{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0001","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,1,3],[1,0,2],[3,2,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 1/2, 是], [0, 2, 1, 3/2, 是], [1, 0, 0, 1/2, 是], [1, 1, 0, 0, 是], [1, 2, 1, 1, 是], [2, 0, 1, 3/2, 是], [2, 1, 1, 1, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 1/2, 是], [0, 2, 1, 3/2, 是], [1, 0, 0, 1/2, 是], [1, 1, 0, 0, 是], [1, 2, 1, 1, 是], [2, 0, 1, 3/2, 是], [2, 1, 1, 1, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161001,"parameters":{"metric":[[0,1,3],[1,0,2],[3,2,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[1,1],[3,1]],[[1,1],[0,1],[2,1]],[[3,1],[2,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0002","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,2,6],[2,0,4],[6,4,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 1, 是], [0, 2, 2, 3, 是], [1, 0, 0, 1, 是], [1, 1, 0, 0, 是], [1, 2, 2, 2, 是], [2, 0, 2, 3, 是], [2, 1, 2, 2, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 1, 是], [0, 2, 2, 3, 是], [1, 0, 0, 1, 是], [1, 1, 0, 0, 是], [1, 2, 2, 2, 是], [2, 0, 2, 3, 是], [2, 1, 2, 2, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161002,"parameters":{"metric":[[0,2,6],[2,0,4],[6,4,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[2,1],[6,1]],[[2,1],[0,1],[4,1]],[[6,1],[4,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0003","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,3,9],[3,0,6],[9,6,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 3/2, 是], [0, 2, 3, 9/2, 是], [1, 0, 0, 3/2, 是], [1, 1, 0, 0, 是], [1, 2, 3, 3, 是], [2, 0, 3, 9/2, 是], [2, 1, 3, 3, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 3/2, 是], [0, 2, 3, 9/2, 是], [1, 0, 0, 3/2, 是], [1, 1, 0, 0, 是], [1, 2, 3, 3, 是], [2, 0, 3, 9/2, 是], [2, 1, 3, 3, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161003,"parameters":{"metric":[[0,3,9],[3,0,6],[9,6,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[3,1],[9,1]],[[3,1],[0,1],[6,1]],[[9,1],[6,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0004","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,4,12],[4,0,8],[12,8,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 2, 是], [0, 2, 4, 6, 是], [1, 0, 0, 2, 是], [1, 1, 0, 0, 是], [1, 2, 4, 4, 是], [2, 0, 4, 6, 是], [2, 1, 4, 4, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 2, 是], [0, 2, 4, 6, 是], [1, 0, 0, 2, 是], [1, 1, 0, 0, 是], [1, 2, 4, 4, 是], [2, 0, 4, 6, 是], [2, 1, 4, 4, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161004,"parameters":{"metric":[[0,4,12],[4,0,8],[12,8,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[4,1],[12,1]],[[4,1],[0,1],[8,1]],[[12,1],[8,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0005","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,5,15],[5,0,10],[15,10,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 5/2, 是], [0, 2, 5, 15/2, 是], [1, 0, 0, 5/2, 是], [1, 1, 0, 0, 是], [1, 2, 5, 5, 是], [2, 0, 5, 15/2, 是], [2, 1, 5, 5, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 5/2, 是], [0, 2, 5, 15/2, 是], [1, 0, 0, 5/2, 是], [1, 1, 0, 0, 是], [1, 2, 5, 5, 是], [2, 0, 5, 15/2, 是], [2, 1, 5, 5, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161005,"parameters":{"metric":[[0,5,15],[5,0,10],[15,10,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[5,1],[15,1]],[[5,1],[0,1],[10,1]],[[15,1],[10,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0006","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,6,18],[6,0,12],[18,12,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 3, 是], [0, 2, 6, 9, 是], [1, 0, 0, 3, 是], [1, 1, 0, 0, 是], [1, 2, 6, 6, 是], [2, 0, 6, 9, 是], [2, 1, 6, 6, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 3, 是], [0, 2, 6, 9, 是], [1, 0, 0, 3, 是], [1, 1, 0, 0, 是], [1, 2, 6, 6, 是], [2, 0, 6, 9, 是], [2, 1, 6, 6, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161006,"parameters":{"metric":[[0,6,18],[6,0,12],[18,12,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[6,1],[18,1]],[[6,1],[0,1],[12,1]],[[18,1],[12,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0007","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,7,21],[7,0,14],[21,14,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 7/2, 是], [0, 2, 7, 21/2, 是], [1, 0, 0, 7/2, 是], [1, 1, 0, 0, 是], [1, 2, 7, 7, 是], [2, 0, 7, 21/2, 是], [2, 1, 7, 7, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 7/2, 是], [0, 2, 7, 21/2, 是], [1, 0, 0, 7/2, 是], [1, 1, 0, 0, 是], [1, 2, 7, 7, 是], [2, 0, 7, 21/2, 是], [2, 1, 7, 7, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161007,"parameters":{"metric":[[0,7,21],[7,0,14],[21,14,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[7,1],[21,1]],[[7,1],[0,1],[14,1]],[[21,1],[14,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0008","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,8,24],[8,0,16],[24,16,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 4, 是], [0, 2, 8, 12, 是], [1, 0, 0, 4, 是], [1, 1, 0, 0, 是], [1, 2, 8, 8, 是], [2, 0, 8, 12, 是], [2, 1, 8, 8, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 4, 是], [0, 2, 8, 12, 是], [1, 0, 0, 4, 是], [1, 1, 0, 0, 是], [1, 2, 8, 8, 是], [2, 0, 8, 12, 是], [2, 1, 8, 8, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161008,"parameters":{"metric":[[0,8,24],[8,0,16],[24,16,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[8,1],[24,1]],[[8,1],[0,1],[16,1]],[[24,1],[16,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0009","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,9,27],[9,0,18],[27,18,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 9/2, 是], [0, 2, 9, 27/2, 是], [1, 0, 0, 9/2, 是], [1, 1, 0, 0, 是], [1, 2, 9, 9, 是], [2, 0, 9, 27/2, 是], [2, 1, 9, 9, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 9/2, 是], [0, 2, 9, 27/2, 是], [1, 0, 0, 9/2, 是], [1, 1, 0, 0, 是], [1, 2, 9, 9, 是], [2, 0, 9, 27/2, 是], [2, 1, 9, 9, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161009,"parameters":{"metric":[[0,9,27],[9,0,18],[27,18,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[9,1],[27,1]],[[9,1],[0,1],[18,1]],[[27,1],[18,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0010","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,10,30],[10,0,20],[30,20,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 5, 是], [0, 2, 10, 15, 是], [1, 0, 0, 5, 是], [1, 1, 0, 0, 是], [1, 2, 10, 10, 是], [2, 0, 10, 15, 是], [2, 1, 10, 10, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 5, 是], [0, 2, 10, 15, 是], [1, 0, 0, 5, 是], [1, 1, 0, 0, 是], [1, 2, 10, 10, 是], [2, 0, 10, 15, 是], [2, 1, 10, 10, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161010,"parameters":{"metric":[[0,10,30],[10,0,20],[30,20,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[10,1],[30,1]],[[10,1],[0,1],[20,1]],[[30,1],[20,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0011","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,11,33],[11,0,22],[33,22,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 11/2, 是], [0, 2, 11, 33/2, 是], [1, 0, 0, 11/2, 是], [1, 1, 0, 0, 是], [1, 2, 11, 11, 是], [2, 0, 11, 33/2, 是], [2, 1, 11, 11, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 11/2, 是], [0, 2, 11, 33/2, 是], [1, 0, 0, 11/2, 是], [1, 1, 0, 0, 是], [1, 2, 11, 11, 是], [2, 0, 11, 33/2, 是], [2, 1, 11, 11, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161011,"parameters":{"metric":[[0,11,33],[11,0,22],[33,22,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[11,1],[33,1]],[[11,1],[0,1],[22,1]],[[33,1],[22,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0012","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,12,36],[12,0,24],[36,24,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 