[0,0,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0094","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4,5],\"codomain_count\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 10, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(6, 10, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075094,"parameters":{"mapping":[0,0,2,3,4,5],"codomain_count":10,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":10,\"mapping\":[0,0,2,3,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0095","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4,5,6],\"codomain_count\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 11, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(7, 11, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075095,"parameters":{"mapping":[0,0,2,3,4,5,6],"codomain_count":11,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":11,\"mapping\":[0,0,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0096","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2],\"codomain_count\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 8, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(3, 8, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075096,"parameters":{"mapping":[0,0,2],"codomain_count":8,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":8,\"mapping\":[0,0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0097","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3],\"codomain_count\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 9, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(4, 9, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075097,"parameters":{"mapping":[0,0,2,3],"codomain_count":9,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":9,\"mapping\":[0,0,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0098","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4],\"codomain_count\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 10, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(5, 10, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075098,"parameters":{"mapping":[0,0,2,3,4],"codomain_count":10,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":10,\"mapping\":[0,0,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0099","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4,5],\"codomain_count\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 11, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(6, 11, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075099,"parameters":{"mapping":[0,0,2,3,4,5],"codomain_count":11,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":11,\"mapping\":[0,0,2,3,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0100","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-pigeonhole-principle"],"title_zh":"鴿籠原理","stem_zh":"完整有限集合大小與total map。操作=pigeon-converse-boundary；資料={\"mapping\":[0,0,2,3,4,5,6],\"codomain_count\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 12, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)","explanation_zh":"Wave6根層核對=(7, 12, 否, 否, size_condition_failure_does_not_imply_given_map_is_injective)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鴿籠原理"]},"variant":{"family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075100,"parameters":{"mapping":[0,0,2,3,4,5,6],"codomain_count":12,"domain":"Complete_finite_total_map_pigeonhole","operation":"pigeon-converse-boundary","semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":12,\"mapping\":[0,0,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}