6, 是], [0, 2, 12, 18, 是], [1, 0, 0, 6, 是], [1, 1, 0, 0, 是], [1, 2, 12, 12, 是], [2, 0, 12, 18, 是], [2, 1, 12, 12, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 6, 是], [0, 2, 12, 18, 是], [1, 0, 0, 6, 是], [1, 1, 0, 0, 是], [1, 2, 12, 12, 是], [2, 0, 12, 18, 是], [2, 1, 12, 12, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161012,"parameters":{"metric":[[0,12,36],[12,0,24],[36,24,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[12,1],[36,1]],[[12,1],[0,1],[24,1]],[[36,1],[24,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0013","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,13,39],[13,0,26],[39,26,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 13/2, 是], [0, 2, 13, 39/2, 是], [1, 0, 0, 13/2, 是], [1, 1, 0, 0, 是], [1, 2, 13, 13, 是], [2, 0, 13, 39/2, 是], [2, 1, 13, 13, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 13/2, 是], [0, 2, 13, 39/2, 是], [1, 0, 0, 13/2, 是], [1, 1, 0, 0, 是], [1, 2, 13, 13, 是], [2, 0, 13, 39/2, 是], [2, 1, 13, 13, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161013,"parameters":{"metric":[[0,13,39],[13,0,26],[39,26,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[13,1],[39,1]],[[13,1],[0,1],[26,1]],[[39,1],[26,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0014","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,14,42],[14,0,28],[42,28,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 7, 是], [0, 2, 14, 21, 是], [1, 0, 0, 7, 是], [1, 1, 0, 0, 是], [1, 2, 14, 14, 是], [2, 0, 14, 21, 是], [2, 1, 14, 14, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 7, 是], [0, 2, 14, 21, 是], [1, 0, 0, 7, 是], [1, 1, 0, 0, 是], [1, 2, 14, 14, 是], [2, 0, 14, 21, 是], [2, 1, 14, 14, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161014,"parameters":{"metric":[[0,14,42],[14,0,28],[42,28,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[14,1],[42,1]],[[14,1],[0,1],[28,1]],[[42,1],[28,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0015","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,15,45],[15,0,30],[45,30,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 15/2, 是], [0, 2, 15, 45/2, 是], [1, 0, 0, 15/2, 是], [1, 1, 0, 0, 是], [1, 2, 15, 15, 是], [2, 0, 15, 45/2, 是], [2, 1, 15, 15, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 15/2, 是], [0, 2, 15, 45/2, 是], [1, 0, 0, 15/2, 是], [1, 1, 0, 0, 是], [1, 2, 15, 15, 是], [2, 0, 15, 45/2, 是], [2, 1, 15, 15, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161015,"parameters":{"metric":[[0,15,45],[15,0,30],[45,30,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[15,1],[45,1]],[[15,1],[0,1],[30,1]],[[45,1],[30,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0016","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,16,48],[16,0,32],[48,32,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 8, 是], [0, 2, 16, 24, 是], [1, 0, 0, 8, 是], [1, 1, 0, 0, 是], [1, 2, 16, 16, 是], [2, 0, 16, 24, 是], [2, 1, 16, 16, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 8, 是], [0, 2, 16, 24, 是], [1, 0, 0, 8, 是], [1, 1, 0, 0, 是], [1, 2, 16, 16, 是], [2, 0, 16, 24, 是], [2, 1, 16, 16, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161016,"parameters":{"metric":[[0,16,48],[16,0,32],[48,32,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[16,1],[48,1]],[[16,1],[0,1],[32,1]],[[48,1],[32,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0017","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,17,51],[17,0,34],[51,34,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 17/2, 是], [0, 2, 17, 51/2, 是], [1, 0, 0, 17/2, 是], [1, 1, 0, 0, 是], [1, 2, 17, 17, 是], [2, 0, 17, 51/2, 是], [2, 1, 17, 17, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 17/2, 是], [0, 2, 17, 51/2, 是], [1, 0, 0, 17/2, 是], [1, 1, 0, 0, 是], [1, 2, 17, 17, 是], [2, 0, 17, 51/2, 是], [2, 1, 17, 17, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161017,"parameters":{"metric":[[0,17,51],[17,0,34],[51,34,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[17,1],[51,1]],[[17,1],[0,1],[34,1]],[[51,1],[34,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0018","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,18,54],[18,0,36],[54,36,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 9, 是], [0, 2, 18, 27, 是], [1, 0, 0, 9, 是], [1, 1, 0, 0, 是], [1, 2, 18, 18, 是], [2, 0, 18, 27, 是], [2, 1, 18, 18, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 9, 是], [0, 2, 18, 27, 是], [1, 0, 0, 9, 是], [1, 1, 0, 0, 是], [1, 2, 18, 18, 是], [2, 0, 18, 27, 是], [2, 1, 18, 18, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161018,"parameters":{"metric":[[0,18,54],[18,0,36],[54,36,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[18,1],[54,1]],[[18,1],[0,1],[36,1]],[[54,1],[36,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0019","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,19,57],[19,0,38],[57,38,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 19/2, 是], [0, 2, 19, 57/2, 是], [1, 0, 0, 19/2, 是], [1, 1, 0, 0, 是], [1, 2, 19, 19, 是], [2, 0, 19, 57/2, 是], [2, 1, 19, 19, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 19/2, 是], [0, 2, 19, 57/2, 是], [1, 0, 0, 19/2, 是], [1, 1, 0, 0, 是], [1, 2, 19, 19, 是], [2, 0, 19, 57/2, 是], [2, 1, 19, 19, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161019,"parameters":{"metric":[[0,19,57],[19,0,38],[57,38,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[19,1],[57,1]],[[19,1],[0,1],[38,1]],[[57,1],[38,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0020","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,20,60],[20,0,40],[60,40,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 10, 是], [0, 2, 20, 30, 是], [1, 0, 0, 10, 是], [1, 1, 0, 0, 是], [1, 2, 20, 20, 是], [2, 0, 20, 30, 是], [2, 1, 20, 20, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 10, 是], [0, 2, 20, 30, 是], [1, 0, 0, 10, 是], [1, 1, 0, 0, 是], [1, 2, 20, 20, 是], [2, 0, 20, 30, 是], [2, 1, 20, 20, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161020,"parameters":{"metric":[[0,20,60],[20,0,40],[60,40,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[20,1],[60,1]],[[20,1],[0,1],[40,1]],[[60,1],[40,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0021","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,21,63],[21,0,42],[63,42,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 21/2, 是], [0, 2, 21, 63/2, 是], [1, 0, 0, 21/2, 是], [1, 1, 0, 0, 是], [1, 2, 21, 21, 是], [2, 0, 21, 63/2, 是], [2, 1, 21, 21, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 21/2, 是], [0, 2, 21, 63/2, 是], [1, 0, 0, 21/2, 是], [1, 1, 0, 0, 是], [1, 2, 21, 21, 是], [2, 0, 21, 63/2, 是], [2, 1, 21, 21, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161021,"parameters":{"metric":[[0,21,63],[21,0,42],[63,42,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[21,1],[63,1]],[[21,1],[0,1],[42,1]],[[63,1],[42,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0022","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,22,66],[22,0,44],[66,44,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 11, 是], [0, 2, 22, 33, 是], [1, 0, 0, 11, 是], [1, 1, 0, 0, 是], [1, 2, 22, 22, 是], [2, 0, 22, 33, 是], [2, 1, 22, 22, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 11, 是], [0, 2, 22, 33, 是], [1, 0, 0, 11, 是], [1, 1, 0, 0, 是], [1, 2, 22, 22, 是], [2, 0, 22, 33, 是], [2, 1, 22, 22, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161022,"parameters":{"metric":[[0,22,66],[22,0,44],[66,44,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[22,1],[66,1]],[[22,1],[0,1],[44,1]],[[66,1],[44,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0023","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,23,69],[23,0,46],[69,46,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 23/2, 是], [0, 2, 23, 69/2, 是], [1, 0, 0, 23/2, 是], [1, 1, 0, 0, 是], [1, 2, 23, 23, 是], [2, 0, 23, 69/2, 是], [2, 1, 23, 23, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 23/2, 是], [0, 2, 23, 69/2, 是], [1, 0, 0, 23/2, 是], [1, 1, 0, 0, 是], [1, 2, 23, 23, 是], [2, 0, 23, 69/2, 是], [2, 1, 23, 23, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161023,"parameters":{"metric":[[0,23,69],[23,0,46],[69,46,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[23,1],[69,1]],[[23,1],[0,1],[46,1]],[[69,1],[46,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0024","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,24,72],[24,0,48],[72,48,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 12, 是], [0, 2, 24, 36, 是], [1, 0, 0, 12, 是], [1, 1, 0, 0, 是], [1, 2, 24, 24, 是], [2, 0, 24, 36, 是], [2, 1, 24, 24, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 12, 是], [0, 2, 24, 36, 是], [1, 0, 0, 12, 是], [1, 1, 0, 0, 是], [1, 2, 24, 24, 是], [2, 0, 24, 36, 是], [2, 1, 24, 24, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161024,"parameters":{"metric":[[0,24,72],[24,0,48],[72,48,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[24,1],[72,1]],[[24,1],[0,1],[48,1]],[[72,1],[48,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0025","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-metric-qualification；資料={\"metric\":[[0,25,75],[25,0,50],[75,50,0]],\"mapping\":[0,0,1],\"q\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 25/2, 是], [0, 2, 25, 75/2, 是], [1, 0, 0, 25/2, 是], [1, 1, 0, 0, 是], [1, 2, 25, 25, 是], [2, 0, 25, 75/2, 是], [2, 1, 25, 25, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation_zh":"Wave6第十層核對=(3, 是, 1/2, 是, [[0, 0, 0, 0, 是], [0, 1, 0, 25/2, 是], [0, 2, 25, 75/2, 是], [1, 0, 0, 25/2, 是], [1, 1, 0, 0, 是], [1, 2, 25, 25, 是], [2, 0, 25, 75/2, 是], [2, 1, 25, 25, 是], [2, 2, 0, 0, 是]], 是, finite_metric_is_complete_but_full_pair_audit_is_still_required)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-metric-qualification","seed":202611161025,"parameters":{"metric":[[0,25,75],[25,0,50],[75,50,0]],"mapping":[0,0,1],"q":[1,2],"domain":"Exact_finite_contraction_mapping","operation":"contract-metric-qualification","semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[25,1],[75,1]],[[25,1],[0,1],[50,1]],[[75,1],[50,1],[0,1]]],\"q\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0026","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,1,3],[1,0,2],[3,2,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [2, 1, 0], 0, [3, 1, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 2, [2, 1, 0], 0, [3, 1, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161026,"parameters":{"metric":[[0,1,3],[1,0,2],[3,2,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":2,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[1,1],[3,1]],[[1,1],[0,1],[2,1]],[[3,1],[2,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0027","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,2,6],[2,0,4],[6,4,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [2, 1, 0, 0], 0, [6, 2, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 3, [2, 1, 0, 0], 0, [6, 2, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161027,"parameters":{"metric":[[0,2,6],[2,0,4],[6,4,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":3,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[2,1],[6,1]],[[2,1],[0,1],[4,1]],[[6,1],[4,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0028","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,3,9],[3,0,6],[9,6,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, [2, 1, 0, 0, 0], 0, [9, 3, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 4, [2, 1, 0, 0, 0], 0, [9, 3, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161028,"parameters":{"metric":[[0,3,9],[3,0,6],[9,6,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":4,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[3,1],[9,1]],[[3,1],[0,1],[6,1]],[[9,1],[6,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0029","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,4,12],[4,0,8],[12,8,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 5, [2, 1, 0, 0, 0, 0], 0, [12, 4, 0, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 5, [2, 1, 0, 0, 0, 0], 0, [12, 4, 0, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161029,"parameters":{"metric":[[0,4,12],[4,0,8],[12,8,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":5,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[4,1],[12,1]],[[4,1],[0,1],[8,1]],[[12,1],[8,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0030","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,5,15],[5,0,10],[15,10,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [15, 5, 0, 0, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [15, 5, 0, 0, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161030,"parameters":{"metric":[[0,5,15],[5,0,10],[15,10,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":6,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[5,1],[15,1]],[[5,1],[0,1],[10,1]],[[15,1],[10,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0031","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,6,18],[6,0,12],[18,12,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [2, 1, 0], 0, [18, 6, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 2, [2, 1, 0], 0, [18, 6, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161031,"parameters":{"metric":[[0,6,18],[6,0,12],[18,12,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":2,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[6,1],[18,1]],[[6,1],[0,1],[12,1]],[[18,1],[12,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0032","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,7,21],[7,0,14],[21,14,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [2, 1, 0, 0], 0, [21, 7, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 3, [2, 1, 0, 0], 0, [21, 7, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161032,"parameters":{"metric":[[0,7,21],[7,0,14],[21,14,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":3,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[7,1],[21,1]],[[7,1],[0,1],[14,1]],[[21,1],[14,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0033","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,8,24],[8,0,16],[24,16,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, [2, 1, 0, 0, 0], 0, [24, 8, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 4, [2, 1, 0, 0, 0], 0, [24, 8, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161033,"parameters":{"metric":[[0,8,24],[8,0,16],[24,16,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":4,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[8,1],[24,1]],[[8,1],[0,1],[16,1]],[[24,1],[16,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0034","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,9,27],[9,0,18],[27,18,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 5, [2, 1, 0, 0, 0, 0], 0, [27, 9, 0, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 5, [2, 1, 0, 0, 0, 0], 0, [27, 9, 0, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161034,"parameters":{"metric":[[0,9,27],[9,0,18],[27,18,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":5,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[9,1],[27,1]],[[9,1],[0,1],[18,1]],[[27,1],[18,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0035","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,10,30],[10,0,20],[30,20,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [30, 10, 0, 0, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [30, 10, 0, 0, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161035,"parameters":{"metric":[[0,10,30],[10,0,20],[30,20,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":6,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[10,1],[30,1]],[[10,1],[0,1],[20,1]],[[30,1],[20,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0036","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,11,33],[11,0,22],[33,22,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [2, 1, 0], 0, [33, 11, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 2, [2, 1, 0], 0, [33, 11, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161036,"parameters":{"metric":[[0,11,33],[11,0,22],[33,22,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":2,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[11,1],[33,1]],[[11,1],[0,1],[22,1]],[[33,1],[22,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0037","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,12,36],[12,0,24],[36,24,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [2, 1, 0, 0], 0, [36, 12, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 3, [2, 1, 0, 0], 0, [36, 12, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161037,"parameters":{"metric":[[0,12,36],[12,0,24],[36,24,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":3,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[12,1],[36,1]],[[12,1],[0,1],[24,1]],[[36,1],[24,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0038","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,13,39],[13,0,26],[39,26,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, [2, 1, 0, 0, 0], 0, [39, 13, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 4, [2, 1, 0, 0, 0], 0, [39, 13, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161038,"parameters":{"metric":[[0,13,39],[13,0,26],[39,26,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":4,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[13,1],[39,1]],[[13,1],[0,1],[26,1]],[[39,1],[26,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0039","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,14,42],[14,0,28],[42,28,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 5, [2, 1, 0, 0, 0, 0], 0, [42, 14, 0, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 5, [2, 1, 0, 0, 0, 0], 0, [42, 14, 0, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161039,"parameters":{"metric":[[0,14,42],[14,0,28],[42,28,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":5,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[14,1],[42,1]],[[14,1],[0,1],[28,1]],[[42,1],[28,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0040","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,15,45],[15,0,30],[45,30,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [45, 15, 0, 0, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [45, 15, 0, 0, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161040,"parameters":{"metric":[[0,15,45],[15,0,30],[45,30,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":6,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[15,1],[45,1]],[[15,1],[0,1],[30,1]],[[45,1],[30,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0041","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,16,48],[16,0,32],[48,32,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [2, 1, 0], 0, [48, 16, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 2, [2, 1, 0], 0, [48, 16, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161041,"parameters":{"metric":[[0,16,48],[16,0,32],[48,32,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":2,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[16,1],[48,1]],[[16,1],[0,1],[32,1]],[[48,1],[32,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0042","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,17,51],[17,0,34],[51,34,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [2, 1, 0, 0], 0, [51, 17, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 3, [2, 1, 0, 0], 0, [51, 17, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161042,"parameters":{"metric":[[0,17,51],[17,0,34],[51,34,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":3,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[17,1],[51,1]],[[17,1],[0,1],[34,1]],[[51,1],[34,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0043","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,18,54],[18,0,36],[54,36,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, [2, 1, 0, 0, 0], 0, [54, 18, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 4, [2, 1, 0, 0, 0], 0, [54, 18, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161043,"parameters":{"metric":[[0,18,54],[18,0,36],[54,36,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":4,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[18,1],[54,1]],[[18,1],[0,1],[36,1]],[[54,1],[36,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0044","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,19,57],[19,0,38],[57,38,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 5, [2, 1, 0, 0, 0, 0], 0, [57, 19, 0, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 5, [2, 1, 0, 0, 0, 0], 0, [57, 19, 0, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161044,"parameters":{"metric":[[0,19,57],[19,0,38],[57,38,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":5,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[19,1],[57,1]],[[19,1],[0,1],[38,1]],[[57,1],[38,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0045","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,20,60],[20,0,40],[60,40,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [60, 20, 0, 0, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [60, 20, 0, 0, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161045,"parameters":{"metric":[[0,20,60],[20,0,40],[60,40,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":6,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[20,1],[60,1]],[[20,1],[0,1],[40,1]],[[60,1],[40,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0046","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,21,63],[21,0,42],[63,42,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [2, 1, 0], 0, [63, 21, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 2, [2, 1, 0], 0, [63, 21, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161046,"parameters":{"metric":[[0,21,63],[21,0,42],[63,42,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":2,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[21,1],[63,1]],[[21,1],[0,1],[42,1]],[[63,1],[42,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0047","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,22,66],[22,0,44],[66,44,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [2, 1, 0, 0], 0, [66, 22, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 3, [2, 1, 0, 0], 0, [66, 22, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161047,"parameters":{"metric":[[0,22,66],[22,0,44],[66,44,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":3,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[22,1],[66,1]],[[22,1],[0,1],[44,1]],[[66,1],[44,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0048","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,23,69],[23,0,46],[69,46,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, [2, 1, 0, 0, 0], 0, [69, 23, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 4, [2, 1, 0, 0, 0], 0, [69, 23, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161048,"parameters":{"metric":[[0,23,69],[23,0,46],[69,46,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":4,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[23,1],[69,1]],[[23,1],[0,1],[46,1]],[[69,1],[46,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0049","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,24,72],[24,0,48],[72,48,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 5, [2, 1, 0, 0, 0, 0], 0, [72, 24, 0, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 5, [2, 1, 0, 0, 0, 0], 0, [72, 24, 0, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161049,"parameters":{"metric":[[0,24,72],[24,0,48],[72,48,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":5,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[24,1],[72,1]],[[24,1],[0,1],[48,1]],[[72,1],[48,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0050","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-picard-orbit；資料={\"metric\":[[0,25,75],[25,0,50],[75,50,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [75, 25, 0, 0, 0, 0, 0], exact_Picard_orbit)","explanation_zh":"Wave6第十層核對=(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [75, 25, 0, 0, 0, 0, 0], exact_Picard_orbit)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-picard-orbit","seed":202611161050,"parameters":{"metric":[[0,25,75],[25,0,50],[75,50,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":6,"domain":"Exact_finite_contraction_mapping","operation":"contract-picard-orbit","semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[25,1],[75,1]],[[25,1],[0,1],[50,1]],[[75,1],[50,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0051","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,1,3],[1,0,2],[3,2,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0], 0, 0, 1, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0], 0, 0, 1, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161051,"parameters":{"metric":[[0,1,3],[1,0,2],[3,2,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":2,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[1,1],[3,1]],[[1,1],[0,1],[2,1]],[[3,1],[2,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0052","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,2,6],[2,0,4],[6,4,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0], 0, 0, 1, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0], 0, 0, 1, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161052,"parameters":{"metric":[[0,2,6],[2,0,4],[6,4,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":3,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[2,1],[6,1]],[[2,1],[0,1],[4,1]],[[6,1],[4,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0053","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,3,9],[3,0,6],[9,6,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], 0, 0, 3/4, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0], 0, 0, 3/4, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161053,"parameters":{"metric":[[0,3,9],[3,0,6],[9,6,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":4,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[3,1],[9,1]],[[3,1],[0,1],[6,1]],[[9,1],[6,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0054","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,4,12],[4,0,8],[12,8,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0], 0, 0, 1/2, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0, 0], 0, 0, 1/2, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161054,"parameters":{"metric":[[0,4,12],[4,0,8],[12,8,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":5,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[4,1],[12,1]],[[4,1],[0,1],[8,1]],[[12,1],[8,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0055","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,5,15],[5,0,10],[15,10,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/16, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/16, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161055,"parameters":{"metric":[[0,5,15],[5,0,10],[15,10,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":6,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[5,1],[15,1]],[[5,1],[0,1],[10,1]],[[15,1],[10,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0056","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,6,18],[6,0,12],[18,12,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0], 0, 0, 6, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0], 0, 0, 6, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161056,"parameters":{"metric":[[0,6,18],[6,0,12],[18,12,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":2,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[6,1],[18,1]],[[6,1],[0,1],[12,1]],[[18,1],[12,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0057","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,7,21],[7,0,14],[21,14,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0], 0, 0, 7/2, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0], 0, 0, 7/2, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161057,"parameters":{"metric":[[0,7,21],[7,0,14],[21,14,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":3,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[7,1],[21,1]],[[7,1],[0,1],[14,1]],[[21,1],[14,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0058","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,8,24],[8,0,16],[24,16,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], 0, 0, 2, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0], 0, 0, 2, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161058,"parameters":{"metric":[[0,8,24],[8,0,16],[24,16,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":4,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[8,1],[24,1]],[[8,1],[0,1],[16,1]],[[24,1],[16,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0059","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,9,27],[9,0,18],[27,18,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0], 0, 0, 9/8, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0, 0], 0, 0, 9/8, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161059,"parameters":{"metric":[[0,9,27],[9,0,18],[27,18,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":5,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[9,1],[27,1]],[[9,1],[0,1],[18,1]],[[27,1],[18,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0060","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,10,30],[10,0,20],[30,20,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/8, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/8, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161060,"parameters":{"metric":[[0,10,30],[10,0,20],[30,20,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":6,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[10,1],[30,1]],[[10,1],[0,1],[20,1]],[[30,1],[20,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0061","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,11,33],[11,0,22],[33,22,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0], 0, 0, 11, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0], 0, 0, 11, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161061,"parameters":{"metric":[[0,11,33],[11,0,22],[33,22,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":2,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[11,1],[33,1]],[[11,1],[0,1],[22,1]],[[33,1],[22,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0062","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,12,36],[12,0,24],[36,24,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0], 0, 0, 6, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0], 0, 0, 6, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161062,"parameters":{"metric":[[0,12,36],[12,0,24],[36,24,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":3,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[12,1],[36,1]],[[12,1],[0,1],[24,1]],[[36,1],[24,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0063","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,13,39],[13,0,26],[39,26,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], 0, 0, 13/4, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0], 0, 0, 13/4, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161063,"parameters":{"metric":[[0,13,39],[13,0,26],[39,26,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":4,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[13,1],[39,1]],[[13,1],[0,1],[26,1]],[[39,1],[26,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0064","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,14,42],[14,0,28],[42,28,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0], 0, 0, 7/4, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0, 0], 0, 0, 7/4, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161064,"parameters":{"metric":[[0,14,42],[14,0,28],[42,28,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":5,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[14,1],[42,1]],[[14,1],[0,1],[28,1]],[[42,1],[28,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0065","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,15,45],[15,0,30],[45,30,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0, 0], 0, 0, 15/16, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0, 0, 0], 0, 0, 15/16, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161065,"parameters":{"metric":[[0,15,45],[15,0,30],[45,30,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":6,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[15,1],[45,1]],[[15,1],[0,1],[30,1]],[[45,1],[30,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0066","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,16,48],[16,0,32],[48,32,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0], 0, 0, 16, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0], 0, 0, 16, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161066,"parameters":{"metric":[[0,16,48],[16,0,32],[48,32,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":2,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[16,1],[48,1]],[[16,1],[0,1],[32,1]],[[48,1],[32,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0067","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,17,51],[17,0,34],[51,34,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0], 0, 0, 17/2, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0], 0, 0, 17/2, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161067,"parameters":{"metric":[[0,17,51],[17,0,34],[51,34,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":3,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[17,1],[51,1]],[[17,1],[0,1],[34,1]],[[51,1],[34,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0068","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,18,54],[18,0,36],[54,36,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], 0, 0, 9/2, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0], 0, 0, 9/2, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161068,"parameters":{"metric":[[0,18,54],[18,0,36],[54,36,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":4,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[18,1],[54,1]],[[18,1],[0,1],[36,1]],[[54,1],[36,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0069","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,19,57],[19,0,38],[57,38,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0], 0, 0, 19/8, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0, 0], 0, 0, 19/8, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161069,"parameters":{"metric":[[0,19,57],[19,0,38],[57,38,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":5,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[19,1],[57,1]],[[19,1],[0,1],[38,1]],[[57,1],[38,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0070","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,20,60],[20,0,40],[60,40,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/4, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/4, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161070,"parameters":{"metric":[[0,20,60],[20,0,40],[60,40,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":6,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[20,1],[60,1]],[[20,1],[0,1],[40,1]],[[60,1],[40,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0071","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,21,63],[21,0,42],[63,42,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0], 0, 0, 21, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0], 0, 0, 21, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161071,"parameters":{"metric":[[0,21,63],[21,0,42],[63,42,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":2,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[21,1],[63,1]],[[21,1],[0,1],[42,1]],[[63,1],[42,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0072","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,22,66],[22,0,44],[66,44,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0], 0, 0, 11, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0], 0, 0, 11, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161072,"parameters":{"metric":[[0,22,66],[22,0,44],[66,44,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":3,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[22,1],[66,1]],[[22,1],[0,1],[44,1]],[[66,1],[44,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0073","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,23,69],[23,0,46],[69,46,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], 0, 0, 23/4, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0], 0, 0, 23/4, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161073,"parameters":{"metric":[[0,23,69],[23,0,46],[69,46,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":4,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[23,1],[69,1]],[[23,1],[0,1],[46,1]],[[69,1],[46,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0074","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,24,72],[24,0,48],[72,48,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0], 0, 0, 3, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0, 0], 0, 0, 3, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161074,"parameters":{"metric":[[0,24,72],[24,0,48],[72,48,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":5,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[24,1],[72,1]],[[24,1],[0,1],[48,1]],[[72,1],[48,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0075","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,25,75],[25,0,50],[75,50,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0, 0], 0, 0, 25/16, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation_zh":"Wave6第十層核對=([2, 1, 0, 0, 0, 0, 0], 0, 0, 25/16, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161075,"parameters":{"metric":[[0,25,75],[25,0,50],[75,50,0]],"mapping":[0,0,1],"q":[1,2],"start":2,"steps":6,"domain":"Exact_finite_contraction_mapping","operation":"contract-fixed-point-bound","semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[25,1],[75,1]],[[25,1],[0,1],[50,1]],[[75,1],[50,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0076","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"q_equal_one\",\"scale\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(q_equal_one, 1, two_point_metric, 1, identity_map_has_two_fixed_points, 否)","explanation_zh":"Wave6第十層核對=(q_equal_one, 1, two_point_metric, 1, identity_map_has_two_fixed_points, 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161076,"parameters":{"mode":"q_equal_one","scale":1,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[1,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0077","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"non_self_map\",\"scale\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_self_map, 2, T_maps_one_point_outside_X, 否)","explanation_zh":"Wave6第十層核對=(non_self_map, 2, T_maps_one_point_outside_X, 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161077,"parameters":{"mode":"non_self_map","scale":2,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"non_self_map\",\"scale\":[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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0078","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"noncomplete_space\",\"scale\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(noncomplete_space, X=(0,3), T(x)=(x+3)/2, fixed_point=3_outside_X, [1, 2], 否)","explanation_zh":"Wave6第十層核對=(noncomplete_space, X=(0,3), T(x)=(x+3)/2, fixed_point=3_outside_X, [1, 2], 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161078,"parameters":{"mode":"noncomplete_space","scale":3,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0079","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"sampled_pairs\",\"scale\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(sampled_pairs, 4, checked_pairs_pass, one_unchecked_pair_violates_global_bound, 否, sampled_pairs_do_not_prove_global_contraction)","explanation_zh":"Wave6第十層核對=(sampled_pairs, 4, checked_pairs_pass, one_unchecked_pair_violates_global_bound, 否, sampled_pairs_do_not_prove_global_contraction)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161079,"parameters":{"mode":"sampled_pairs","scale":4,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"sampled_pairs\",\"scale\":[4,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0080","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"q_equal_one\",\"scale\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(q_equal_one, 5, two_point_metric, 1, identity_map_has_two_fixed_points, 否)","explanation_zh":"Wave6第十層核對=(q_equal_one, 5, two_point_metric, 1, identity_map_has_two_fixed_points, 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161080,"parameters":{"mode":"q_equal_one","scale":5,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[5,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0081","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"non_self_map\",\"scale\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_self_map, 6, T_maps_one_point_outside_X, 否)","explanation_zh":"Wave6第十層核對=(non_self_map, 6, T_maps_one_point_outside_X, 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161081,"parameters":{"mode":"non_self_map","scale":6,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"non_self_map\",\"scale\":[6,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0082","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"noncomplete_space\",\"scale\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(noncomplete_space, X=(0,7), T(x)=(x+7)/2, fixed_point=7_outside_X, [1, 2], 否)","explanation_zh":"Wave6第十層核對=(noncomplete_space, X=(0,7), T(x)=(x+7)/2, fixed_point=7_outside_X, [1, 2], 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161082,"parameters":{"mode":"noncomplete_space","scale":7,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[7,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0083","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"sampled_pairs\",\"scale\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(sampled_pairs, 8, checked_pairs_pass, one_unchecked_pair_violates_global_bound, 否, sampled_pairs_do_not_prove_global_contraction)","explanation_zh":"Wave6第十層核對=(sampled_pairs, 8, checked_pairs_pass, one_unchecked_pair_violates_global_bound, 否, sampled_pairs_do_not_prove_global_contraction)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161083,"parameters":{"mode":"sampled_pairs","scale":8,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"sampled_pairs\",\"scale\":[8,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0084","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"q_equal_one\",\"scale\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(q_equal_one, 9, two_point_metric, 1, identity_map_has_two_fixed_points, 否)","explanation_zh":"Wave6第十層核對=(q_equal_one, 9, two_point_metric, 1, identity_map_has_two_fixed_points, 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161084,"parameters":{"mode":"q_equal_one","scale":9,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[9,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0085","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"non_self_map\",\"scale\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_self_map, 10, T_maps_one_point_outside_X, 否)","explanation_zh":"Wave6第十層核對=(non_self_map, 10, T_maps_one_point_outside_X, 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161085,"parameters":{"mode":"non_self_map","scale":10,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"non_self_map\",\"scale\":[10,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0086","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"noncomplete_space\",\"scale\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(noncomplete_space, X=(0,11), T(x)=(x+11)/2, fixed_point=11_outside_X, [1, 2], 否)","explanation_zh":"Wave6第十層核對=(noncomplete_space, X=(0,11), T(x)=(x+11)/2, fixed_point=11_outside_X, [1, 2], 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161086,"parameters":{"mode":"noncomplete_space","scale":11,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[11,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0087","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"sampled_pairs\",\"scale\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(sampled_pairs, 12, checked_pairs_pass, one_unchecked_pair_violates_global_bound, 否, sampled_pairs_do_not_prove_global_contraction)","explanation_zh":"Wave6第十層核對=(sampled_pairs, 12, checked_pairs_pass, one_unchecked_pair_violates_global_bound, 否, sampled_pairs_do_not_prove_global_contraction)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161087,"parameters":{"mode":"sampled_pairs","scale":12,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"sampled_pairs\",\"scale\":[12,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0088","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"q_equal_one\",\"scale\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(q_equal_one, 13, two_point_metric, 1, identity_map_has_two_fixed_points, 否)","explanation_zh":"Wave6第十層核對=(q_equal_one, 13, two_point_metric, 1, identity_map_has_two_fixed_points, 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161088,"parameters":{"mode":"q_equal_one","scale":13,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[13,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0089","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"non_self_map\",\"scale\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_self_map, 14, T_maps_one_point_outside_X, 否)","explanation_zh":"Wave6第十層核對=(non_self_map, 14, T_maps_one_point_outside_X, 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161089,"parameters":{"mode":"non_self_map","scale":14,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"non_self_map\",\"scale\":[14,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0090","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"noncomplete_space\",\"scale\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(noncomplete_space, X=(0,15), T(x)=(x+15)/2, fixed_point=15_outside_X, [1, 2], 否)","explanation_zh":"Wave6第十層核對=(noncomplete_space, X=(0,15), T(x)=(x+15)/2, fixed_point=15_outside_X, [1, 2], 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161090,"parameters":{"mode":"noncomplete_space","scale":15,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[15,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0091","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"sampled_pairs\",\"scale\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(sampled_pairs, 16, checked_pairs_pass, one_unchecked_pair_violates_global_bound, 否, sampled_pairs_do_not_prove_global_contraction)","explanation_zh":"Wave6第十層核對=(sampled_pairs, 16, checked_pairs_pass, one_unchecked_pair_violates_global_bound, 否, sampled_pairs_do_not_prove_global_contraction)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161091,"parameters":{"mode":"sampled_pairs","scale":16,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"sampled_pairs\",\"scale\":[16,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0092","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"q_equal_one\",\"scale\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(q_equal_one, 17, two_point_metric, 1, identity_map_has_two_fixed_points, 否)","explanation_zh":"Wave6第十層核對=(q_equal_one, 17, two_point_metric, 1, identity_map_has_two_fixed_points, 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161092,"parameters":{"mode":"q_equal_one","scale":17,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[17,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0093","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"non_self_map\",\"scale\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_self_map, 18, T_maps_one_point_outside_X, 否)","explanation_zh":"Wave6第十層核對=(non_self_map, 18, T_maps_one_point_outside_X, 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161093,"parameters":{"mode":"non_self_map","scale":18,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"non_self_map\",\"scale\":[18,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0094","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"noncomplete_space\",\"scale\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(noncomplete_space, X=(0,19), T(x)=(x+19)/2, fixed_point=19_outside_X, [1, 2], 否)","explanation_zh":"Wave6第十層核對=(noncomplete_space, X=(0,19), T(x)=(x+19)/2, fixed_point=19_outside_X, [1, 2], 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161094,"parameters":{"mode":"noncomplete_space","scale":19,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[19,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0095","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"sampled_pairs\",\"scale\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(sampled_pairs, 20, checked_pairs_pass, one_unchecked_pair_violates_global_bound, 否, sampled_pairs_do_not_prove_global_contraction)","explanation_zh":"Wave6第十層核對=(sampled_pairs, 20, checked_pairs_pass, one_unchecked_pair_violates_global_bound, 否, sampled_pairs_do_not_prove_global_contraction)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161095,"parameters":{"mode":"sampled_pairs","scale":20,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"sampled_pairs\",\"scale\":[20,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0096","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"q_equal_one\",\"scale\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(q_equal_one, 21, two_point_metric, 1, identity_map_has_two_fixed_points, 否)","explanation_zh":"Wave6第十層核對=(q_equal_one, 21, two_point_metric, 1, identity_map_has_two_fixed_points, 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161096,"parameters":{"mode":"q_equal_one","scale":21,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[21,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0097","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"non_self_map\",\"scale\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_self_map, 22, T_maps_one_point_outside_X, 否)","explanation_zh":"Wave6第十層核對=(non_self_map, 22, T_maps_one_point_outside_X, 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161097,"parameters":{"mode":"non_self_map","scale":22,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"non_self_map\",\"scale\":[22,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0098","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"noncomplete_space\",\"scale\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(noncomplete_space, X=(0,23), T(x)=(x+23)/2, fixed_point=23_outside_X, [1, 2], 否)","explanation_zh":"Wave6第十層核對=(noncomplete_space, X=(0,23), T(x)=(x+23)/2, fixed_point=23_outside_X, [1, 2], 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161098,"parameters":{"mode":"noncomplete_space","scale":23,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[23,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0099","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"sampled_pairs\",\"scale\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(sampled_pairs, 24, checked_pairs_pass, one_unchecked_pair_violates_global_bound, 否, sampled_pairs_do_not_prove_global_contraction)","explanation_zh":"Wave6第十層核對=(sampled_pairs, 24, checked_pairs_pass, one_unchecked_pair_violates_global_bound, 否, sampled_pairs_do_not_prove_global_contraction)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161099,"parameters":{"mode":"sampled_pairs","scale":24,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"sampled_pairs\",\"scale\":[24,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0100","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-contraction-mapping-theorem"],"title_zh":"壓縮映射定理","stem_zh":"完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-boundary；資料={\"mode\":\"q_equal_one\",\"scale\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(q_equal_one, 25, two_point_metric, 1, identity_map_has_two_fixed_points, 否)","explanation_zh":"Wave6第十層核對=(q_equal_one, 25, two_point_metric, 1, identity_map_has_two_fixed_points, 否)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["壓縮映射定理"]},"variant":{"family_id":"qf-mod47-contract-boundary","seed":202611161100,"parameters":{"mode":"q_equal_one","scale":25,"domain":"Exact_finite_contraction_mapping","operation":"contract-boundary","semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[25,1]}"}},"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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}